Welcome to TiddlyWiki created by Jeremy Ruston; Copyright © 2004-2007 Jeremy Ruston, Copyright © 2007-2011 UnaMesa Association
[<img[images/png/1-form.png]]A ''1-form'', or ''//cotangent vector//'', $\f{f}$, is a geometric object that acts on a [[tangent vector]] at a point, $p$, to give a real number. It may be written in terms of the [[coordinate basis 1-forms]] as
\[ \f{f} = f_i \f{dx^i} \in T_{p}^{*}M \]
It is a linear operator, and so may be written as a function of a vector or more simply as a [[vector-form contraction|vector-form algebra]] (product),
\[ \f{f}(\ve{v}) = {\bf i}_{\ve{v}} \f{f} = \ve{v} \f{f} = v^j f_i \ve{\pa_j} \f{dx^i} = v^j f_i \de_j^i= v^i f_i \in \Re \]
The vector space of [[1-form]]s at each point, $p$, of a [[manifold]], $M$, is the ''cotangent space'', $T_{p}^* M$, and is spanned by the $\f{dx^i}$. The space of 1-forms is the [[dual space]] to the space of tangent vectors. There is a unique basis 1-form, $\f{dx^i}$, corresponding to each basis vector, $\ve{\pa_i}$, such that $\f{dx^i}(\ve{\pa_j}) = \ve{\pa_j} \f{dx^i} =\de^i_j$. As a dual space, there is also an induced [[metric]] between 1-forms, $\lp \f{dx^i}, \f{dx^j} \rp = g^{ij}$.
A ''2-sphere'' embedded in flat 3-space is a two dimensional [[manifold]], $M$, defined by the equation $x^e x^f \de_{ef} = r^2$ -- it is the surface of constant $r=r$ in [[spherical coordinates]]. The angular spherical coordinates, $(\th,\ph)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 2-sphere is $g_{ij} = {\rm diag}(r^2, r^2 \sin^2(\th))$. The simplest [[frame]] compatible with this metric is
$$
\f{e} = \f{d \th} \, r \si_1 + \f{d \ph} \, r \sin(\th) \si_2
$$
in which $\si_{1/2}$ are the [[Clifford basis vectors]] for [[Cl(2,0)|Clifford matrix representation]]. The coframe is
$$
\ve{e} = \si^1 \fr{1}{r} \ve{\pa_\th} + \si^2 \fr{1}{r \sin(\th)} \ve{\pa_\ph}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
$$
\f{\om} = - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp = - \f{d \ph} \cos(\th) \si_{12}
$$
The [[Clifford vector bundle]] curvature is
$$
\ff{F} = \f{d} \f{\om} + \ha \f{\om} \f{\om} = \f{d \th} \f{d \ph} \sin(\th) \si_{12}
$$
The [[Clifford-Ricci curvature]] is
$$
\f{R} = \ve{e} \times \ff{F} = \f{d \ph} \fr{1}{r} \sin(\th) \si_2 + \f{d \th} \fr{1}{r} \si_1 = \fr{1}{r^2} \f{e}
$$
showing that the 2-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\La = 0$ (as do all two dimensional spaces). The [[Clifford curvature scalar]] is $R = \ve{e} \cdot \f{R} = \fr{2}{r^2}$.
A ''3-sphere'' embedded in flat 4-space of positive signature is a three dimensional [[manifold]], $M$, defined by the equation $x^w x^x \de_{wx} = r^2$ -- it is the surface of constant $r=r$ in $4d$ [[hyperspherical coordinates]],
\begin{eqnarray}
x^1 &=& r \cos(a^1) \\
x^2 &=& r \sin(a^1) \cos(a^2) \\
x^3 &=& r \sin(a^1) \sin(a^2) \cos(a^3) \\
x^4 &=& r \sin(a^1) \sin(a^2) \sin(a^3)
\end{eqnarray}
The angular hyperspherical coordinates, $(a^1,a^2,a^3)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 3-sphere is $g_{ij} = {\rm diag}(r^2, r^2 \sin^2(a^1),r^2 \sin^2(a^1) \sin^2(a^2))$. The simplest [[frame]] compatible with this metric is
$$
\f{e} = \f{d a^1} \, r \si_1 + \f{d a^2} \, r \sin(a^1) \si_2 + \f{d a^3} \, r \sin(a^1) \sin(a^2) \si_3
$$
in which $\si_{1/2/3}$ are the [[Clifford basis vectors]] for [[Cl(3,0)|Cl(3)]]. The coframe is
$$
\ve{e} = \si^1 \fr{1}{r} \ve{\pa_1} + \si^2 \fr{1}{r \sin(a^1)} \ve{\pa_2} + \si^3 \fr{1}{r \sin(a^1) \sin(a^2)} \ve{\pa_3}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& - \f{d a^2} \cos(a^1) \si_{12} - \f{d a^3} \cos(a^1) \sin(a^2) \si_{13} - \f{d a^3} \cos(a^2) \si_{23}
\end{eqnarray}
The [[Clifford vector bundle]] curvature is
\begin{eqnarray}
\ff{F} &=& \f{d} \f{\om} + \ha \f{\om} \f{\om} \\
&=& \f{d a^1} \f{d a^2} \sin(a^1) \si_{12} + \f{d a^1} \f{d a^3} \sin(a^1) \sin(a^2) \si_{13} + \f{d a^2} \f{d a^3} \sin^2(a^1) \sin(a^2) \si_{23}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
\begin{eqnarray}
\f{R} &=& \ve{e} \times \ff{F} \\
&=& \f{d a^1} \fr{2}{r} \si_1 + \f{d a^2} \fr{2}{r} \sin(a^1) \si_2 + \f{d a^3} \fr{2}{r} \sin(a^1) \sin(a^2) \si_3 \\
&=& \fr{2}{r^2} \f{e}
\end{eqnarray}
showing that the 3-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\La = \fr{1}{2 r^2}$. The [[Clifford curvature scalar]] is $R = \ve{e} \cdot \f{R} = \fr{8}{r^2}$.
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[[Garrett Lisi]] '86 Cate Centennial@@
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[[Garrett Lisi]] IEEE Aerospace Conference, Big Sky, Montana, March 6, 2011@@
<<tiddler HideTags>>$$
\begin{array}{rclclc}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^k} \ha \om_k^{\p{k}\mu\nu} \ga_{\mu\nu} \!&\!\! \in \!\!&\! \f{Cl}^2(3,1)
&
\quad
\f{e} = \f{dx^k} (e_k)^\mu \ga_\mu \, \in \, \f{Cl}^1(3,1) \vp{|_{(}} \\
\f{W} \!\!&\!\!=\!\!&\!\! \f{dx^k} W_k^{\p{i}\pi} \fr{i}{2} \si_\pi \!&\!\! \in \!\!&\! \f{su}(2)
&
\quad
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\qquad
\lb \matrix{
\nu_{eL} \\ e_L
} \rb
\\
\f{B} \!\!&\!\!=\!\!&\!\! \f{dx^k} B_k i \!&\!\! \in \!\!&\! \f{u}(1)
&
\quad
Y \\
\f{g} \!\!&\!\!=\!\!&\!\! \f{dx^k} g_k^{\p{k}A} \fr{i}{2} \la_A \!&\!\! \in \!\!&\! \f{su}(3)
&
\quad
\lb u^r, u^g, u^b \rb \vp{|^{(^(}_{(}}
\end{array}
\begin{array}{c}
\quad
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\; \\
\; \\
\end{array}
$$
$$
\updownarrow \vp{{\huge(}_{\big(}}
$$
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\! {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{W} + \f{B} + \f{g} + ( \ud{\nu}{}_e + \ud{e} + \ud{u} + \ud{d} ) \\
&& + \, (\ud{\nu}{}_\mu + \ud{\mu} + \ud{c} + \ud{s}) + (\ud{\nu}{}_\ta + \ud{\ta} + \ud{t} + \ud{b}) \vp{|_{\Big(}}
\end{array}
$$
$$
\udff{F} = \f{d} \udf{A} + {\scriptsize \frac{1}{2}} \big[ \udf{A}, \udf{A} \big]
$$
This hasn't been around long enough for any questions asked to be frequent, but I'll try to anticipate some.
!!Who?
The site is principally authored by me, [[Garrett Lisi]]. I may open it up for collaboration in the future.
As to who it's good for... well, mostly it's for my own use. But if you have a background in physics and math, with at least some graduate level work under your belt, most of what's here should be accessible to you, and some may even be of interest.
!!What?
It's sort of a "choose your own adventure" book in theoretical physics — only the book is being written day-by-day and no one knows the ending, or if there is one. It's my real-time research notebook, made available to public view. I hope to make it comparable to an open ended [[Living Reviews in Relativity|http://relativity.livingreviews.org/]] article in spirit and quality, but updated more frequently and navigable as a wiki. My long term goal is to construct a concise and beautiful theoretical description of reality unifying General Relativity, Quantum Field Theory, and the Standard Model using the foundations and language of basic differential geometry. Such a theory may not exist, but that's what I'm after. And here you can watch me walk down every dark alleyway looking for it — until I find it, or at least some interesting stuff along the way. This evolving search tree will grow and be pruned in ways I can't now predict. But I expect the information contained to be equivalent to a book and several overlapping research papers, wikified and presented as they are written. It's open source physics.
!!Why?
I needed a way to organize my physics notes. And I was simultaneously contemplating the best way to present and navigate theoretical physics. [[Semantic Networks|http://www.jfsowa.com/pubs/semnet.htm]] provide a natural structure for relating abstract conceptual information, and I considered building a graphical system to do what I want along those lines... but a wiki is a good practical equivalent. It allows a reader to quickly learn new concepts in digestible pieces, and trace forwards or backwards to the implications or foundations of those concepts — while allowing an author, or authors, to easily expand the content. I am very impressed with the way [[Wikipedia|http://en.wikipedia.org/wiki/Main_Page]] works and evolves, and this is my own personal version, for research.
!!How?
The two main pieces from which this site is built are [[TiddlyWiki|http://www.tiddlywiki.com/]], created by Jeremy Ruston, and [[jsMath|http://www.math.union.edu/~dpvc/jsMath/]], made by Davide P. Cervone. Both of these excellent open source software packages are under continuing development and have supportive communities. I owe thanks to many people for building the pieces used for this site, and for helping out with technical details. To delve more into the nitty-gritty of how it's put together, and see who contributed which plugins, go check out the [[Configuration]]. If things don't work perfectly, it's probably my fault. You can get everything you need to set up a similar wiki for yourself from this [[downloads directory|http://deferentialgeometry.org/download/]].
To use it... try things. Click on buttons and see what happens, you'll figure it out.
!!What about money for food?
My research is supported entirely by private contributions, mostly mine. Support is always appreciated, is tax deductible via a 501(c)(3) corporation, and may be contributed by contacting me.
!!Where
This site is served from a closet in San Jose, California (thanks Rich!). It's currently mirrored from a laptop on a volcanic island in the middle of the Pacific, but the laptop follows its owner everywhere... except out surfing — it hates that.
!!When
Now.
----
<<slider chkSliderAbout 'About (slider)' 'More questions and answers >' 'Click to see more questions and answers'>>
!!How did you come up with the title, "Deferential Geometry"?
My favorite interpretation is that it's about geometry in the service of physics. There is a lot of bad theoretical physics out there without math, and a lot of good math without physics; good physics uses math, and this site is about using only the math needed by physics. There shouldn't be any mathematical tangents here without physics ideas motivating them — the geometry is deferential to the physics.
<<tiddler HideTags>>Bosonic connection:
$$\f{H} = {\textstyle \ha} \f{\om} + {\textstyle \fr{1}{4}} \f{e} \ph + \f{G} \;\;\;\; \in spin(3,1) + 4 \! \times \! 10 + spin(10) = spin(3,11)$$
Curvature:
$$\ff{F} = \f{d} \f{H} + \f{H} \f{H} = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^G \vp{A_{\big(}}$$
Riemann: $\;\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} \s$ Torsion: $\;\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om} \s $ Covariant: $\;\; \f{D} \ph = \f{d} \ph + \f{G} \ph - \ph \f{G} \vp{A_{\big(}}$
$spin(3,1)$ duality operator: $\;\; \ep = \Ga_1 \Ga_2 \Ga_3 \Ga_4 \s\;\;$ Hodge duality operator: $\;\; * = \ff{\vv{\ep}} = \big< \f{e} \f{e} \ep \ve{e} \ve{e} \big> \vp{A_{\Big(}}$
Boson action:
\begin{eqnarray}
S_H &=& \int \big< {\textstyle \fr{-1}{\pi G}} \ff{F} \ep \ff{F} + {\textstyle \fr{1}{4g^2}} \ff{F} \ff{\vv{\ep}} \ff{F} \big> \\
&=& \int \big< {\textstyle \fr{-1}{4 \pi G}} \ff{R} \ff{R} \ep + {\textstyle \fr{1}{16 \pi G}} \ph^2 \ff{R} \f{e} \f{e} \ep + {\textstyle \fr{3}{32\pi G}} \ph^4 \nf{e}
+ {\textstyle \fr{1}{64 g^2}} \ph^2 \ff{T} \ff{\vv{\ep}} \ff{T} + {\textstyle \fr{1}{64 g^2}} \f{e} \f{D} \ph \ff{\vv{\ep}} \f{e} \f{D} \ph
+ {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{\vv{\ep}} \ff{F}^G \big> \\
&\sim& \int \big< {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big>
\end{eqnarray}
A possible $spin(3,11)$ invariant action:
$$
S_H = \int \big< \ff{B} \ff{F} - \ff{B} \Ph \ff{B} + {\textstyle \fr{\al}{3}} \ff{B} \Ph^3 \ff{B} \big>
$$
Symmetry breaking: $\s\;\;\; \Ph^0 = \fr{\pi G}{4} \ep - g^2 \ff{\vv{\ep}} \s\;\;\;\; \f{H}^0 = \fr{1}{4}\f{e}\ph^0$
<<tiddler HideTags>>Bosonic connection:
$$\f{H} = {\textstyle \ha} \f{\om} + {\textstyle \fr{1}{4}} \f{e} \ph + \f{A} \;\;\;\; \in spin(1,3) \,\oplus\, 4 \! \otimes \! 10 \,\oplus\, spin(10) = spin(11,3)$$
Curvature:
$$\ff{F} = \f{d} \f{H} + \f{H} \f{H} = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^A \vp{A_{\big(}}$$
Riemann: $\;\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} \s$ Torsion: $\;\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om} \s $ Covariant: $\;\; \f{D} \ph = \f{d} \ph + \f{A} \ph - \ph \f{A} \vp{A_{\big(}}$
$spin(1,3)$ duality operator: $\;\; \ep = \Ga_1 \Ga_2 \Ga_3 \Ga_4 \s\;\;$ Hodge duality operator: $\;\; * = \ff{\vv{\ep}} = \big< \f{e} \f{e} \ep \ve{e} \ve{e} \big> \vp{A_{\Big(}}$
Boson action:
\begin{eqnarray}
S_H &=& \int \big< {\textstyle \fr{-1}{\pi G}} \ff{F} \ep \ff{F} + {\textstyle \fr{1}{4g^2}} \ff{F} \ff{\vv{\ep}} \ff{F} \big> \\
&=& \int \big< {\textstyle \fr{-1}{4 \pi G}} \ff{R} \ff{R} \ep + {\textstyle \fr{1}{16 \pi G}} \ph^2 \ff{R} \f{e} \f{e} \ep + {\textstyle \fr{3}{32\pi G}} \ph^4 \nf{e}
+ {\textstyle \fr{1}{64 g^2}} \ph^2 \ff{T} \ff{\vv{\ep}} \ff{T} + {\textstyle \fr{1}{64 g^2}} \f{e} \f{D} \ph \ff{\vv{\ep}} \f{e} \f{D} \ph
+ {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{\vv{\ep}} \ff{F}^A \big> \\
&\sim& \int \big< {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big>
\end{eqnarray}
A possible $spin(11,3)$ invariant action:
$$
S_H = {\textstyle \fr{1}{g}} \int \big< \ff{B} \ff{F} + \ff{B} \Ph \ff{B} + {\textstyle \fr{1}{3}} \ff{B} \Ph^3 \ff{B} \big>
$$
Symmetry breaking: $\s\;\;\; \Ph_0 = \ff{\vv{\ep}} \s\;\;\;\; \f{H}{}_0 = \fr{1}{4}\f{e}{}_0 \ph_0$
Modified BF action, using $\ff{\od{B}} = \ff{B} + \fff{\od{B}} \,$:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F} + \nf{\Phi} ( \f{H}{}_1, \f{H}{}_2, \ff{B} ) \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \ff{B} \ff{F} + {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} \fr{1}{16 \pi G} \ph^2 \big( R - \fr{3}{2} \ph^2 \big) + \fr{1}{4} \ff{F'} \ff{*F'} \big>
\end{eqnarray}
Cosmological constant from the Higgs VEV: $\quad \La = \fr{3}{4} \ph^2$
Implies frame VEV is de Sitter: $\quad \ff{R} = \fr{\La}{6} \f{e} \f{e} \qquad R = 4 \La$
Vacuum expectation value of the curvature vanishes: $\quad \udff{F} = 0$
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \f{G} + \f{G} \f{G} \big) + \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big)
$$
Modified BF action for everything, using $\ff{\od{B}} = \ff{B} + \fff{\od{B}} \,$:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F} + \nf{\Phi} ( \f{H}, \f{G}, \ff{B} ) \big> \\
&=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big)
+ \ff{B} \ff{F} - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga + \ff{B_{m,h,G}} \ff{*B_{m,h,G}} \big>
\end{eqnarray}
Fermionic part, using [[anti-ghost|BRST technique]] [[Grassmann|Grassmann number]] 3-form, $\fff{\od{B}} = \nf{e} \od{\ps} \ve{e} \,$:
\begin{eqnarray}
S_f &=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} \big) \big> \\
&=& \int \big< \nf{e} \od{\ps} \ve{e} \big( \f{d} \ud{\ps} + {\scriptsize \frac{1}{2}} \f{\om} \ud{\ps} + {\scriptsize \frac{1}{4}} \f{e} \ph \ud{\ps} + \f{B} \ud{\ps} + \f{W} \ud{\ps} + \ud{\ps} \f{G} \big) \big> \\
&=& \int \nf{d^4 x} |e| \, \big< \od{\ps} \ga^\mu (e_\mu)^i \big( \pa_i \ud{\ps} + {\scriptsize \frac{1}{4}} \om_i^{\p{i} \mu \nu} \ga_{\mu \nu} \ud{\ps} + B_i \ud{\ps} + W_i \ud{\ps} - \ud{\ps} G_i \big) + \od{\ps} \, \ph \, \ud{\ps} \big>
\end{eqnarray}
<<tiddler HideTags>>
$$
\begin{array}{rcl}
L \!\!&\!\!=\!\!&\!\! \bar{\ps} \ve{e} \lp \f{\pa} + {\small \frac{1}{4}} \f{\om}^{a b} \ga_{a b} + \f{A} + \f{W} + \f{g} \rp \ps + \bar{\ps} \ph \ps
\end{array}
$$
<html>
<table class="gtable">
<tr>
<td>
<img SRC="images/png/fermion photon vertex.png" height=100px>
</td>
<td> </td>
<td>
<SPAN class="math">\ps =
\lb \matrix{
\ps_1 \\ \ps_2 \\ \ps_3 \\ \vdots
} \rb
</SPAN>
</td>
</tr>
</table>
</html>
/***
name: AllTagsExceptPlugin
author: Garrett
version: 0.1.0
This is a revision of Clint Checketts' allTagsExcept plugin, which lists all tags except those listed.
<<option chkDisableExcept>> show hidden system tags
!!Usage
{{{
<<AllTagsExcept tag1 tag2 ...>>
}}}
!!!Code
***/
/*{{{*/
version.extensions.AllTagsExcept = {major: 0, minor: 1, revision: 0};
if (!config.options.chkDisableExcept) config.options.chkDisableExcept=false; // default to standard action
config.macros.AllTagsExcept = {tooltip: "Show notes tagged with '%0'",noTags: "There are no tags to display"};
config.macros.AllTagsExcept.handler = function(place,macroName,params)
{
var tags = store.getTags();
var theDateList = createTiddlyElement(place,"ul");
if(tags.length == 0)
createTiddlyElement(theDateList,"li",null,"listTitle",this.noTags);
for(var t=0; t<tags.length; t++)
{
var includeTag = true;
for (var p=0;p<params.length; p++) if ((tags[t][0] == params[p])&&(!config.options.chkDisableExcept)) includeTag = false;
if (includeTag)
{
var theListItem =createTiddlyElement(theDateList,"li");
var theTag = createTiddlyButton(theListItem,tags[t][0] + " (" + tags[t][1] + ")",this.tooltip.format([tags[t][0]]),onClickTag);
theTag.setAttribute("tag",tags[t][0]);
}
}
}
/*}}}*/
author: [[Garrett Lisi]]
arxiv: http://arxiv.org/abs/0711.0770
locally: [[AESToE|papers/AESToE.pdf]]
abstract:
All fields of the [[standard model]] and [[gravity|modified BF gravity]] are unified as an [[E8]] [[principal bundle]] [[connection]]. A non-compact real form of the [[E8 Lie algebra|e8]] has [[G2]] and [[F4]] subalgebras which break down to strong [[su(3)]], electroweak [[su(2)]] x u(1), gravitational [[so(3,1)|spacetime]], the [[frame]]-Higgs, and three generations of fermions related by [[triality]]. The interactions and dynamics of these [[1-form]] and [[Grassmann|Grassmann number]] valued parts of an E8 [[superconnection]] are described by the [[curvature]] and action over a four dimensional base [[manifold]].
A [[talk for ILQGS 07]] on 11/13/07 was presented on this paper.
Internet discussion of the paper, in chronological order:
*[[Backreaction|http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html]]
*[[Physics Forums|http://www.physicsforums.com/showthread.php?t=196498]]
*[[The Reference Frame|http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html]]
*[[Hidden Variables|http://blog.domenicdenicola.com/post/2007/11/Criteria-for-a-Theory-of-Everything.aspx]]
*[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=617]]
*[[Arcadian Functor|https://www.blogger.com/comment.g?blogID=28857369&postID=5548882952979522971]]
*[[Freedom of Science|http://globalpioneering.com/wp02/an-exceptionally-simple-theory-of-everything/]]
*[[Theoreman Egregium|http://egregium.wordpress.com/2007/11/10/physics-needs-independent-thinkers/]]
*[[Science Forums|http://www.scienceforums.net/forum/showthread.php?t=29522]]
And previously:
*[[This Week's Finds 253]]
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<<tiddler HideTags>>The BRST technique accounts for gauge symmetries by introducing new fields with anticommuting coefficients and dynamics that fixes the original local gauge symmetry and introduces a new global (super) symmetry -- the BRST transformation.
Connection: $\;\;\;\; \f{A} = \f{H} + \f{K} \;\; \in \; H + K = G\vp{A_{\big(}} \;\;\;\;$ with $H$ reductive in $G$.
Action: $\;\;\;\; S = \int \big< \ff{B} \ff{F} + \nf{V}(B^H) \big> \vp{A_{\big(}} \;\;\;\;$ purely gauge (topological) in $\f{K}$.
BRST transformation: (make gauge parameter, $\ps = \ps^{\, \io}(x) T_\io \in K$, an anticommuting BRST field)
$$
\begin{array}{rclcrclcrcl}
\ud{\de} \f{K} \!\!&\!\!=\!\!&\!\! -\f{D} \ud{\ps} & \;\;\; & \ud{\de} \ud{\ps} \!\!&\!\!=\!\!&\!\! -\ha \big[ \ud{\ps}, \ud{\ps} \big] & & & & \\
\ud{\de} \ff{B} \!\!&\!\!=\!\!&\!\! \big[ \ff{B}, \ud{\ps} \big] & \;\;\; & \ud{\de} \fff{\od{B}} \!\!&\!\!=\!\!&\!\! \fff{\la} & \;\; & \ud{\de} \fff{\la} \!\!&\!\!=\!\!&\!\! 0
\end{array}
\s\s\; \Longrightarrow \;\;\;
\ud{\de} S = 0, \;\; \ud{\de} \ud{\de} = 0
$$
Choose a ''BRST potential'', $\od{\Ps} = \int \big< \fff{\od{B}} \f{K} \big>$, and use it to make the BRST action,
$$
S' = \ud{\de} \od{\Ps} + S = \int \big< \fff{\la} \f{K} + \fff{\od{B}} \f{D} \ud{\ps} + \ff{B} \ff{F} + \nf{V}(B^H) \big>
$$
Varying $\fff{\la}$ fixes the gauge to $\f{K} = 0$, giving the effective action,
$$
S^{\mbox{eff}} = \int \big< \fff{\od{B}} \f{D} \ud{\ps} + \ff{B} \ff{F}{}^H + \nf{V}(B^H) \big>
$$
which can be the Dirac action for a suitable algebra, and $\fff{\od{B}}= \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep$. The literature mentions the BRST superconnection, $\udf{A} = \f{H} + \ud{\ps}$, with $\ud{\ps}$ a "1-form in the space of connections," related to TQFT, and $(\f{d}+\ud{\de})$ relating to BRST cohomology and anomalies.
<<tiddler HideTags>>Start with $E8$ principal bundle connection and its curvature,
$$
\f{A} = \f{H} + \f{\Ps} \qquad \quad
\ff{F} = (\f{d} \f{H} + \f{H} \f{H} + \f{\Ps} \f{\Ps})
+ (\f{d} \f{\Ps} + \f{H} \f{\Ps} + \f{\Ps} \f{H})
$$
Action such that $\f{\Ps}$ part is pure gauge,
$$
S = \int \big< \ff{B} \ff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big>
$$
BRST: Replace $\f{\Ps}$ part with ghosts, $\ud{\Ps}$, in extended connection,
$$
\udf{A} = \f{H} + \ud{\Ps} \qquad \quad
\udff{F} = \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\Ps} + [ \f{H}, \ud{\Ps} ] \big)
= \ff{F}{}_H + \f{D} \ud{\Ps}
$$
Effective action for gauge fields, ghosts, and anti-ghosts:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big> \\
&=& \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} \fr{1}{16 \pi G} \ph^2 \big( R - \fr{3}{2} \ph^2 \big) + \fr{1}{4} \ff{F'} \ff{*F'} \big>
\end{eqnarray}
<<tiddler HideTags>>$\de \nf{L} = 0$ under [[gauge transformation]]: $\de \f{A} = - \f{\na} C = -\f{d} C - \big[ \f{A}, C \big]$
Account for gauge part of $\f{A}$ by introducing [[Grassmann|Grassmann number]] valued ''ghosts'', $\ud{C} \in \ud{\rm Lie}(G)_g$, ''anti-ghosts'', $\nf{\od{B}}$, ''partners'', $\nf{\la}$, and [[BRST transformation|BRST technique]]:
$$
\begin{array}{rclcrcl}
\ud{\de} \f{A} &=& - \f{\na} \ud{C} & \s\;\;\; & \ud{\de} \ud{C} &=& - \ha \big[ \ud{C}, \ud{C} \big] \\
\ud{\de} \ff{B} &=& \big[ \ff{B}, \ud{C} \big] & \s\;\;\; & \ud{\de} \nf{\od{B}} &=& \nf{\la} \\
\ud{\de} \nf{\la} &=& 0 & \s\;\;\; & & &
\end{array}
$$
This satisfies $\ud{\de} \nf{L} = 0_{\phantom{\big(}}$and $\ud{\de} \ud{\de} = 0$.
Choose a ''BRST potential'', $\nf{\od{\Ps}} = \big< \nf{\od{B}} \f{A} \big>$, to get new Lagrangian:
$$
\nf{L'} = \nf{L} + \ud{\de} \nf{\od{\Ps}} = \nf{L} + \big< \nf{\la} \f{A_g} \big> + \big< \nf{\od{B}} \f{\na} \ud{C} \big>
$$
BRST partners act as Lagrange multipliers; ''effective Lagrangian'':
$$
\nf{L^{\rm eff}} = \nf{L}[\ff{B'},\f{A'}] + \big< \nf{\od{B}} \f{\na'} \ud{C} \big>
$$
//This is a speculative description of the [[BRST technique]] based on conversations with [[Michael Edwards]]//
Start with a [[connection]] 1-form, $\f{\om}$, defined over the entire space of a [[fiber bundle]], and some fiber bundle section, $\si$. The connection field over the base manifold is the pullback of the connection along the section,
$$
\f{A} = \si^* \f{\om}
$$
A BRST transformation may be a way of describing how $\f{A}$ changes under a change of section. (Though I think this is just a gauge transformation, with a funny pair of Grassmann valued parameters.) Consider a vector field,
$$
\ve{\va} = \va^A(x) \ve{\xi_A}(p)
$$
on the entire space, with $\ve{\xi_A}$ the flow fields corresponding to the group generators, $T_A$. The gauge transformation parameters can be written in terms of a Grassmann valued parameter and Grassmann valued ghost fields as $\va^A(x)= \va C^A(x)$. The BRST transformation then is
$$
\f{\de A} = - \si^* L_{\ve{\va}} \f{\om} = \va s \f{A} = - \va (\f{d} C + \f{A} \times C)
$$
in which $C=C^A T_A$.
//Hmm, this seems to give the change in A from flowing $\om$ under $\si$...//
Another idea, from Picken. Instead of pulling the [[Ehresmann connection]] back along a section, use the surface [[vector projection]] on the E conn to project to the gauge 1-form on the surface and a ''ghost'' -- a 1-form off of the surface. This ghost is the projection of the [[Maurer-Cartan form]], and its value determines the shape of the section. May be able to connect this with other descriptions, like the one above.
Nah, none of this is going to work right. Have to work in the space of connections.
variational bicomplex
Refs:
*M. Ghiotti
**[[Gauge fixing and BRST formalism in non-Abelian gauge theories|papers/Ghiotti - Gauge fixing and BRST formalism in non-Abelian gauge theories.pdf]]
***Excellent new thesis.
*G. Catren and J. Devoto
**[[Extended Connection in Yang-Mills Theory|http://arxiv.org/abs/0710.5698]]
*Bonora and Cotta-Ramusino
**[[Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations|papers/1103922136.pdf]]
***This is one of the first, and probably the best, description of ghosts as 1-forms in the space of connections.
***${\cal G}$ is group of vertical [[automorphism]]s of $E$, equals group of [[gauge transformation]]s.
*Stora and Kastler
**[[A Differential Geometric Setting for BRS Transformations and Anomalies|papers/A Differential Geometric Setting for BRS Transformations and Anomalies.pdf]]
***detailed exposition.
***gauge transformation bundle
***old (hard to read scanned text) but good. lengthy.
***same gauge trasf as Viallet (below)
*Viallet
**[[The Geometry of the Space of Fields in Yang-Mills theory|papers/The Geometry of the Space of Fields in Yang-Mills theory.pdf]]
***space of fields as bundle, physical fields as base
***strange definition for group automorphism, which disagrees with mine and Wikipedia's.
***gauge transf are ''equivariant'' automorphisms, $f(p)=p\ph(p)$, satisfying $f(ph)=f(p)h$ and hence $\ph(ph)=h^- \ph(p) h$.
*J.W. van Holten
**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]
***excellent elementary practical intro
***ghost to M-C form mapping not an identification
**[[The BRST Complex and the Cohomology of Compact Lie Algebras|papers/The BRST Complex and the Cohomology of Compact Lie Algebras.pdf]]
***BRST analysis analogous to [[Hodge decomposition]]
***this paper's content is included in the paper above
*http://en.wikipedia.org/wiki/BRST_Quantization
*[[Principal Bundles, Connections and BRST Cohomology|papers/9408003.pdf]]
**(//read this now//)
**BRST cohomology in the space of connections
**mathematically dense, but I'm hacking it so far
**[[lots of geometric brst papers from spires|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+c+cmpha,87,589&SKIP=0]]
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***for particle physicists
***ghosts are 1-forms in the space of gauge transformations
***BRST operator is exterior derivative in this space
*Barnich, Brandt, and Henneaux
**[[Local BRST cohomology in gauge theories|papers/0002245.pdf]]
***these guys are the big shots in the field, but I don't like this antifield approach yet.
*[[Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory|papers/9705123.pdf]]
**pretty good and succinct intros, plus treats BF
**agrees with Viallet's def of equivariant automorphism.
*Moritsch, Sorella, et al
**[[Algebraic characterization of gauge anamolies on a nontrivial bundle|papers/9611168.pdf]]
***nice algebraic treatment, with generalized connection
**[[Algebraic characterization of the Wess-Zumino consistency conditions in gauge theory|papers/9302136.pdf]]
***Sorrella's paper introducing $\de$. Seems to work with jets without calling them that.
**[[Algebraic structure of gravity in Ashtekar variables|papers/9409046.pdf]]
***Blaga, using $\de$.
*Jim Stasheff
**[[The (secret?) homological algebra of the Batalin-Vilkovisky approach|papers/9712157.pdf]]
***abstract mathematical overview (jets) of relations between physics and math structures
***ghost = Chevelley-Eilenberg generator
***anti-ghost = Tate generator
***anti-field = Koszul generator
*Yang, Lee
**[[Lie algebra cohomology and group structure of gauge theories|papers/9503204.pdf]]
***maybe or maybe not useful
*Kelnhofer
**[[On the Geometrical Structure of Covariant Anomalies in Yang-Mills Theory|paper/9302012.pdf]]
***damn this is a (unavoidable) mess
***universal bundle
*Thomas Schucker
**see his book on Amazon for introductory material:
***[[Differential Geometry, Gauge Theories and Gravity|http://www.amazon.com/Differential-Geometry-Cambridge-Monographs-Mathematical/dp/0521378214/ref=si3_rdr_bb_product/104-9709999-3726336]]
**[[The Cohomological Construction of Stora's Solutions|papers/The Cohomological Construction of Stora's Solutions.pdf]]
*Picken: http://www.iop.org/EJ/abstract/0305-4470/19/5/001
**//(scan this)//
*J. P. Zwart
**[[BRST Reduction and Quantization of Constrained Hamiltonian Systems|papers/zwart98brst.pdf]]
*Witten
**[[Topological Quantum Field Theory|papers/1104161738.pdf]]
**differential forms on the space of connections
*Laurent Baulieu
**[[On the Cohomological Structure of Gauge Theories|papers/On the Cohomological Structure of Gauge Theories.pdf]]
***adds two Grassmann coordinates to spacetime
*Rudolf Schmid
**[[Local Cohomology in Gauge Theories BRST Tansformations and Anomalies|papers/localbrst.pdf]]
***mathematically abstract, but geometric
***ack, [[jet]]s.
***connects ghosts to [[Maurer-Cartan form]]
**[[A Few BRST Bicomplexes|papers/nankai.pdf]]
*discussion with [[Michael Edwards]] on
**[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=436]]
*Ian Anderson
**[[The Variational Bicomplex|papers/The Variational Bicomplex.pdf]]
The [[BRST technique]] fixes and accounts for [[gauge symmetries|gauge transformation]] by introducing new fields with [[Grassmann valued|Grassmann number]] coefficients having dynamics and interactions with existing fields that breaks the original local gauge symmetry but includes a new global (super) symmetry -- the BRST transformation -- that's a "rotation" between old and new fields. This method of gauge fixing is an indispensable tool in the application of path integral methods in the quantum field theory of non-abelian gauge fields ([[principal bundle]] connections), and has a natural extension to describe the existence and dynamics of fermionic [[spinor]] fields.
A restricted BF [[Lagrangian|action]],
$$
\nf{\cal L} = \li \nf{B} \ff{F} + \nf{\Phi}(\f{A},\nf{B}) \ri
$$
invariant, $\delta_{G} \nf{\cal L} = 0$, under some subset of the gauge transformation, $G \in {\frak h} \subset {\frak g}$, is amenable to the BRST technique. A ''ghost field'', $\ud{C} = \ud{C^A} T_A \in \ud{\frak h}$, is introduced with Grassmann coefficients multiplying [[Lie algebra]] elements, along with an anti-Grassmann valued $(n-1)$-form ''antighost field'', $\nf{\od{B}} = \nf{\od{B}{}^A} T_A$, and a real valued $(n-1)$-form ''BRST partner field'', $\nf{\lambda} = \nf{\lambda^A} T_A$. This new system is equipped with a global ''BRST transformation'' -- a ''supersymmetry rotation'' between real and Grassmann valued variables,
$$
\begin{array}{rclcrcl}
\ud{\de} \f{A} &=& -\f{\nabla} \ud{C} & \;\;\; & \ud{\de} \ud{C} &=& -\ha \lb \ud{C}, \ud{C} \rb \\
\ud{\de} \nf{B} &=& \lb \nf{B}, \ud{C} \rb & \;\;\; & \ud{\de} \nf{\od{B}} &=& \nf{\la} \\
\ud{\de} \nf{\la} &=& 0 & & & &
\end{array}
$$
that is nilpotent, $\ud{\de} \ud{\de} = 0$, and leaves the Lagrangian invariant (''BRST [[closed]]''), $\ud{\de} \nf{\cal L} = 0$. Physical observables are in the [[cohomology]] of this ''BRST operator'', $\ud{\de}$. Dynamics are introduced for the ghosts by adding a ''BRST [[exact]]'' term to get a ''BRST extended Lagrangian'',
$$
\nf{\cal L'} = \nf{\cal L} + \ud{\de} \nf{\od{\Psi}}
$$
with some ''BRST potential'', $\nf{\od{\Psi}}$, chosen. For example, choosing
$$
\nf{\od{\Psi}} = \li \nf{\od{B}} \f{A} \ri
$$
gives
$$
\ud{\de} \nf{\od{\Psi}} = \li \nf{\la} \f{A} \ri + \li \nf{\od{B}} \f{\nabla} \ud{C} \ri
$$
The BRST partner field, $\nf{\la}$, acts as a Lagrange multiplier constraining the gauge freedom of the connection, so the ''gauge fixed connection'' is $\f{A} = \f{A'}$, with $\nf{\la} \f{A'} = 0$. The resulting ''effective Lagrangian'' is
$$
\nf{\cal L^{\rm eff}} = \li \nf{B'} \ff{F'} + \nf{\Phi}(\f{A'},\nf{B'}) \ri
+ \li \nf{\od{B}} \f{\nabla'} \ud{C} \ri
$$
This form of the Lagrangian suggests the introduction of a ''BRST extended connection'',
$$
\udf{A} = \f{A'} + \ud{C}
$$
with ''BRST extended curvature'',
$$
\udff{F} = \f{d} \udf{A} + \ha \lb \udf{A} , \udf{A} \rb = \ff{F'} + \f{\nabla'} \ud{C} + \ha \lb \ud{C} , \ud{C} \rb
$$
allowing the effective Larangian to be written as
$$
\nf{\cal L^{\rm eff}} = \li \nf{\od{B'}} \udff{F} + \nf{\Phi}(\f{A'},\nf{B'}) \ri
$$
with $\nf{\od{B'}} = \nf{B'} + \nf{\od{B}}$.
Ref:
*J.W. van Holten
**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]
***Good modern introduction.
*Laurent Baulieu
**[[Perturbative Gauge Theories|papers/LBaulieu_BRST.pdf]]
***Early reference, including superconnection.
<<tiddler HideTags>>
Equate $Pin(1,3)$ spinor representation space with left-chiral ''biquaternions'' -- complex quaternions,
$$
\ps = \lb \ba{c} \ps_1 \\ \ps_2 \\ \ps_3 \\ \ps_4 \ea \rb \; \;
\ps^C = i \ga_2 \ps^* = \lb \ba{c} -\ps_4^* \\ \ps_3^* \\ \ps_2^* \\ -\ps_1^* \ea \rb \; \;
\sim \; \;
\ps_Q = \lb \ba{cc} \ps_1 & -\ps_4^* \\ \ps_2 & \ps_3^* \ea \rb = \lb \ps_L \; \bar{\ps}_L \rb \in GL(2,\mathbb{C})
$$
Using the Pauli matrix representation of quaternions, $\{ e_0 = \si_0, \, e_\pi = -i \si_\pi\}$, we have the isomorphism to biquaternionic spinors,
$$
\ps
\s \sim \s
\ps_Q = \ps_Q^0 \si_0 + \ps_Q^\pi (-i \si_\pi)
\s \sim \s
\ps_{\mathbb{H}} = \ps_{\mathbb{H}}^\mu e_\mu \, \in \, \mathbb{C} \otimes \mathbb{H}
$$
The action of Lorentz generators (rotations and boosts) on biquaternionic spinors is
$$
J_\pi \ps = \lp - \fr{i}{2} \si_0 \otimes \si_\pi \rp \ps \;\; \sim \;\; J_\pi \ps_\mathbb{H} = \fr{1}{2} e_\pi \ps_\mathbb{H} \s \s
K_\pi \ps = \lp \fr{1}{2} \si_3 \otimes \si_\pi \rp \ps \;\; \sim \;\; K_\pi \ps_\mathbb{H} = \fr{i}{2} e_\pi \ps_\mathbb{H}
$$
showing $Spin^+(1,3) = SL(2,\mathbb{C}) = \mathbb{C} \otimes \mathbb{H}^\mathbb{I}$.
The $C$, $P$, and $T$ generators become:
$$
\ba{rclcrclcrclcrcl}
\ps^C \ae i \ga_2 \ps^* &\s& \ps_{Q}^C \ae \ps_{Q} \si_1 &\s& \ps_{\mathbb{H}}^C \ae i \ps_{\mathbb{H}} e_1 &\s& C &\sim& i e_1 \\[2pt]
\ps^P \ae i \ga_0 \ps &\s& \ps_{Q}^P \ae - \si_2 \ps_{Q}^* \si_1 &\s& \ps_{\mathbb{H}}^P \ae - \ps_{\mathbb{H}}^* e_3 &\s& P &\sim& -K e_3 \\[2pt]
\ps^{T} \ae \ga_{13} \ps^* &\s& \ps_{Q}^{T} \ae i \si_2 \ps_{Q}^* &\s& \ps_{\mathbb{H}}^{T} \ae - \ps_{\mathbb{H}}^* e_2 &\s& T &\sim& -K e_2
\ea
$$
with the complex conjugation and quaternion multiplication in $C$, $P$, and $T$ acting to the left. These combine to give $CPT \sim -i$.
A ''Borcherds algbera'', or ''Generalized Lie algebra'', is similar to a [[Kac-Moody algbera]], such as an [[affine Kac-Moody Lie algebra|affine Lie algebra]], but is allowed to have imaginary (or null) simple roots, such as $<\!\al_B,\al_B\!>\,=0$. The ''generalized [[Chevalley-Serre relations|Lie algebra structure]]'' gives the presentation of a Borcherds algebra,
$$
\begin{array}{rcl}
\big[ H_a , H_b \big] \!\!&\!\!=\!\!&\!\! 0 \\
\big[ H_a , E_{\pm b} \big] \!\!&\!\!=\!\!&\!\! \pm S_{ab} E_{\pm b} \\
\big[ E_{+a} , E_{-a} \big] \!\!&\!\!=\!\!&\!\! H_a \\
\big[ E_{\pm i} , E_{\pm j} \big] \!\!&\!\!=\!\!&\!\! N_{\pm i, \pm j} \, E_{\pm k} \\
Ad_{(E_{\pm a})^{1-S_{ab}}} \lp E_{\pm b} \rp &=& 0 \;\;\;\; \text{ for } S_{aa} \ne 0 \text{ and } a \ne b
\end{array}
$$
As a simple example, consider the symmetric ''generalized symmeterized [[Cartan matrix|root system]]'', $S_{ab} = \,<\!\al_a,\al_b\!>\,=(H_a,H_b)$, for $su(2)^B$, with simple roots $\al_1$ and $\al_B$,
$$
S_{ab} =
\lb
\begin{array}{cc}
2 & -1 \\
-1 & 0 \\
\end{array}
\rb
$$
The [[Cartan subalgebra|Lie algebra structure]], $\mathfrak{C}$, is spanned by $H_1 = T'_3$ and $H_B = \ha ( N - T'_3 )$, in which $T'_3$ is a normalized [[su(2)]] generator, $T'_3 = \fr{i}{2} T_3$, and $N$ is the ''number operator'', chosen to be orthonormal: $(T'_3,T'_3)=2$, $(T'_3,N)=0$, and $(N,N)=-2$. The fundamental [[weights]], $\om^1$ and $\om^B$, satisfy $<\! \om_1,\al_1\!>\, = 1$, $<\! \om_1,\al_B \!>\, =0$, $<\! \om_B,\al_B \!>\, = 1$, and $<\! \om_B,\al_1 \!>\, =0$, and are therefore $\om^1 = - \ha \al_B$ and $\om^B = -\ha \al_1 -\ha \al_B = -\ha \al_N$. If we choose the fundamental representation space, with highest weight $\La=\om_B$, the representation space looks like the [[Fock space]] for a [[spinor]].
Further generalization: Change Serre relations to restrict to symmetric or antisymmetric applications of creation operators, instead of ignoring restriction on Borcherds generators, in order to act like bosons and fermions?
<<tiddler HideTags>>$$\begin{array}{rcl}
\f{H} \!\!&\!\!=\!\!&\!\! \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W} =
{\scriptsize
\lb \begin{array}{cccc}
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_R} \ph_0^* & \fr{1}{4} \f{e_R} \ph_+ \\
i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \p{-} \fr{1}{4} \f{e_R} \ph_+^* & \fr{1}{4} \f{e_R} \ph_0 \\
- \fr{1}{4} \f{e_L} \ph_0 & \fr{1}{4} \f{e_L} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & \\
\p{-} \fr{1}{4} \f{e_L} \ph_+^* & \fr{1}{4} \f{e_L} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B}
\end{array} \rb_{\p{(}}
} \\
\!\!&\!\!=\!\!&\!\! \f{dx^a} \ha h_a^{\p{a} \al\be} \ga_{\al\be} \;\; \in \;\; \f{so}(1,7) = \f{Cl}^2(1,7) \subset \f{\mathbb{C}}(8\times8)
\end{array}$$
@@display:block;text-align:center;[[Clifford bivector|Clifford algebra]] parts:@@$$
\begin{array}{rcl}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^a} \ha \om_a^{\p{a} \mu \nu} \ga_{\mu \nu}
\s\s\s\s\s\s\s\s
\leftarrow \text{spin connection} \\
\f{e} \ph \!\!&\!\!=\!\!&\!\! \f{dx^a} \lp e_a \rp^\mu \ph^\ph \ga_{\mu \ph}
\, \left\{
\begin{array}{rcl}
\f{e} \!\!&\!\!=\!\!&\!\! \f{dx^a} \lp e_a \rp^\mu \ga_\mu
\s\s\;
\leftarrow \text{frame (vierbein)} \\
\ph \!\!&\!\!=\!\!&\!\! \ph^\ph \ga_\ph
\, \left\{
\begin{array}{rcl}
\ph_+ \!\!&\!\!=\!\!&\!\! (-\ph^5 \!+\! i \ph^6) \\
\ph_0 \!\!&\!\!=\!\!&\!\! (\ph^7 \!+\! i \ph^8)
\end{array}
\right\}
\begin{array}{c}
\;\;\;\;
\leftarrow \text{Higgs} \\
\ph \ph = -M^2
\end{array}
\end{array}
\rd
\end{array}
$$
$$
\begin{array}{rcl}
\f{B} \!\!&\!\!=\!\!&\!\! - \! \f{dx^a} \ha B_a \big( \ga_{56} - \ga_{78} \big)
\s\s\s\;\;\,
\leftarrow \; \downarrow \text{electroweak gauge fields} \\
\f{W} \!\!&\!\!=\!\!&\!\! - \! \ha \f{W^1} \big( \ga_{67} + \ga_{58} \big)
- \ha \f{W^2} \big(-\ga_{57} + \ga_{68} \big)
- \ha \f{W^3} \big( \ga_{56} + \ga_{78} \big)
\s\;\, \\
\end{array}
$$
@@display:block;text-align:center;[[indices]]: $\;\;\;\; 0 \le a,b \le 3 \s 0 \le \mu,\nu \le 3 \s 5 \le \ph,\ps \le 8 $@@
The [[table of Clifford matrix representations]] repeats with eight-fold ''Bott periodicity'', $Cl(p+8,q) = Cl(p,q) \otimes Cl(8)$, because $Cl(8)$ can be represented by $\mathbb{R}(16)$ and has $\Ga^2 = 1$.
Explicitly, if $\ga_\mu$ are the basis elements of $Cl(p,q)$, with $n = p + q$ even and $\ga \ga = +1$, then
$$
\begin{array}{rcl}
\ga'_\mu \!\!&\!\!=\!\!&\!\! \Ga_0 \otimes \ga_\mu \\
\ga'_{n + \al} \!\!&\!\!=\!\!&\!\! \Ga_\al \otimes 1 \\
\ga'_{n + 8} \!\!&\!\!=\!\!&\!\! \Ga_0 \otimes \ga \\
\end{array}
$$
with $1 \le \al \le 7$, is a basis for $Cl(p+8,q)$.
This also goes through similarly for other $Cl(p,q)$, and/or $Cl(4,4)$ and $Cl(0,8)$.
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>
</center></html>@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>
</center></html>@@
Creating bulleted lists is simple.
* Just add an asterisk
* at the beginning of a line.
** If you want to create sub-bullets
** start the line with two asterisks
*** And if you want yet another level
*** use three asterisks
* You can also do [[Numbered Lists]]
{{{
Creating bulleted lists is simple.
* Just add an asterisk
* at the beginning of a line.
** If you want to create sub-bullets
** start the line with two asterisks
*** And if you want yet another level
*** use three asterisks
* You can also do [[Numbered Lists]]
}}}
<<tiddler HideTags>>
<html><center>
<table class="gtable">
<tr><td>
</td></tr>
<tr><td><SPAN class="math">$\p{a}$</SPAN></td></tr>
<tr><td>
<img SRC="talks/ToE24/CPTti.png" height=400px>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>Since every fermion has an anti-fermion, there is a ''charge symmetry'', $C$, that transforms between them. This symmetry is not in $Pin(1,3)$, but operates on the complex representation space of $Pin(1,3)$ spinors as an anti-unitary operator,
$$
U_C = i \ga_2 K
$$
in which $K$ is complex conjugation. We can combine this with our unitary time operator to get ''anti-unitary time conjugation'', and add a phase to parity conjugation,
$$
U_T = U_C \, i U'_T = (i \ga_2 K)(i \ga_0 \ga) = \ga_1 \ga_3 K \s \s \s
U_P = i U'_P= i \ga_0
$$
These three conjugations, $C$, $P$, and $T$, combine to give
$$
U_{CP} = -\ga_0 \ga_2 K \s
U_{CT} = - i \ga_1 \ga_2 \ga_3 \s
U_{PT} = i \ga_0 \ga_1 \ga_3 K \s
U_{CPT} = \ga
$$
The ''CPT Group'', $G_{CPT}$, of order $16$, thus has multiplication table:
| $\;1\;$ | $C$ | $P$ | $T$ | $CP$ | $CT$ | $PT$ | $CPT$ |
| $\;C\;$ | $+1$ | $CP$ | $CT$ | $P$ | $T$ | $CPT$ | $PT$ |
| $\;P\;$ | $CP$ | $ -1$ | $PT$ | $-C$ | $CPT$ | $-T$ | $-CT$ |
| $\;T\;$ | $CT$ | $-PT$ | $-1$ | $-CPT$ | $-C$ | $P$ | $CP$ |
| $\;CP\;$ | $P$ | $-C$ | $CPT$ | $-1$ | $PT$ | $-CT'$ | $-T$ |
| $\;CT\;$ | $T$ | $-CPT$ | $-C$ | $-PT$ | $-1$ | $CP $ | $P$ |
| $\;PT\;$ | $CPT$ | $T$ | $-P$ | $CT $ | $-CP $ | $-1$ | $-C$ |
| $\;CPT\;$ | $PT$ | $CT$ | $-CP$ | $T'$ | $-P$ | $-C $ | $-1$ |
Since $C$ commutes, this is identifiable as the split-biquaternion group, $G_{CPT} = Q_8 \times \mathbb{Z}_2$, the direct product of the quaternion group, $Q_8 = \{ \pm 1, \pm P, \pm T, \pm PT \}$, and $\mathbb{Z}_2 = \{ 1, C \}$.
A [[quantum Dirac spinor]] in Minkowski spacetime,
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} u_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} v_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp
$$
and its adjoint,
$$
\ud{\hat{\bar{\Psi}}} = \ud{\hat{\Psi}}^\da \ga^0
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee \, \da} {\bar{u}}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee} {\bar{v}}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp
$$
include creation and annihilation operators for particles, $\ud{\hat{a}}_{\, p}^{L/R}$, and antiparticles, $\ud{\hat{b}}_{\, p}^{L/R}$, of left and right chirality, for all possible momenta.
The charge, parity, and time conjugates of Dirac solutions are also Dirac solutions, and correspond to conjugations of a quantum Dirac spinor. A faithful representative group operator, $O(g)$, on Dirac spinors has a corresponding operator, $\hat{O}(g)$, on the infinite-dimensional representation space operators (i.e. creation and annihilation operators) of QFT,
$$
\ud{\hat{\Ps}}^g(x) = \hat{O}(g) \, \ud{\hat{\Ps}}(x) \, \hat{O}^-(g) = O(g) \, \ud{\hat{\Ps}}(x'_g(x))
$$
The [[charge conjugate]], $\Ps^C = i \ga_2 \Ps^*$, is:
$$
\ba{rcl}
\ud{\hat{\Ps}}^C
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} \rp^C u_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \lp \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} \rp^C v_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp \\[5pt]
&=& i \ga_2 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} u_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} v_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp^* \\[5pt]
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{\wedge/\vee \, \da} v_p^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{\wedge/\vee} u_p^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp
\ea
$$
using Dirac solution identities, $i \ga_2 \, u_p^{\wedge/\vee \, *} = v_p^{\wedge/\vee}$ and $i \ga_2 \, v_p^{\wedge/\vee \, *} = u_p^{\wedge/\vee}$. The charge conjugation transformations of the creation and annihilation operators, using the corresponding operation on the infinite-dimensional representation, $\ud{\hat{\Ps}}^C = \hat{C} \ud{\hat{\Ps}} \hat{C}^-$, are thus:
$$
\lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} \rp^C = \ud{\hat{b}}_{\, p}^{\wedge/\vee}
\s \s
\lp \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} \rp^C = \ud{\hat{a}}_{\, p}^{\wedge/\vee \, \da}
$$
The [[parity conjugate]], $\Ps^P = i \ga_0 \Ps(t,-x)$, is
$$
\ba{rcl}
\ud{\hat{\Ps}}^P &=& i \ga_0 \int{\fr{-d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, -p}^{\wedge/\vee} u_{-p}^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, -p}^{\wedge/\vee \, \da} v_{-p}^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp \\[5pt]
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp -i \, \ud{\hat{a}}_{\, -p}^{\wedge/\vee} u_p^{\wedge/\vee} e^{-i p_\mu x^\mu} +i \, \ud{\hat{b}}_{\, -p}^{\wedge/\vee \, \da} v_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp
\ea
$$
using $i \ga_0 \, u_{-p}^{\wedge/\vee} = + i \, u_p^{\wedge/\vee}$ and $i \ga_0 \, v_{-p}^{\wedge/\vee} = - i \, v_p^{\wedge/\vee}$. The parity conjugation transformations of the creation and annihilation operators are thus:
$$
\lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} \rp^P = -i \, \ud{\hat{a}}_{\, -p}^{\wedge/\vee}
\s \s
\lp \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} \rp^P = +i \, \ud{\hat{b}}_{\, -p}^{\wedge/\vee \, \da}
$$
The [[time conjugate]], $\Ps^T = \ga_{13} \Ps(-t,x)$, of a massless quantum Dirac spinor corresponds to an antiunitary operator on Fock space,
$$
\ba{rcl}
\ud{\hat{\Ps}}^T &=& \hat{T}' \ud{\hat{\Ps}} \hat{T}'^-
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} \rp^T u_p^{\wedge/\vee \, *} e^{+i p_\mu x^\mu} + \lp \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} \rp^T v_p^{\wedge/\vee \, *} e^{-i p_\mu x^\mu} \rp \\[5pt]
&=& \ga_{13} \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, -p}^{\wedge/\vee} u_{-p}^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{\, -p}^{\wedge/\vee \, \da} v_{-p}^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp \\[5pt]
&=& \int{\fr{-d^3p}{(2\pi)^3 (2E)}} \lp \mp \ud{\hat{a}}_{\, -p}^{\wedge/\vee} u_p^{\vee/\wedge \, *} e^{+i p_\mu x^\mu} \mp \ud{\hat{b}}_{\, -p}^{\wedge/\vee \, \da} v_p^{\vee/\wedge \, *} e^{-i p_\mu x^\mu} \rp \\
\ea
$$
using $\ga_{13} \, u_{-p}^{\wedge/\vee} = \mp \, u_p^{\vee/\wedge \, *}$ and $\ga_{13} \, v_{-p}^{\wedge/\vee} = \mp \, v_p^{\vee/\wedge \, *}$. The time conjugation transformations of the creation and annihilation operators for particles and antiparticles are thus:
$$
\lp \ud{\hat{a}}_{\, p}^{\wedge/\vee} \rp^T = \mp \, \ud{\hat{a}}_{\, -p}^{\vee/\wedge}
\s \s
\lp \ud{\hat{b}}_{\, p}^{\wedge/\vee \, \da} \rp^T = \mp \, \ud{\hat{b}}_{\, -p}^{\vee/\wedge \, \da}
$$
Ref:
*M. Berg, C. DeWitt-Morette, S. Gwo, E. Kramer, [[The Pin Groups in Physics: C, P, and T|papers/The Pin Groups in Physics- C, P, and T.pdf]]
<<tiddler HideTags>>
<html><center>Charge, parity, and time conjugation operators act on fermion states and their weights:</center></html>
$$
\ba{rclcrcl}
(a_{L/R}^{\wedge/\vee})^C \ae \bar{a}_{L/R}^{\wedge/\vee} & \s & C \, : \, (\om_T, \om_S, h, q) & \mapsto & (\;\;\, \om_T, \;\;\, \om_S, \;\;\, h, - q) \\[10pt]
(a_{L/R}^{\wedge/\vee})^P \ae -i \, a_{R/L}^{\wedge/\vee} & & P \, : \, (\om_T, \om_S, h, q) & \mapsto & (-\om_T, \;\;\, \om_S, - h, \;\;\, q) \\[10pt]
(a_{L/R}^{\wedge/\vee})^T \ae \mp \, a_{L/R}^{\vee/\wedge} & & T \, : \, (\om_T, \om_S, h, q) & \mapsto & (-\om_T, - \om_S, \;\;\, h, \;\;\, q) \\
\ea
$$
<html>
<center>
<img src="talks/ToE24/CPTcube.png" width="240px">
</center>
</html>
There is a finite [[group]], the ''CPT group'', $G_{CPT}$, corresponding to the actions of three generators, [[charge conjugation|charge conjugate]], $\hat{C}$, [[parity conjugation|parity conjugate]], $\hat{P}$, and [[time conjugation|time conjugate]], $\hat{T}$, on [[Fock space]]. These correspond to generators operating on the [[Dirac spinor]] [[representation space]],
$$
\hat{C} \sim C = i \ga_2 K \s \s \hat{P} \sim P = i \ga_0 \s \s \hat{T} \sim T' = - C T_U = K T = \ga_{13} K
$$
in which $K$ is complex conjugation. It is necessary to use the antiunitary time conjugation operator, $T' \sim \hat{T}$, here instead of $T$ in order to properly match the fact that $\hat{T}$ is [[antiunitary]] when acting on Fock space. These generators square to
$$
\hat{C}^2 \sim C^2 = +1 \s \s \hat{P}^2 \sim P^2 = -1 \s \s \hat{T}^2 \sim {T'}^2 = -1
$$
and cross multiply to
$$
C P = P C = - \ga_{02} K \s \s C T' = T' C = - i \ga_{123} \s \s P T' = - T' P = i \ga_{013} K \s \s CPT' = \ga
$$
This closes the group, of order 16,
$$
\ba{rcl}
G_{CPT} \ae \{ \pm 1, \pm C, \pm P, \pm T', \pm CP, \pm CT', \pm PT', \pm CPT' \} \\
\ae \{ \pm 1, \pm i \ga_2 K, \pm i \ga_0, \pm \ga_{13}K, \mp \ga_{02} K, \mp i \ga_{123}, \pm i \ga_{013}K, \pm \ga \}
\subset Pin(3,1)
\ea
$$
which is a [[subgroup]] of the [[spacetime pin group|spacetime spin group]], $Pin(3,1)$. The group multiplication table (which is the same whether we use $T'$ or $T_U$) is
| $\;1\;$ | $C$ | $P$ | $T'$ | $CP$ | $CT'$ | $PT'$ | $CPT'$ |
| $\;C\;$ | $+1$ | $CP$ | $CT'$ | $P$ | $T'$ | $CPT'$ | $PT'$ |
| $\;P\;$ | $CP$ | $ -1$ | $PT'$ | $-C$ | $CPT'$ | $-T'$ | $-CT'$ |
| $\;T'\;$ | $CT'$ | $-PT'$ | $-1$ | $-CPT'$ | $-C$ | $P$ | $CP$ |
| $\;CP\;$ | $P$ | $-C$ | $CPT'$ | $-1$ | $PT'$ | $-CT'$ | $-T'$ |
| $\;CT'\;$ | $T'$ | $-CPT'$ | $-C$ | $-PT'$ | $-1$ | $CP $ | $P$ |
| $\;PT'\;$ | $CPT'$ | $T'$ | $-P$ | $CT' $ | $-CP $ | $-1$ | $-C $ |
| $\;CPT'\;$ | $PT'$ | $CT'$ | $-CP$ | $T'$ | $-P$ | $-C $ | $-1$ |
which relates to [[quaternion]]s. If we identify
$$
C \sim I \s \s P \sim k \s \s T' \sim j
$$
with $I^2=1$ a [[split-complex number]] commuting with basis quaternions, $j k = i$, then $CPT'$ generate the ''split-biquaternion group'' (''CPT group''), $G_{CPT } = Q_8 \otimes \mathbb{Z}_2$, with $Q_8$ the ''quaternion group''. This CPT group has application in physics as [[CPT symmetry]].
The [[charge conjugation|charge conjugate]], $C$, [[parity conjugation|parity conjugate]], $P$, and [[time conjugation|time conjugate]], $T$, operators are elements of a finite [[group]], the [[CPT group]], $G_{CPT}$, which act on several [[representation space]]s of interest to physics. The parity and unitary time conjugation operators, $P$ and $T_U$, are elements of the [[pin group|spacetime spin group]], $\mbox{Pin}(3,1)$, which may be represented as [[Cl(3,1)]]${}^*$ [[Clifford group]] [[Clifford matrix representation]] elements via the [[Dirac matrices]]. Since we have so far preferred to work in [[spacetime]] of signature $(1,3)$, we can work in $Cl(3,1)$ by multiplying the usual [[Cl(1,3)]] basis vectors, $\ga_\mu$, by $i$. We extend the Clifford group by including the operation of [[complex conjugation|complex structure]], $K$, on Clifford matrix representatives and complex representation spaces. Some representative generators are
$$
C = i \ga_2 K \s \s P = i \ga_0 \s \s T_U = i \ga_0 \ga \s \s T = -K C T_U = \ga_{13} \s \s T' = - C T_U = TK = \ga_{13} K
$$
which can act on a [[Dirac spinor]],
$$
\Ps^C = i \ga_2 \Ps^* \s \s \Ps^P = i \ga_0 \Ps(t,-x) \s \s \Ps^{T_U} = i \ga_0 \ga \, \Ps(-t,x) \s \s \Ps^{T'} = \ga_{13} \Ps(-t,x)^*
$$
Note that here we have $\Ps^T = (((\Ps^K)^C)^{T_U}$ for the time conjugation operator. The [[Clifford reflection]]s, $P$ and $T_U$, are [[Lorentz transformation]]s, while complex and charge conjugations, $K$ and $C$, are [[gauge transformation]]s that may or may not be elements of the [[standard model gauge group]].
The $C$, $P$, and $T$ operators, as group elements, also have an [[infinite-dimensional unitary representation]]. The [[quantum Dirac spinor]] field,
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} u_p^{\wedge/\vee} e^{-i (Et-p \cdot x)} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} v_p^{\wedge/\vee} e^{+i (Et-p \cdot x)} \rp
$$
includes [[creation and annihilation operators]] for anti-particles and particles with different momentum and spin, and acts on multi-particle [[Fock space]]. Through the correspondence,
$$
\hat{U}(g) \hat{\Ps}(x) \hat{U}(g)^- = O(g) \hat{\Ps}(g^- x)
$$
the $CPT$ operators induce maps between creation and annihilations operators:
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^C = \ud{\hat{b}}_p^{\wedge/\vee} \s \s \lp \ud{\hat{b}}_p^{\wedge/\vee} \rp^C = \ud{\hat{a}}_p^{\wedge/\vee}
$$
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^P = i \, \ud{\hat{a}}_{-p}^{\wedge/\vee} \s \s \lp \ud{\hat{b}}_p^{\wedge/\vee} \rp^P = i \, \ud{\hat{b}}_{-p}^{\wedge/\vee}
$$
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^T = \pm \ud{\hat{a}}_{-p}^{\vee/\wedge} \s \s \lp \ud{\hat{b}}_p^{\wedge/\vee} \rp^T = \pm \ud{\hat{b}}_{-p}^{\vee/\wedge}
$$
in which $C$ and $P$ are [[unitary]] operators and $T$ is [[antiunitary]], so $(i)^T = -i$. These operators on Fock space interrelate in the [[CPT group]], which is equivalent to the interrelation between $C$, $P$, and $T'$ acting on Dirac spinors.
It is a bit confusing, but the action on Dirac spinors is [[antiunitary]] for $C$ and $T'$ and [[unitary]] for $P$, $T_U$, and $T$, while the action on Fock space is unitary for $\hat{C}$, $\hat{P}$, and $\hat{T}_{\! U}$ and antiunitary for the [[time conjugation|time conjugate]] operator, $\hat{T}$.
Combining $C$, $P$, and $T$ or $T'$, we have the antiunitary operator, $CPT = \ga K$, or the unitary operator, $CPT' = \ga$, and the corresponding antiunitary operator on Fock space,
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^{CPT} = \mp i \, \ud{\hat{b}}_p^{\vee/\wedge} \s \s \lp \ud{\hat{b}}_p^{\wedge/\vee} \rp^{CPT} = \mp i \, \ud{\hat{a}}_p^{\vee/\wedge}
$$
[[arxiv|http://arxiv.org/abs/0706.0217]]
S. Raby, A. Wingerter
Abstract: We investigate whether the hypercharge assignments in the Standard Model can be interpreted as a hint at Grand Unification in the context of heterotic string theory. To this end, we introduce a general method to calculate U(1)_Y for any heterotic orbifold and compare our findings to the cases where hypercharge arises from a GUT. Surprisingly, in the overwhelming majority of 3-2 Standard Models, a non-anomalous hypercharge direction can be defined, for which the spectrum is vector-like. For these models, we calculate sin^2 theta to see how well it agrees with the standard GUT value. We find that 12% have sin^2 theta = 3/8, while all others have values which are less. Finally, 89% of the models with sin^2 theta = 3/8 have U(1)_Y in SU(5).
*computation to find hypercharge directions in E8xE8 root system
[>img[images/person/Carlo Rovelli.jpg]]Homepage: http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
*Location: Marseille
*CV: http://www.cpt.univ-mrs.fr/%7Erovelli/vita.pdf
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Rovelli_C/0/1/0/all/0/1
Selected work:
*[[Quantum Gravity|http://www.cpt.univ-mrs.fr/%7Erovelli/book.pdf]]
*[[Graviton propagator in loop quantum gravity|http://arxiv.org/abs/gr-qc/0604044]]
**nice treatment. includes basic example of canonical and path integral QM, field theory, then does LQG via GFT.
A ''Cartan H-bundle'', with total space $E_H$, is a [[principal bundle]] with $n_H$ dimensional [[Lie group]], $H$, as the typical fiber (and structure group) and $n_M$ dimensional base, $M$. This bundle is not [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, since the structure group of $E_H$ is $H \subset G$; however, $E_H$ does serve as a base space under $E_G$, and the [[Ehresmann Cartan connection]] over $E_G$ does pull back to give a connection over $E_H$.
The Cartan H-bundle is mapped into a section (a [[submanifold]]), $E'_H$, of the Ehresmann Cartan geometry, $E_G$, by the reference section of the [[Cartan homogeneous space bundle]],
\begin{eqnarray}
\si'^S &:& E_H \to E_G \\
\si'^S(x,y) &=& (x,x_{s\si}(x),y)
\end{eqnarray}
The [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Ehresmann Cartan H-connection form'' over $E_H$,
\begin{eqnarray}
\f{ {\cal C}_H}(x,y) &=& \si'^{S*} \f{\cal C} = \lp \f{C^J}(x) L^I{}_J(x_{s\si}(x),y) + \si'^{S*} \f{\xi_R^I}(x_{s\si}(x),y) \rp T_I \\
&=& g^-(x_{s\si}(x),y) \, \f{C}(x) \, g(x_{s\si}(x),y) + g^-(x_{s\si}(x),y) \, \f{d} \, g(x_{s\si}(x),y) \\
&=& h^-(y) \Big( r^-(x_{s\si}(x)) \, \f{C}(x) \, r(x_{s\si}(x)) + r^-(x_{s\si}(x)) \, \f{d} \, r(x_{s\si}(x)) \Big) h(y) + h^-(y) \, \f{d} h(y)
\end{eqnarray}
in which the [[coset representative section|homogeneous space]], $r:G/H \to G$, is used to write $g(x_s,y)=r(x_s) \, h(y)$. Choosing the homogeneous space bundle zero reference section, $r(x_{s\si_0}(x)) = r(0) = 1$, this gives
$$
\f{{\cal C}_H}(x,y) = h^-(y) \, \f{C}(x) \, h(y) + h^-(y) \, \f{d} \, h(y)
$$
Pulling this back along the [[canonical reference section|Ehresmann principal bundle connection]] gives the [[Cartan connection|Cartan geometry]], $\si_0^{H*} \f{{\cal C}_H} = \f{C}$, over $M$.
If $H$ is [[reductive]] in $G$ (as is usually assumed) the Ehresmann Cartan H-connection form splits into the ''Ehresmann Cartan H-connection frame form'' and '' Ehresmann H-connection form'',
\begin{eqnarray}
\f{ {\cal C}_H} &=& \f{ {\cal E}_H} + \f{ {\cal A}_H} \\
\f{ {\cal E}_H} &=& h^-(y) \, \f{e}(x) \, h(y) \in \f{\mathfrak{g}/\mathfrak{h}} \\
\f{ {\cal A}_H} &=& h^-(y) \, \f{A}(x) \, h(y) + h^-(y) \, \f{d} \, h(y) \in \f{\mathfrak{h}}
\end{eqnarray}
The Ehresmann H-connection form, $\f{{\cal A}_H}(x,y)$, over $E_H$ is an [[Ehresmann principal bundle connection]] form for the bundle.
When the Ehresmann Cartan H-connection form equals the [[Maurer-Cartan form]], $\f{{\cal C}_H} = \f{\cal I}$, the Cartan H-bundle is an [[Ehresmann homogeneous space geometry]], $E_H = G$. In this way, the Cartan H-bundle may be considered to be a [[reductive Lie group geometry]], $G$, that has gone wavy along $G/H$ -- with $\f{{\cal C}_H}$ deviating from $\f{\cal I}$ to give the new [[frame]] 1-forms, $\f{{\cal C}_H^J} = \f{E^J}$, of the [[Cartan tangent bundle geometry]] over what was $G$.
A real [[Lie algebra]], ${\frak g}$, with a [[Killing form]], can be spanned by a set of orthogonal generators, $T_A^{\frak k}$ and $T_M^{\frak p}$, satisfying
$$
\lp T_A^{\frak k}, T_A^{\frak k} \rp = -1 \s \s
\lp T_M^{\frak p}, T_M^{\frak p} \rp = +1 \s \s
$$
These ''compact generator''s and ''noncompact generator''s generate a compact subgroup of the Lie group $G$ and a noncompact subspace, and span a ''compact subalgebra'', ${\frak k}$, and ''noncompact subspace'', ${\frak p}$, such that there is a ''Cartan decomposition'', ${\frak g} = {\frak k} \oplus {\frak p}$, and the Lie brackets between Lie algebra elements in these subspaces are necessarily
$$
\lb {\frak k}, {\frak k} \rb = {\frak k} \s \s
\lb {\frak k}, {\frak p} \rb = {\frak p} \s \s
\lb {\frak p}, {\frak p} \rb = {\frak k} \s \s
$$
A ''Cartan involution'' is a [[Lie algebra involution]], with $\th : {\frak k} \to {\frak k}$ and $\th : {\frak p} \to - {\frak p}$. The compact version of the Lie algebra can be defined as ${\frak g}' = {\frak k} \oplus i \, {\frak p}$.
A ''Cartan geometry'' is a [[Lie group geometry]], $G$, that's allowed to go wavy while maintaining some of its symmetry, represented by a subgroup, $H \subset G$, usually assumed to be [[reductive]] in $G$. The wavy ''Cartan geometry base manifold'', $M$, is ''modeled'' on the [[homogeneous space]], $M \sim S=G/H$, and has the same dimension, $n_S = (n_G - n_H)$. The ''Cartan connection'' over $M$,
$$
\f{C}(x) = \f{e} + \f{A} \in \f{\mathfrak{g}}
$$
is a [[Lieform]] modeled on the [[Maurer-Cartan frame|homogeneous space]], $\f{C} \sim \f{I} = r^- \f{d} r(x)$, and splits (for $H$ reductive in $G$) into the ''Cartan frame'', $\f{e}(x) = \f{e^A} K_A \in \f{\mathfrak{g}/\mathfrak{h}}$, and ''Cartan H-connection'', $\f{A}(x) = \f{A^P} H_P \in \f{\mathfrak{h}}$, which (unlike their homogeneous space counterparts) may vary freely.
The ''Cartan [[curvature]]'' of the connection is
\begin{eqnarray}
\ff{F}(x) &=& \f{d} \f{C} + \ha \lb \f{C}, \f{C} \rb \\
&=& \f{d} \f{e} + \f{d} \f{A} + \ha \lb \f{e}, \f{e} \rb + \lb \f{A}, \f{e} \rb + \ha \lb \f{A}, \f{A} \rb \\
&=& \ff{F^A} K_A + \ff{F^P} H_P
\end{eqnarray}
which (like the [[homogeneous space curvature|homogeneous space tangent bundle geometry]]) splits into
\begin{eqnarray}
\ff{F^A} &=& \f{d} \f{e^A} + \f{A^P} \f{e^B} C_{PB}{}^A + \ha \f{e^C} \f{e^B} C_{CB}{}^A \\
\ff{F^P} &=& \ff{F_H^P} + \ha \f{e^C} \f{e^D} C_{CD}{}^P
\end{eqnarray}
with the ''curvature of the Cartan H-connection'' defined by:
$$
\ff{F_H^P} = \f{d} \f{A^P} + \ha \f{A^Q} \f{A^R} C_{QR}{}^P
$$
Note that $\ha \f{e^C} \f{e^B} C_{CB}{}^A = 0$ if $G/H$ is a [[symmetric space]].
There are many relationships between a Cartan geometry and other structures. A [[natural]] [[Ehresmann Cartan geometry]] is a description of a Cartan geometry as an [[Ehresmann principal bundle connection]] for a [[G-bundle|principal bundle]] -- and this description splits via $G/H$ into the [[Cartan H-bundle]] and [[Cartan homogeneous space bundle]]. These two bundles relate to the way the [[Lie group tangent bundle geometry]] of $G$ turns wavy, described by the [[Cartan tangent bundle geometry]].
Refs:
*http://en.wikipedia.org/wiki/Cartan_connection
*[[Differential Geometry of Cartan Connections|papers/9412232.pdf]]
**by [[Peter Michor]] and Alekseevsky (tiddler $G \subset H$)
*[[The Works of Charles Ehresmann on Connections: From Cartan Connections to Connections on Fibre Bundles|papers/CMMarle.pdf]]
**Ehresmann version of Cartan, nicely explained. See p9 for main def.
**http://www.math.jussieu.fr/~marle/
*[[MacDowell-Mansouri Gravity and Cartan Geometry|papers/0611154.pdf]]
**a new paper by Derek Wise
*[[Natural Operations on the Bundle of Cartan Connections|papers/Natural Operations on the Bundle of Cartan Connections.pdf]]
*[[The Existance of Cartan Connections and Geometrizable Principal Bundles|papers/0206136.pdf]]
**a very concise and interesting mathematical treatment.
*[[Gravity, Cartan geometry, and idealized waywisers|http://arxiv.org/abs/1203.5709]]
**a nice description by Hans Westman and Tom Zlosnik
A ''Cartan homogeneous space bundle'', with total space $E_S$, is a [[fiber bundle]] with $n_S$ dimensional [[homogeneous space]], $F=S=G/H$, as the typical fiber and $n_M = n_S$ dimensional base, $M$. This bundle may be visualized as the set of homogeneous spaces tangent to the base space. (If $n_M \neq n_S$ this is a ''generalized Cartan homogeneous space bundle''.) The structure group, $G$, of the bundle is the subset of [[homogeneous space geometry symmetries]] corresponding to the [[left action|group]] of $G$ on the space.
The Cartan homogeneous space bundle, $E_S$, is [[associated]] to the [[Ehresmann Cartan geometry]], $E_G$, and $E_S$ also serves as a base space under $E_G$. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, and the $n_S$ homogeneous space coordinates, $x_s^a$, cover patches of $S$ -- the combined coordinates, $(x,x_s)$, cover patches of $E_S$. The [[reference section|Ehresmann gauge transformation]], $\si^S : M \to E_S$, of the Cartan homogeneous space bundle determines the ''points of tangency'' -- the points, $x_{s\si}(x)$, of the $S_x$ in contact with $x$. Since a homogeneous space has a natural zero point, we often use the ''zero point reference section'', $\si_0^S(x) = (x,0)$.
The Cartan homogeneous space bundle, $E_S$, is mapped into a section (a [[submanifold]]), $E'_S$, of the Ehressmann Cartan geometry, $E_G$, by the reference section of the [[Cartan H-bundle]],
$$
\si'^H(x,x_s) = (x,x_s,y_\si(x))
$$
The [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Cartan homogeneous space connection form'' over $E_S$,
$$
\begin{eqnarray}
\f{{\cal C}_S}(x,x_s) &=& \si'^{H*} \f{\cal C} = \lp \f{C^J}(x) L^I{}_J(x_s,y_\si(x)) + \si'^{H*} \f{\xi_R^I}(x_s,y_\si(x)) \rp T_I \\
&=& g^-(x_s,y_\si(x)) \, \f{C}(x) \, g(x_s,y_\si(x)) + g^-(x_s,y_\si(x)) \, \f{d} \, g(x_s,y_\si(x)) \\
&=& h^-(y_\si(x)) \Big( r^-(x_s) \, \f{C}(x) \, r(x_s) + r^-(x_s) \, \f{d} \, r(x_s) \Big) h(y_\si(x)) + h^-(y_\si(x)) \, \f{d} h(y_\si(x))
\end{eqnarray}
$$
in which the [[coset representative section|homogeneous space]], $r:S \to G$, is used to write $g(x_s,y)=r(x_s) \, h(y)$. Choosing the canonical H-bundle reference section, $h(y_{\si_0}(x)) = h(0) = 1$, this gives
$$
\f{{\cal C}_S}(x,x_s) = r^-(x_s) \, \f{C}(x) \, r(x_s) + r^-(x_s) \, \f{d} \, r(x_s)
$$
Pulling this back along the zero point reference section gives the [[Cartan connection|Cartan geometry]], $\si_0^{S*} \f{{\cal C}_S} = \f{C}$, over $M$.
<<tiddler HideTags>>Mutually [[commuting|commutator]] set of $r$ [[Lie algebra]] generators:
$$
\left\{ T_1, T_2, ..., T_r \right\} \s\s\;\;\;\; [ T_i, T_j ] = 0
$$
[[Cartan subalgebra|Lie algebra structure]]: $\;\;\; C=c^J T_J \;\; \in {\rm Lie(G)} \p{{}_{(}}$
[[Eigenvalues|eigen]], $\al^a$, and [[eigenvectors|eigen]], $V_a \in {\rm Lie(G)}$, using the Lie bracket:
$$
[ C , V_a ] = \al^a V_a = \sum_J c^J \al_J^a V_a
$$
Unique eigenvalue for each of the $(n-r)$ eigenvectors, corresponding to $(n-r)$ ''roots'', $\al_J^a$, in $r$ dimensional vector space.
Cartan subalgebra of the standard model and gravity:
$$
C = {\scriptsize \frac{1}{2}} \om^{01} \ga_{01} + {\scriptsize \frac{1}{2}} \om^{12} \ga_{12} + W^3 i \Si_3 + B i Y + G^3 i \la_3 + G^8 i \la_8
$$
Eigenvectors are elementary particles, roots are their charges:
$$
\al(e_L) = ( \pm {\scriptsize \frac{1}{2}}, \mp {\scriptsize \frac{1}{2}}, -1, -1, 0, 0 ) \p{{}_{\Big(}}
$$
The [[Riemann curvature]] for the [[Cartan tangent bundle geometry]] is calculated from the [[Cartan tangent bundle spin connection]],
$$
\ff{R}^J{}_I = \f{d} \f{W}^J{}_I + \f{W}^J{}_K \f{W}^K{}_I
$$
We'll tackle this in pieces. Using a [[left-right rotator]] identity,
$$
\f{d} \lp L^h \rp^J{}_I = \f{e_H^P} C_P{}^J{}_K \lp L^h \rp^K{}_I
$$
the [[exterior derivative]]s are:
\begin{eqnarray}
\f{d} \f{W}^B{}_A &=& \f{d} \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E - \f{e_H^P} C_P{}^B{}_A \rp \\
&=& \lp \f{d} \f{\nu}^E{}_F \rp \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \f{e_H^P} C_P{}^B{}_D \lp L^h \rp^D{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \lp L^h \rp^B{}_E \f{e_H^P} C_P{}^F{}_C \lp L^h \rp^C{}_A \\
&-& \ha \lp \f{d} \f{A^Q} + \lp \f{d} \f{e_H^P} \rp \lp L^h \rp_P{}^Q - \f{e_H^P} \f{e_H^R} C_{RP}{}^T \lp L^h \rp_T{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E \\
&+& \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp
\lp \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \f{e_H^R} C_R{}^B{}_C \lp L^h \rp^{CD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \lp L^h \rp^{BD} \f{e_H^R} C_R{}_{AC} \lp L^h \rp^{CE} \rp \\
&-& \f{d} \f{e_H^P} C_P{}^B{}_A \\
\\
\f{d} \f{W}^B{}_R &=& \f{d} \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q \rp \\
&=& \ha \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q
- \ha \f{e^D} F^H_{DEQ} \f{e_H^P} C_P{}^B{}_C \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q
- \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \f{e_H^P} C_{PRS} \lp L^h \rp^{SQ} \\
\\
\f{d} \f{W}^Q{}_R &=& - \ha \f{d} \lp \f{A^S} \, \lp L^h \rp^P{}_S + \f{e_H^P} \rp C_P{}^Q{}_R \\
&=& - \ha \lp \lp \f{d} \f{A^S} \rp \lp L^h \rp^P{}_S
- \f{A^S} \f{e_H^U} C_U{}^P{}_T \lp L^h \rp^T{}_S
+ \f{d} \f{e_H^P} \rp C_P{}^Q{}_R
\end{eqnarray}
The pieces quadratic in the spin connection are:
\begin{eqnarray}
\f{W}^B{}_K \f{W}^K{}_A &=& \f{W}^B{}_C \f{W}^C{}_A + \f{W}^B{}_P \f{W}^P{}_A \\
&=&
\lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp \\
&\times&
\lp \f{\nu}^G{}_H \lp L^h \rp^C{}_G \lp L^h \rp_A{}^H - \ha \lp \f{A^R} + \f{e_H^S} \lp L^h \rp_S{}^R \rp F^H_{HGR} \lp L^h \rp^{CH} \lp L^h \rp_A{}^G - \f{e_H^Q} C_Q{}^C{}_A \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\lp \ha \f{e^C} F^H_{CFR} \lp L^h \rp_A{}^F \lp L^h \rp^{PR} \rp \\
\\
\f{W}^B{}_K \f{W}^K{}_R &=& \f{W}^B{}_C \f{W}^C{}_R + \f{W}^B{}_P \f{W}^P{}_R \\
&=& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp
\lp \ha \f{e^G} F^H_{GHS} \lp L^h \rp^{CH} \lp L^h \rp_R{}^S \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^P{}_R \\
\\
\f{W}^Q{}_K \f{W}^K{}_R &=& \f{W}^Q{}_C \f{W}^C{}_R + \f{W}^Q{}_P \f{W}^P{}_R \\
&=& - \lp \ha \f{e^A} F^H_{AFR} \lp L^h \rp_C{}^F \lp L^h \rp^{QR} \rp
\lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q \rp \\
&+&
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^Q{}_P
\ha \lp \f{A^U} \, \lp L^h \rp^V{}_U + \f{e_H^V} \rp C_V{}^P{}_R
\end{eqnarray}
Combining these gives the curvature,
\begin{eqnarray}
\ff{R}^B{}_A &=& \lp \f{d} \f{\nu}^E{}_F \rp \lp L^h \rp^B{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \f{e_H^P} C_P{}^B{}_D \lp L^h \rp^D{}_E \lp L^h \rp_A{}^F - \f{\nu}^E{}_F \lp L^h \rp^B{}_E \f{e_H^P} C_P{}^F{}_C \lp L^h \rp^C{}_A \\
&-& \ha \lp \f{d} \f{A^Q} + \lp \f{d} \f{e_H^P} \rp \lp L^h \rp_P{}^Q - \f{e_H^P} \f{e_H^R} C_{RP}{}^T \lp L^h \rp_T{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_A{}^E \\
&+& \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp
\lp \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \f{e_H^R} C_R{}^B{}_C \lp L^h \rp^{CD} \lp L^h \rp_A{}^E
+ F^H_{DEQ} \lp L^h \rp^{BD} \f{e_H^R} C_R{}_{AC} \lp L^h \rp^{CE} \rp \\
&-& \f{d} \f{e_H^P} C_P{}^B{}_A \\
&+& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp \\
&\times&
\lp \f{\nu}^G{}_H \lp L^h \rp^C{}_G \lp L^h \rp_A{}^H - \ha \lp \f{A^R} + \f{e_H^S} \lp L^h \rp_S{}^R \rp F^H_{HGR} \lp L^h \rp^{CH} \lp L^h \rp_A{}^G - \f{e_H^Q} C_Q{}^C{}_A \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\lp \ha \f{e^C} F^H_{CFR} \lp L^h \rp_A{}^F \lp L^h \rp^{PR} \rp \\
\\
\ff{R}^B{}_R &=& \ha \lp \f{d} F^H_{DEQ} \rp \lp L^h \rp^{BE} \lp L^h \rp_R{}^Q
- \ha \f{e^D} F^H_{DEQ} \f{e_H^P} C_P{}^B{}_C \lp L^h \rp^{CE} \lp L^h \rp_R{}^Q
- \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \f{e_H^P} C_{PRS} \lp L^h \rp^{SQ} \\
&+& \lp \f{\nu}^E{}_F \lp L^h \rp^B{}_E \lp L^h \rp_C{}^F - \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp^{BD} \lp L^h \rp_C{}^E - \f{e_H^P} C_P{}^B{}_C \rp
\lp \ha \f{e^G} F^H_{GHS} \lp L^h \rp^{CH} \lp L^h \rp_R{}^S \rp \\
&-& \lp \ha \f{e^D} F^H_{DEQ} \lp L^h \rp^{BE} \lp L^h \rp_P{}^Q \rp
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^P{}_R \\
\\
\ff{R}^Q{}_R &=& - \ha \lp \lp \f{d} \f{A^S} \rp \lp L^h \rp^P{}_S
- \f{A^S} \f{e_H^U} C_U{}^P{}_T \lp L^h \rp^T{}_S
+ \f{d} \f{e_H^P} \rp C_P{}^Q{}_R \\
&+&
\ha \lp \f{A^S} \, \lp L^h \rp^T{}_S + \f{e_H^T} \rp C_T{}^Q{}_P
\ha \lp \f{A^U} \, \lp L^h \rp^V{}_U + \f{e_H^V} \rp C_V{}^P{}_R
\end{eqnarray}
From this ugly mess, the [[Ricci curvature]], $\f{R}{}_I = \ve{E_J} \ff{R}^J{}_I$, is
\begin{eqnarray}
\f{R}{}_B &=& \ve{E_A} \ff{R}^A{}_B + \ve{E_R} \ff{R}^R{}_B \\
\f{R}{}_R &=& \ve{E_B} \ff{R}^B{}_R + \ve{E_Q} \ff{R}^Q{}_R \\
\end{eqnarray}
//ack, I've got to work on something else for awhile.//
[[curvature scalar]]
Check that it matches [[reductive Lie group tangent bundle geometry]] as special case.
A ''Cartan tangent bundle geometry'' is a [[reductive Lie group tangent bundle geometry]] that has gone a little wavy. The [[frame]] 1-forms, $\f{E^J}$, over what was the Lie group manifold, $E_H \sim G$, split in adapted coordinates as
\begin{eqnarray}
\f{E^A}(x,y) &=& \f{e^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(x,y) &=& \f{A^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
in which $\f{e^B}$ and $\f{A^Q}$ are the [[Cartan frame|Cartan geometry]] forms and [[Cartan H-connection|Cartan geometry]] forms, $\lp L^h \rp^A{}_B = \lp H^A, h^- H_B h(y) \rp$ is the [[left-right rotator]] over $H$, and $\f{e_H^P}$ are the frame 1-forms over $H$. These frame 1-forms are components of the Ehresmann Cartan H-connection form, $\f{E^J} = \f{{\cal C}_H^J}$, over the [[Cartan H-bundle]], $E_H$, with $\f{E^A} = \f{{\cal E}_H^A}$ and $\f{E^P} = \f{{\cal A}_H^P}$. The Cartan tangent bundle, $TE_H$ IS the bundle of tangent vectors over the Cartan H-bundle, $E^H$, but from the point of view of treating the Ehresmann Cartan H-connection forms as a frame. Holding this point of view, we need to figure out what the [[Cartan tangent bundle spin connection]], $\f{W}{}^J{}_K$, is from this frame, and its curvature.
The [[tangent bundle spin connection|tangent bundle connection]] for a [[Cartan tangent bundle geometry]] is determined by insisting the [[torsion]] vanishes over $E_H$, giving [[Cartan's equation]],
$$
\ff{T^J} = 0 = \f{d} \f{E^J} + \f{W}{}^J{}_K \f{E^K}
$$
which may be solved for the ''Cartan tangent bundle spin connection'', $\f{W}{}^J{}_K$. To construct the solution, we first compute the [[exterior derivative]] of the [[frame]] 1-forms,
\begin{eqnarray}
\f{d} \f{E^A} &=& \lp \f{d} \f{e^B} \rp \lp L^h\rp^A{}_B - \f{e^B} \f{d} \lp L^h\rp^A{}_B \\
&=& \lp \f{d} \f{e^B} \rp \lp L^h\rp^A{}_B - \f{e^B} \f{e_H^P} \lp L^h \rp^D{}_B C_{DP}{}^A \\
\f{d} \f{E^P} &=& \lp \f{d} \f{A^Q} \rp \lp L^h \rp^P{}_Q - \f{A^Q} \f{d} \lp L^h \rp^P{}_Q + \f{d} \f{e_H^P} \\
&=& \lp \f{d} \f{A^Q} \rp \lp L^h \rp^P{}_Q - \f{A^Q} \f{e_H^R} \lp L^h \rp^T{}_Q C_{TR}{}^P - \ha \f{e_H^Q} \f{e_H^R} C_{QR}{}^P
\end{eqnarray}
using the [[left-right rotator]], $\lp L^h\rp^I{}_J = \lp T^I, h^- T_J h \rp$, and the [[Maurer-Cartan equation|Maurer-Cartan form]] over $H$. Next we write down the [[orthonormal basis vectors|frame]] (satisfying $\ve{E_J} \f{E^K} = \de_J^K$),
\begin{eqnarray}
\ve{E_A}(x,y) &=& \lp L^h \rp_A{}^B \, \ve{e_B}(x) - \lp L^h \rp_A{}^C \lp \ve{e_C} \f{A^Q} \rp \lp L^h \rp^P{}_Q \, \ve{e^H_P} \\
\ve{E_P}(x,y) &=& \ve{e^H_P}(y)
\end{eqnarray}
and compute the [[anholonomy|Cartan's equation]] coefficients, $f_{IJ}{}^K = \ve{E_J} \ve{E_I} \lp \f{d} \f{E^K} \rp$, getting
\begin{eqnarray}
f_{AB}{}^C &=& \ve{e_E} \ve{e_D} \lp \f{d} \f{e^F} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^C{}_F
+ 2 \lp - \lp L^h \rp_{\lb B \rd}{}^E \, \ve{e_E} \lp L^h \rp_{\ld A \rb}{}^G \lp \ve{e_G} \f{A^Q} \rp \lp L^h \rp^R{}_Q \, \ve{e^H_R} \rp \lp - \f{e^F} \f{e_H^P} \lp L^h \rp^D{}_F C_{DP}{}^C \rp \\
&=& f^M_{DE}{}^F \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp^C{}_F
- 2 \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^R{}_Q \lp L^h \rp_{\lb A \rd}{}^E \, C_{\ld B \rb R}{}^C \\
f_{AQ}{}^C &=& - C_{AQ}{}^C \\
f_{AB}{}^R &=& \ve{e_E} \ve{e_D} \lp \f{d} \f{A^Q} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^R{}_Q
- \lp L^h \rp_A{}^E \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^P{}_Q \lp L^h \rp_B{}^D \lp \ve{e_D} \f{A^S} \rp \lp L^h \rp^T{}_S C_{TP}{}^R \\
&=& \ve{e_E} \ve{e_D} \lp \ff{F_H^Q} \rp \lp L^h \rp_B{}^E \lp L^h \rp_A{}^D \lp L^h \rp^R{}_Q \\
&=& F^H_{DE}{}^Q \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp^R{}_Q \\
f_{AQ}{}^R &=& 0 \\
f_{PQ}{}^C &=& 0 \\
f_{PQ}{}^R &=& - C_{PQ}{}^R
\end{eqnarray}
in which $f^M_{DE}{}^F(x)$ is the anholonomy for $\f{e^A}$ and $\ff{F_H^Q}(x) = \f{d} \f{A^Q} + \ha \f{A^P} \f{A^R} C_{PR}{}^Q$ is the [[curvature]] for $\f{A^Q}$. Using these, the solution to Cartan's equation, $W_{IJK} = \ha \lp f_{IJK} - f_{JKI} + f_{KIJ} \rp$, gives the Cartan tangent bundle spin connection coefficients,
\begin{eqnarray}
W_{ABC} &=& \nu_{DEF} \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F - \lp L^h \rp_A{}^E \lp \ve{e_E} \f{A^Q} \rp \lp L^h \rp^R{}_Q C_{B R C} \\
W_{ABR} &=& \ha f_{ABR} = \ha F^H_{DEQ} \lp L^h \rp_A{}^D \lp L^h \rp_B{}^E \lp L^h \rp_R{}^Q \\
W_{AQR} &=& 0 \\
W_{PBC} &=& - C_{PBC} - \ha f_{BCP} = - C_{PBC} - \ha F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E \lp L^h \rp_P{}^Q \\
W_{PQC} &=& 0 \\
W_{PQR} &=& - \ha C_{PQR}
\end{eqnarray}
with $\nu_{DEF}(x)$ the coefficients of the torsionless spin connection for $\f{e^A}$. From these, the ''Cartan tangent bundle spin connection'', $\f{W}{}_{JK} = \f{E^I} W_{IJK}$, is
\begin{eqnarray}
\f{W}{}_{BC} &=& \f{E^A} W_{ABC} + \f{E^P} W_{PBC} \\
&=& \f{e^D} \lp \nu_{DEF} \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F - \lp \ve{e_D} \f{A^Q} \rp \lp L^h \rp^R{}_Q C_{B R C} \rp
+ \lp \f{A^R} \, \lp L^h \rp^P{}_R + \f{e_H^P} \rp \lp - C_{PBC} - \ha F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E \lp L^h \rp_P{}^Q \rp \\
&=& \f{\nu}{}_{EF} \lp L^h \rp_B{}^E \lp L^h \rp_C{}^F
- \ha \lp \f{A^Q} + \f{e_H^P} \lp L^h \rp_P{}^Q \rp F^H_{DEQ} \lp L^h \rp_B{}^D \lp L^h \rp_C{}^E - \f{e_H^P} C_{PBC} \\
\f{W}{}_{BR} &=& \f{E^A} W_{ABR} = \ha \f{e^D} F^H_{DEQ} \lp L^h \rp_B{}^E \lp L^h \rp_R{}^Q \\
\f{W}{}_{QR} &=& \f{E^P} W_{PQR} = - \ha \lp \f{A^S} \, \lp L^h \rp^P{}_S + \f{e_H^P} \rp C_{PQR}
\end{eqnarray}
This may be used to calculate the [[Cartan tangent bundle curvature]].
''Cartan's equation'' relates the [[spin connection]] to the [[exterior derivative]] of the [[frame]] by asserting that the [[torsion]] is zero,
$$
0 = \f{d} \f{e} + \f{\om} \times \f{e}
$$
or, equivalently,
$$
0 = \f{d} \f{e^\al} + \f{\om}^\al{}_\be \f{e^\be}
$$
This equation may be solved in closed form for the spin or [[tangent bundle connection]] coefficients. To generalize, lets solve
$$
\f{\om} \times \f{e} = -\ff{f}
$$
for $\f{\om}$ in terms of the frame and an arbitrary Clifford vector valued 2-form, $\ff{f}$. In components, using the [[index bracket]], this is
$$
\om_{\lb i \rd}{}^{\al \be} \lp e_{\ld j \rb} \rp_\be = -\ha f_{ij}{}^\al
$$
Using the frame to change from coordinate to Clifford indices, and using antisymmetry of the last two spin connection coefficient indices, this may be expressed simply as
$$
\om_{\lb \be \ga \rb \al} = \ha f_{\be \ga \al}
$$
By once again juggling spin connection indices, we see from this that
$$
\om_{\lp \be \ga \rp \al} = \om_{\lb \al \be \rb \ga} + \om_{\lb \al \ga \rb \be} = f_{\al \lp \be \ga \rp}
$$
Adding these last two expressions gives the explicit solution for the spin connection coefficients:
$$
\om_{\al \be \ga} = \om_{\lb \al \be \rb \ga} + \om_{\lp \al \be \rp \ga} = \ha \lp f_{\al \be \ga} - f_{\be \ga \al} + f_{\ga \al \be} \rp
$$
Putting these indices in their more familiar positions is done using the frame and [[Minkowski metric]]: $\om_{i}{}^{\de \ep} = \lp e_i \rp^\al \et^{\de \be} \et^{\ep \ga} \om_{\al \be \ga}$. Cartan's equation is solved by simply plugging $\ff{f}=\f{d} \f{e}$ into the above equation -- in coefficients,
$$
f_{\al \be \ga} = \lp e_\al \rp^i \lp e_\be \rp^j \et_{\ga \de} \lp \pa_i \lp e_j \rp^\de - \pa_j \lp e_i \rp^\de \rp
= \ve{e_\be} \ve{e_\al} \lp \f{d} \f{e}{}_\ga \rp
$$
Note that this last tensor, the ''anholonomy'', may also be expressed using the [[Lie bracket|Lie derivative]] of the orthonormal basis vectors,
$$
\lb \ve{e_\al} , \ve{e_\be} \rb_L = 2 \lb \lp e_{\lb \al \rd} \rp^j \pa_j \lp e_{\ld \be \rb} \rp^i \rb \ve{\pa_i}
= - 2 \lb \lp e_{\lb \al \rd} \rp^j \lp e_{\ld \be \rb} \rp^k \lp e_\ga \rp^i \pa_j \lp e_k \rp^\ga \rb \ve{\pa_i}
= - f_{\al \be}{}^\ga \ve{e_\ga}
$$
It is possible to express the solution to Cartan's equation in a particularly pretty, index free way using [[Clifform algebra]]:
$$
\f{\om} = - \ve{e} \times \ff{f} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \ff{f} \rp
$$
(A cute, if not particularly useful expression.)
[[Lie algebra]]
http://en.wikipedia.org/wiki/Casimir_invariant
related to [[Laplacian]]
For a $2 \times 2$ square [[matrix|linear operator]] or $2$ or $3$ dimensional [[Clifford algebra]] element, $A$, using the [[trace]] and products gives
$$
0 = A^2 - \li A \ri A + \ha \lp \li A \ri^2 - \li A^2 \ri \rp
$$
For a $3 \times 3$ square matrix, $A$,
$$
0 = A^3 - \li A \ri A^2 + \ha \lp \li A \ri^2 - \li A^2 \ri \rp A - \fr{1}{6} \lp \li A \ri^3 - 3 \li A^2 \ri \li A \ri + 2 \li A^3 \ri \rp
$$
This generalizes to formula for larger matrices,
http://arxiv.org/hep-th/0701116
A ''Chern-Simons form'', $\nf{\om_p}$, is a grade $p$ [[differential form]] defined (for odd $p$) to satisfy
$$
\f{d} \nf{\om_p} = Tr\lp \ff{F}^{\fr{p+1}{2}} \rp
$$
in which $\ff{F}=\f{d} \f{A} + \f{A} \f{A}$ is the [[curvature]] for some [[principal bundle]] [[connection]], $\f{A}$. The first few Chern-Simons forms are
\begin{eqnarray}
\f{\om_1} &=& Tr\lp \f{A} \rp \\
\nf{\om_3} &=& Tr\lp \ff{F} \f{A} - \fr{1}{3} \f{A} \f{A} \f{A} \rp \\
\nf{\om_5} &=& Tr\lp \ff{F} \ff{F} \f{A} - \fr{1}{2} \ff{F} \f{A} \f{A} \f{A} + \fr{1}{10} \f{A} \f{A} \f{A} \f{A} \f{A} \rp \\
\nf{\om_7} &=& Tr\lp \ff{F} \ff{F} \ff{F} \f{A} + ? \ff{F} \ff{F} \f{A} \f{A} \f{A} + ? \ff{F} \f{A} \f{A} \f{A} \f{A} \f{A} + ? \f{A} \f{A} \f{A} \f{A} \f{A} \f{A} \f{A} \rp
\end{eqnarray}
The [[integral|integration]] of a Chern-Simons p-form over a $p$ dimensional [[manifold]] is a homotopy invariant called the ''Chern number'',
$$
c_p = \int \nf{\om_p}
$$
corresponding to the topology of the manifold. For a $(p+1)$ dimensional manifold with a boundary,
$$
\int Tr\lp \ff{F}^{\fr{p+1}{2}} \rp = \int \f{d} \nf{\om_p} = \int_\pa \nf{\om_p} = c_p
$$
Also of potential interest is the relationship to the ''Pfaffian'',
$$
\ff{F}^{\fr{p+1}{2}} = Pf\lp F \rp \nf{d^{p+1}x}
$$
where $Pf(F) = \sqrt{\ll F \rl}$
ref:
http://en.wikipedia.org/wiki/Chern-Simons_form
Using the alternative notation for the [[covariant derivative]] employing the [[tangent bundle connection]] and [[cotangent bundle connection]], the ''covariant derivative'' of a suitably indexed tensor is written as
$$
D_{i}T^{k}{}_{j}=\partial_{i}T^{k}{}_{j}+\Gamma^{k}{}_{im}T^{m}{}_{j}-\Gamma^{m}{}_{ij}T^{k}{}_{m}
$$
with the ''Christoffel symbols'', $\Ga^k{}_{ij}$, defined as tangent bundle connection coefficients,
$$
\na_i \ve{\pa_j} = \Ga^k{}_{ij} \ve{\pa_k}
$$
The Christoffel symbols are determined from the assumptions that the [[torsion]] vanishes,
$$
\Ga^k{}_{\lb ij \rb} = 0
$$
and that the covariant derivative is ''[[metric]] compatible'',
$$
0 = D_i g_{jk} = \pa_i g_{jk} - \Gamma^{m}{}_{ij} g_{mk} - \Gamma^{m}{}_{ik} g_{jm}
$$
It is then computed explicitly in terms of the metric, metric inverse, and its partial derivatives as
$$
\Ga^k{}_{ij} = \ha g^{km} \lp \pa_j g_{mi} + \pa_i g_{jm} - \pa_m g_{ij} \rp
$$
Computing the Christoffel symbols from vanishing torsion and metric compatibility is equivalent to calculating the [[spin connection]] from [[Cartan's equation]].
The Christoffel symbols (with torsion) may alternatively be computed from the [[tangent bundle spin connection|tangent bundle connection]], using the expression for the covariant derivative of the [[orthonormal basis vectors|frame]],
$$
\na_i \ve{e_\al} = \lp \pa_i \lp e_\al \rp^j + \lp e_\al \rp^k \Ga^j{}_{ik} \rp \ve{\pa_j} = w_{i}{}^\be{}_\al \ve{e_\be}
= w_{i}{}^\be{}_\al \lp e_\be \rp^j \ve{\pa_j}
$$
to get
$$
\Ga^j{}_{ik} = \lp e_k \rp^\al \lp w_{i}{}^\be{}_\al \lp e_\be \rp^j - \pa_i \lp e_\al \rp^j \rp = \lp e_\be \rp^j \lp w_{i}{}^\be{}_\al \lp e_k \rp^\al + \pa_i \lp e_k \rp^\be \rp
$$
This last expression may be used to easily determine how the Christoffel symbols, which do not constitute a tensor, transform under a [[coordinate change]] to
\begin{eqnarray}
\Ga^n{}_{ml} &=& \fr{\pa x^k}{\pa y^l} \lp e_k \rp^\al \fr{\pa x^i}{\pa y^m} \lp w_{i}{}^\be{}_\al \lp e_\be \rp^j \fr{\pa y^n}{\pa x^j} - \pa_i \lp \lp e_\al \rp^j \fr{\pa y^n}{\pa x^j} \rp \rp \\
&=& \fr{\pa y^n}{\pa x^j} \fr{\pa x^i}{\pa y^m} \fr{\pa x^k}{\pa y^l} \Ga^j{}_{ik} - \fr{\pa x^i}{\pa y^m} \fr{\pa x^j}{\pa y^l} \fr{\pa^2 y^n}{\pa x^i \pa x^j}
\end{eqnarray}
From the last term we see that it's possible to choose a set of coordinates in which the Christoffel symbols vanish if and only if the torsion vanishes, $\Ga^k{}_{\lb ij \rb} = 0$. It is sometimes argued, along the lines of the [[equivalence principle|frame]], that such a choice should be possible and hence torsion should vanish.
Using the Christoffel symbols is quite old fashioned, but sometimes practical. Things may be fancied up a bit by defining the ''Christoffel 1-form''s, $\f{\Ga^k{}_j} = \f{dx^i} \Ga^k{}_{ij}$, and using the [[vector-form algebra]] and [[partial derivative]] to get $\f{\na} \ve{\pa_j} = \f{\Ga^k{}_j} \ve{\pa_k}$ and
$$
\f{\na} \ve{e_\al} = \f{\na} \lp e_\al \rp^k \ve{\pa_k} = \f{\pa} \ve{e_\al} + \lp e_\al \rp^k \f{\Ga^j{}_k} \ve{\pa_j} = \f{\om^\be{}_\al} \ve{e_\be}
$$
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The $n=6$ dimensional [[Clifford algebra]], ''Cl(0,6)'', is built from 6 negative-norm [[Clifford basis vectors]], $\Ga_\al$.
This algebra has many [[Clifford matrix representation]]s in real or complex $8\times8$ matrices. One particularly nice real representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \, \si_1 \otimes \si_2 \otimes \si_1 \\
\Ga_2 &=& i \, \si_3 \otimes \si_2 \otimes \si_0 \\
\Ga_3 &=& i \, \si_1 \otimes \si_2 \otimes \si_3 \\
\Ga_4 &=& i \, \si_3 \otimes \si_1 \otimes \si_2 \\
\Ga_5 &=& i \, \si_1 \otimes \si_0 \otimes \si_2 \\
\Ga_6 &=& i \, \si_3 \otimes \si_3 \otimes \si_2
\end{eqnarray}
Another nice real representation, adapted to $so(6)=su(4)$, is
\begin{eqnarray}
\Ga_1 &=& i \, \si_3 \otimes \si_2 \otimes \si_1 \\
\Ga_2 &=& i \, \si_0 \otimes \si_2 \otimes \si_3 \\
\Ga_3 &=& i \, \si_2 \otimes \si_0 \otimes \si_1 \\
\Ga_4 &=& i \, \si_2 \otimes \si_3 \otimes \si_3 \\
\Ga_5 &=& i \, \si_1 \otimes \si_2 \otimes \si_1 \\
\Ga_6 &=& i \, \si_2 \otimes \si_1 \otimes \si_3
\end{eqnarray}
A choice of three bivectors in a [[Cartan subalgra|Lie algebra structure]] is (up to signs)
$$
\begin{eqnarray}
\Ga_{12} &=& i \, \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_{34} &=& i \, \si_0 \otimes \si_3 \otimes \si_2 \\
\Ga_{56} &=& i \, \si_3 \otimes \si_3 \otimes \si_2 \\
\end{eqnarray}
$$
Multiplying these together gives (up to a sign)
$$
J = \Ga_{123456} = i \, \si_0 \otimes \si_0 \otimes \si_2
$$
This Clifford algebra element provides the [[complex structure]] when acting on an 8-[[spinor]] of spin(0,6) via the Clifford algebra [[antisymmetric bracket]], and commutes with the three bivectors chosen for the Cartan subalgebra, which allows us to identify Cl(0,6) bivectors with elements of su(4).
A real, chiral representation of $Cl(0,8)$ is
\begin{eqnarray}
\Ga_1 &=& i \si_1 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_2 &=& i \si_1 \otimes \si_1 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& i \si_1 \otimes \si_2 \otimes \si_3 \otimes \si_0 \\
\Ga_4 &=& i \si_1 \otimes \si_2 \otimes \si_1 \otimes \si_0 \\
\Ga_5 &=& i \si_1 \otimes \si_0 \otimes \si_2 \otimes \si_3 \\
\Ga_6 &=& i \si_1 \otimes \si_0 \otimes \si_2 \otimes \si_1 \\
\Ga_7 &=& i \si_1 \otimes \si_2 \otimes \si_2 \otimes \si_2 \\
\Ga_8 &=& i \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0
\end{eqnarray}
There is also a nice [[Clifford division algebra representation]] of $Cl(0,8)$.
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', ''Cl(1,3)'', is built from 4 anti-commuting, [[Clifford basis vectors]], $\ga_\mu$, with positive time [[Minkowski norm|Minkowski metric]],
$$
\ga_\mu \cdot \ga_\nu = \ha \lp \ga_\mu \ga_\nu + \ga_\nu \ga_\mu \rp = \et_{\mu \nu}
$$
The full algebra has $2^4 = 16$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Dimensions |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\ga_\mu$ | $1$ | $4$ |vector |
| $\ga_{\mu \nu}$ | $2$ | $6$ |[[bivector|Cl(1,3) bivector]] |
| $\ga_{\mu \nu \ka } = \fr{1}{3!} \ep_{\mu \nu \ka \la} \ga^\la \ga $ | $3$ | $4$ |trivector |
| $\ga_{\mu \nu \ka \la} = \ep_{\mu \nu \ka \la} \ga$ | 4 | $1$ |4-vector, pseudoscalar |
The spacetime [[pseudoscalar]], $\ga = \ga_0 \ga_1 \ga_2 \ga_3$, satisfies $\ga \ga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\times4$ matrices -- the [[Dirac matrices]].
The 6 bivector [[Clifford basis elements]] of the spacetime Clifford algebra, [[Cl(1,3)]], may be represented by multiplying the 4 basis vectors, in the [[Weyl representation|Dirac matrices]], to get:
\begin{eqnarray}
\ga_{01} &=& \;\;\;\; \si_3 \otimes \si_1 \\
\ga_{02} &=& \;\;\;\; \si_3 \otimes \si_2 \\
\ga_{03} &=& \;\;\;\; \si_3 \otimes \si_3 \\
\ga_{12} &=& -i \, \si_0 \otimes \si_3 \\
\ga_{13} &=& +i \, \si_0 \otimes \si_2 \\
\ga_{23} &=& -i \, \si_0 \otimes \si_1
\end{eqnarray}
Any ''Cl(1,3) bivector'' may thus be represented as
\begin{eqnarray}
B &=& \ha B^{\mu \nu} \ga_{\mu \nu} =
\lb \begin{array}{cc}
B_L & 0 \\
0 & B_R
\end{array} \rb
=
\lb \begin{array}{cc}
B^{0 \va} \si_\va - i \ha B^{\va \ze} \ep_{\va \ze \ta} \si_\ta & 0 \\
0 & - B^{0 \va} \si_\va - i \ha B^{\va \ze} \ep_{\va \ze \ta} \si_\ta
\end{array} \rb \\
&=&
\lb \begin{array}{cccc}
B^{03}- i B^{12} & B^{01}+B^{13}-i B^{02}- i B^{23} & 0 & 0 \\
B^{01}- B^{13}+i B^{02}-i B^{23} & -B^{03}+i B^{12} & 0 & 0 \\
0 & 0 & -B^{03}- i B^{12} & -B^{01}+B^{13}+i B^{02}- i B^{23} \\
0 & 0 & -B^{01}- B^{13}-i B^{02}-i B^{23} & B^{03}+i B^{12}
\end{array} \rb
\end{eqnarray}
in which
$$
B_{L/R} = B_{L/R}^\va \si_\va = \pm B_{\mathbb{R}}^\va \si_\va - i B_{\mathbb{I}}^\va \si_\va
$$
are the ''left and right [[chiral]] bivector parts'', projected out by the [[left/right chirality projector]], and $\ep_{\va \ze \ta}$ is the three dimensional [[permutation symbol]]. These $2\times2$ matrices satisfy $B_L^\da = - B_R$, using the [[Hermitian]] conjugate, so their complex coefficients are related by $B_L^{\va*}=-B_R^\va$. Note that a bivector is completely determined by one of its chiral parts. The bivectors of [[Cl(3,1)]] have the same expression, with signs reversed since the expressions of all vectors pick up $i$'s. These $4 \times 4$ and $2 \times 2$ matrices are [[representation]]s of the [[spin(1,3)]] [[Lorentz algebra]].
The adjoint of a Cl(1,3) bivector is
$$
B^\da = B^{0 \ep} \ga^\da_{0 \ep} + \ha B^{\ep \ze} \ga^\da_{\ep \ze}
= B^{0 \ep} \ga_{0 \ep} - \ha B^{\ep \ze} \ga_{\ep \ze}
= - \ga^0 B \ga_0
$$
A Cl(1,3) bivector can be exponentiated to give a [[Clifford rotation]] operator, $U = e^{\ha B}$. It is not [[unitary]], but
$$
U^- = e^{- \ha B} = \ga_0 U^\da \ga^0
$$
The $n=14$ dimensional [[Clifford algebra]], ''Cl(11,3)'', is built from 11 positive-norm and 3 negative-norm [[Clifford basis vectors]], $\Ga_\al$.
This algebra has many [[Clifford matrix representation]]s in real or complex $8\times8$ matrices. One particularly nice, [[chiral]], real representation, related to the [[spin(11,3) GraviGUT fermions]], built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& \; \; \, i \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_1 \otimes \si_2 \\
\Ga_2 &=& - i \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_2 \otimes \si_0 \\
\Ga_3 &=& \; \; \, i \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_3 \otimes \si_2 \\
\Ga_4 &=& \;\;\;\; \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_0 \otimes \si_2 \\
\Ga_5 &=& \; \;\;\; \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
\Ga_6 &=& \; \;\;\; \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_7 &=& \; \;\;\; \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
\Ga_8 &=& \;\, - \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\Ga_9 &=& \; \;\;\; \, \si_2 \otimes \si_3 \otimes \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_1 \\
\Ga_{10} &=& \;\, - \, \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_3 \\
\Ga_{11} &=& \;\, - \, \si_2 \otimes \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_1 \\
\Ga_{12} &=& \; \;\;\; \, \si_2 \otimes \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_3 \\
\Ga_{13} &=& \; \;\;\; \, \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_1 \\
\Ga_{14} &=& \;\, - \, \si_2 \otimes \si_2 \otimes \si_1 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_3 \\
\end{eqnarray}
A nice choice of 7 bivectors in a [[Cartan subalgebra|Lie algebra structure]] of ''spin(11,3)'' is
Multiplying these together gives the psuedoscalar,
The $n=16$ dimensional [[Clifford algebra]], $Cl(16,0)$, is built from 16 anti-commuting, positive norm [[Clifford basis vectors]], $\ga^{\lp16\rp}_\al$. The full algebra has $2^{16} = 65,536$ [[Clifford basis elements]]. This algebra has many [[Clifford matrix representation]]s in real or complex $256\times256$ matrices. One particularly nice representation, built using the [[Kronecker product]] of [[Cl(8)]] elements, is
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga \otimes \Ga_\al
\end{eqnarray}
with $1 \le \al \le 8$. (Note that this rep is not [[chiral]].) These $16$ ''Cl(16) basis vectors'' may be multiplied to get the $120$ ''Cl(16) basis bivectors'',
\begin{eqnarray}
\ga^{\lp16\rp}_{\al \be} &=& \Ga_{\al \be} \otimes 1 \\
\ga^{\lp16\rp}_{\lp\al+8\rp \lp\be+8\rp} &=& 1 \otimes \Ga_{\al \be} \\
\ga^{\lp16\rp}_{\al \lp\be+8\rp} = -\ga^{\lp16\rp}_{\lp\be+8\rp \al} &=& \lp \Ga_\al \Ga \rp \otimes \Ga_{\be}
\end{eqnarray}
The [[pseudoscalar]] in this rep, $\ga^{\lp16\rp} = \Ga \otimes \Ga$, satisfies $\ga^{\lp16\rp} \ga^{\lp16\rp} = 1$ and anti-commutes with odd-graded elements.
A chiral representation for $Cl(16)$ may be built by starting with a chiral [[Cl(8)]] rep and picking out one of the vectors, such as $\Ga_8$, and using it to build the ''chiral Cl(16) basis vectors'':
\begin{eqnarray}
\ga^{\lp16\rp}_\al &=& \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_8 &=& \Ga_8 \otimes \Ga \\
\ga^{\lp16\rp}_{\lp\al+8\rp} &=& \Ga_8 \otimes \Ga_\al \\
\ga^{\lp16\rp}_{(16)} &=& \Ga_8 \otimes \Ga_8
\end{eqnarray}
with $1 \le \al \le 7$. The pseudoscalar in this rep is
$$
\ga^{\lp16\rp} = ( \Ga \otimes \Ga ) ( 1 \otimes \Ga ) = \Ga \otimes 1
$$
The (1st level) chirality projector is $P_{\pm} = \ha \lp 1 \mp \ga^{\lp16\rp} \rp = \ha \lp 1 \mp \Ga \rp \otimes 1$. The basis vectors may be used to build the ''chiral Cl(16) basis bivectors'',
\begin{eqnarray}
\ga^{\lp16\rp}_{\al \be} &=& \Ga_{\al \be} \otimes 1 \\
\ga^{\lp16\rp}_{\al 8} &=& \Ga_{\al 8} \otimes \Ga \\
\ga^{\lp16\rp}_{\al \lp\be+8\rp} &=& \Ga_{\al 8} \otimes \Ga_\be \\
\ga^{\lp16\rp}_{\al (16)} &=& \Ga_{\al 8} \otimes \Ga_8 \\
\ga^{\lp16\rp}_{\lp\al+8\rp \lp\be+8\rp} &=& 1 \otimes \Ga_{\al \be} \\
\ga^{\lp16\rp}_{\lp\al+8\rp (16)} &=& 1 \otimes \Ga_{\al 8} \\
\ga^{\lp16\rp}_{8 \lp\be+8\rp} &=& 1 \otimes \Ga \Ga_\be \\
\ga^{\lp16\rp}_{8 (16)} &=& 1 \otimes \Ga \Ga_8
\end{eqnarray}
A complex, [[chiral]] representation for the [[Clifford algebra]] with 2 positive and 4 negative norm basis vectors, ''Cl(2,4)'', may be built by starting with the chiral representation of the [[Cl(1,3)]] [[Dirac matrices]] and using [[Clifford representation doubling]],
\begin{eqnarray}
\ga'_\mu &=& \;\;\, \, \si_1 \otimes \ga_\mu \\
\ga'_{5} &=& - \, \si_1 \otimes \ga \\
\ga'_{6} &=& \;\;\, \, \si_2 \otimes 1 \\
\end{eqnarray}
(We labeled the added vectors "$5$" and "$6$" because "$4$" is sometimes used for the time vector instead of "$0$".) So we have
\begin{eqnarray}
\ga'_4 = \ga'_0 &=& \;\;\; \; \, \si_1 \otimes \si_1 \otimes \si_0 \\
\ga'_\va &=& - i \, \si_1 \otimes \si_2 \otimes \si_\va \\
\ga'_5 &=& \;\;\, i \, \si_1 \otimes \si_3 \otimes \si_0 \\
\ga'_6 &=& \;\;\;\; \,\si_2 \otimes \si_0 \otimes \si_0 \\
\end{eqnarray}
A $Cl(2,4)$ vector may thus be written as
\begin{eqnarray}
v' &=& v^\al \ga'_\al = \lb \begin{array}{cc} 0 & v'_- \\ v'_+ & 0 \end{array} \rb
= \lb \begin{array}{cc}
0 & v^\mu \ga_\mu - v^5 \ga - i \, v^6 \\
v^\mu \ga_\mu - v^5 \ga + i \, v^6 & 0 \\
\end{array} \rb \\
& =&
\lb \begin{array}{cccc}
0 & 0 & i \, v_- & v_L \\
0 & 0 & v_R & - i \, v_+ \\
i \, v_+ & v_L & 0 & 0 \\
v_R & - i \, v_- & 0 & 0 \\
\end{array} \rb
\end{eqnarray}
with ''chiral Cl(2,4) vector''s defined as
$$
v'_\pm = v^\mu \ga_\mu - v^5 \ga \pm i \, v^6 = v + i \, v_\pm P_L - i \, v_\mp P_R
$$
using the [[left/right chirality projector]]s, with $v_\pm = v^5 \pm v^6$ defined for fun (and, as scalars, not likely to be confused with the chiral vectors). The $Cl(2,4)$ [[pseudoscalar]] is
$$
\ga' = \ga'_0 \ga'_1 \ga'_2 \ga'_3 \ga'_5 \ga'_6 = i \, \si_3 \otimes \si_0 \otimes \si_0
$$
The Clifford vectors satisfy $\ga'_\al{}^* = \ga'_2 \ga'_\al \ga'_2$ and $\ga'^\da_\al = \ga'_{06} \ga'_\al \ga'_{06}$, with [[spinor metric|conjugate spinor]]
$$
\ga'_{06} = i \, \si_3 \otimes \si_1 \otimes \si_0 = i \, \si_3 \otimes \ga_0
$$
The spinor metric provides a [[Hermitian form]] on the eight complex dimensional fundamental [[representation space]] of $Cl(2,4)$, with signature $(4,4)$.
Alternatively, there is a [[Clifford compound division algebra representation of Cl(2,4)]].
The three dimensional [[Clifford algebra]], Cl(3,0), is generated by three [[Clifford basis vectors]], $\si_\ep$. These basis vectors have a [[matrix representation|Clifford matrix representation]] as the three [[Pauli matrices]], $\si_\ep$. The eight [[Clifford basis elements]] are formed by all possible products of these Clifford basis vectors. The complete multiplication table for the algebra, calculated from the general [[Clifford basis identities]], is (row header times column header equals entry):
| | !$1$ | !$\si_1$ | !$\si_2$ | !$\si_3$ | !$\si_{12}$ | !$\si_{13}$ | !$\si_{23}$ | !$\si$ |
| !$1$ | $1$ | $\si_1$ | $\si_2$ | $\si_3$ | $\si_{12}$ | $\si_{13}$ | $\si_{23}$ | $\si$ |
| !$\si_1$ | $\si_1$ | $1$ |bgcolor(#a0ffa0): $\si_{12}$ |bgcolor(#a0ffa0): $\si_{13}$ | $\si_2$ | $\si_3$ | $\si$ |bgcolor(#a0ffa0): $\si_{23}$ |
| !$\si_2$ | $\si_2$ |bgcolor(#a0ffa0): $-\si_{12}$ | $1$ |bgcolor(#a0ffa0): $\si_{23}$ | $-\si_1$ | $-\si$ | $\si_3$ |bgcolor(#a0ffa0): $-\si_{13}$ |
| !$\si_3$ | $\si_3$ |bgcolor(#a0ffa0): $-\si_{13}$ |bgcolor(#a0ffa0): $-\si_{23}$ | $1$ | $\si$ | $-\si_1$ | $-\si_2$ |bgcolor(#a0ffa0): $\si_{12}$ |
| !$\si_{12}$ | $\si_{12}$ | $-\si_2$ | $\si_1$ | $\si$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\si_{23}$ |bgcolor(#88ccff): $\si_{13}$ | $-\si_3$ |
| !$\si_{13}$ | $\si_{13}$ | $-\si_3$ | $-\si$ | $\si_1$ |bgcolor(#88ccff): $\si_{23}$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\si_{12}$ | $\si_2$ |
| !$\si_{23}$ | $\si_{23}$ | $\si$ | $-\si_3$ | $\si_2$ |bgcolor(#88ccff): $-\si_{13}$ |bgcolor(#88ccff): $\si_{12}$ |bgcolor(#88ccff): $-1$ | $-\si_1$ |
| !$\si$ | $\si$ |bgcolor(#a0ffa0): $\si_{23}$ |bgcolor(#a0ffa0): $-\si_{13}$ |bgcolor(#a0ffa0): $\si_{12}$ | $-\si_3$ | $\si_2$ | $-\si_1$ | $-1$ |
The blue square shows the bivector subalgebra. This bivector subalgebra is the [[three dimensional special unitary group Lie algebra|su(2)]], with an identification giving the three [[su(2)]] generators,
$$
\begin{array}{ccc}
T_1 = i \sigma_{1} = \si_{23}
&
T_2 = - i \sigma_{2} = \si_{13}
&
T_3 = i \sigma_{3} = \si_{12}
\end{array}
$$
which form a closed subalgebra under the [[commutator]]. The green entries illustrate the two ways the bivector basis elements can be represented -- as the product of vectors, or as the product of vector and pseudoscalar. The [[pseudoscalar]], $\si=\si_1 \si_2 \si_3$, squares to $-1$ and has the matrix representation $\si = i$.
The ''three dimensional Clifford algebra of negative signature'', $Cl(0,3)$, is obtained by using $\si'_\va = i \si_\va$ as the basis vectors.
The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(3,1), is built from 4 anti-commuting, [[Clifford basis vectors]], $\ga_\mu$, with negative time [[Minkowski norm|Minkowski metric]],
$$
\ga_\mu \cdot \ga_\nu = \ha \lp \ga_\mu \ga_\nu + \ga_\nu \ga_\mu \rp = \et_{\mu \nu}
$$
The full algebra has $2^4 = 16$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Dimensions |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\ga_\mu$ | $1$ | $4$ |vector |
| $\ga_{\mu \nu}$ | $2$ | $6$ |bivector |
| $\ga_{\mu \nu \ka } = \fr{1}{3!} \ep_{\mu \nu \ka \la} \ga^\la \ga $ | $3$ | $4$ |trivector |
| $\ga_{\mu \nu \ka \la} = \ep_{\mu \nu \ka \la} \ga$ | 4 | $1$ |4-vector, pseudoscalar |
The ''spacetime [[pseudoscalar]]'', $\ga = \ga_0 \ga_1 \ga_2 \ga_3$, satisfies $\ga \ga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\times4$ matrices -- the [[Dirac matrices]].
The $n=6$ dimensional [[Clifford algebra]], ''Cl(3,3)'', is built from 3 positive norm and 3 negative norm [[Clifford basis vectors]], $\Ga_\al$.
This algebra has many [[Clifford matrix representation]]s in real or complex $8\times8$ matrices. One particularly nice, [[chiral]], real representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \, \si_2 \otimes \si_1 \otimes \si_0 \\
\Ga_2 &=& i \, \si_2 \otimes \si_2 \otimes \si_2 \\
\Ga_3 &=& i \, \si_2 \otimes \si_3 \otimes \si_0 \\
\Ga_4 &=& \p{i \,} \si_1 \otimes \si_0 \otimes \si_0 \\
\Ga_5 &=& \p{i \,} \si_2 \otimes \si_2 \otimes \si_3 \\
\Ga_6 &=& \p{i \,} \si_2 \otimes \si_2 \otimes \si_1
\end{eqnarray}
The first four of these come from realifying the Weyl representation of the [[Dirac matrices]].
Similar to [[Cl(1,3)]], the $n=4$ dimensional ''Euclidean spacetime [[Clifford algebra]]'', ''Cl(4)'', is built from 4 anti-commuting, [[Clifford basis vectors]], $\ga_\mu$, with [[metric]],
$$
\ga_\mu \cdot \ga_\nu = \ha \lp \ga_\mu \ga_\nu + \ga_\nu \ga_\mu \rp = \de_{\mu \nu}
$$
The ([[chiral]]) ''Weylish representation'' of the [[Dirac matrices]] of Cl(4), using [[Pauli matrices]], is:
\begin{eqnarray}
\ga_0 &=& \;\;\, \si_1 \otimes \si_0 \\
\ga_1 &=& - \si_2 \otimes \si_1 \\
\ga_2 &=& - \si_2 \otimes \si_2 \\
\ga_3 &=& - \si_2 \otimes \si_3
\end{eqnarray}
giving a complex rep for ''Cl(4) vectors'',
\begin{eqnarray}
v &=& v^\mu \ga_\mu =
\lb \begin{array}{cc}
0 & v_L \\
v_R & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & v^0 \si_0 + v^\va i \si_\va \\
v^0 \si_0 - v^\va i \si_\va & 0
\end{array} \rb
\\
&=&
\lb \begin{array}{cccc}
0 & 0 & v^0+i v^3 & i v^1+ v^2 \\
0 & 0 & i v^1 - v^2 & v^0 - i v^3 \\
v^0-i v^3 & -i v^1- v^2 & 0 & 0 \\
-i v^1+v^2 & v^0+i v^3 & 0 & 0
\end{array} \rb
\end{eqnarray}
This is related to the Weyl representation by $\ga_\va \to -i \ga_\va$. The resulting ''Euclidean spacetime [[pseudoscalar]]'' is $\ga = \si_3 \otimes 1$, satisfying $\ga \ga = 1$ and anti-commuting with odd-graded elements.
The $v_{L/R}$ are ''left and right chiral Euclidean vector'' parts -- $2\times2$ complex matrices projected out by the [[left/right chirality projector]]. Note that a vector is completely determined by one of its chiral parts, $v_R = \bar{v}_L = v_L^\da$. Also note the identification between a right chiral vector and a [[quaternion]], $v_R = v^0 \si_0 - v^\va i \si_\va = v^a e_a$.
A [[spinor]] of ''spin(4)'' (acted on by Cl(4) bivectors), has left and right [[chiral]] ''Weylish spinor'' parts, $\ps_{L/R}$.
The $n=8$ dimensional [[Clifford algebra]], ''Cl(4,4)'', is built from 4 positive norm and 4 negative norm [[Clifford basis vectors]], $\Ga'_\al$. It is the same as [[Cl(8)]] except for the signature.
It has nice, real, [[chiral]] representation, from [[realify]]ing the representation of [[Cl(2,4)]] and adding two vectors to match the realified [[complex structure]],
$$
\begin{array}{rcrlcrl}
e'_5 = \Ga'_1 \!\!&\!\!=\!\!&\!\! -i \!\!&\!\! \si_1 \otimes \si_2 \otimes \si_1 \otimes \si_0 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \ga_1 \\
e'_6 = \Ga'_2 \!\!&\!\!=\!\!&\!\! i \!\!&\!\! \si_1 \otimes \si_2 \otimes \si_2 \otimes \si_2 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \ga_2 \\
e'_7 = \Ga'_3 \!\!&\!\!=\!\!&\!\! -i \!\!&\!\! \si_1 \otimes \si_2 \otimes \si_3 \otimes \si_0 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \ga_3 \\
e'_0 = \Ga'_0 = \Ga'_4 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \si_1 \otimes \si_0 \otimes \si_0 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \ga_0 \\
e'_4= \Ga'_5 \!\!&\!\!=\!\!&\!\! -i \!\!&\!\! \si_1 \otimes \si_3 \otimes \si_0 \otimes \si_2 \!\!&\!\!=\!\!&\!\! - \!\!&\!\! \si_1 \otimes \ga \\
e'_2 = \Ga'_6 \!\!&\!\!=\!\!&\!\! - \!\!&\!\! \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \ep \otimes J \\
e'_1 = \Ga'_7 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_1 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \ep \otimes \ga \ga_2 \, JK \\
e'_3 = \Ga'_8 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_3 \!\!&\!\!=\!\!&\!\! \!\!&\!\! \ep \otimes \ga \ga_2 \, K
\end{array}
$$
with $\ep = -i \si_2$ the [[skew]] matrix, giving $\Ga' = \Ga'_1 \Ga'_2 \Ga'_3 \Ga'_4 \Ga'_5 \Ga'_6 \Ga'_7 \Ga'_8 = \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0$, as well as corresponding to the [[split-octonion]]s. The additional basis vectors, $\Ga'_7$ and $\Ga'_8$, combine to give the [[complex structure]], $\Ga'_{78} = \si_0 \otimes \si_0 \otimes \si_0 \otimes J$, with $J = - i \si_2$ a [[sl(2)]] generator representative, and $K$ the complex conjugation operator. Note that $\Ga'_2$ and $\Ga'_5$ also have $J$'s in them, replacing the $i$'s. There is a [[similarity transformation for Cl(4,4)]] relating this representation to the [[split-octonionic representation of Cl(4,4)]].
The [[chiral]] parts of a $Cl(4,4)$ vector are a realified $Cl(2,4)$ vector,
$$
v'_\pm = v^\mu \ga_\mu - v^5 \ga \pm v^6 J \pm \ga \ga_2 (v^7 J + v^8) K = v + v_\pm P_L J - v_\mp P_R J \pm \ga \ga_2 v_c K
$$
using the [[left/right chirality projector]]s, with, $v_\pm = v^5 \pm v^6$, plus a new part, $v_c = v^7 J + v^8$. The $Cl(4,4)$ bivectors generate [[spin(4,4)]].
The $n=8$ dimensional [[Clifford algebra]], ''Cl(5,3)'', is built from 5 positive norm and 3 negative norm [[Clifford basis vectors]], $\Ga_\al$. It is the same as [[Cl(8)]] except for the signature.
This algebra has many [[Clifford matrix representation]]s in real or complex $16\times16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& i \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_1 \\
\Ga_2 &=& i \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& i \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_3 \\
\Ga_0 = \Ga_4 &=& \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
\Ga_5 &=& \si_2 \otimes \si_1 \otimes \si_1 \otimes \si_0 \\
\Ga_6 &=& \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_0 \\
\Ga_7 &=& \si_2 \otimes \si_1 \otimes \si_3 \otimes \si_0 \\
\Ga_8 &=& \si_2 \otimes \si_2 \otimes \si_0 \otimes \si_0
\end{eqnarray}
giving $\Ga = - i \, \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0$.
Standard model using this Cl(5,3) rep:
Has correct Higgs.
Ah, no good -- would give negative [[cosmological constant]].
$$
{\scriptsize
\lb \begin{array}{cccccccc}
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_L} \ph_0^* & \fr{1}{4} \f{e_L} \ph_+ &
\ud{\nu_L} & \ud{u_L^r} & \ud{u_L^b} & \ud{u_L^g} \\
i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \fr{1}{4} \f{e_L} \ph_+^* & \fr{1}{4} \f{e_L} \ph_0 &
\ud{e_L} & \ud{d_L^r} & \ud{d_L^b} & \ud{d_L^g} \\
\fr{1}{4} \f{e_R} \ph_0 & -\fr{1}{4} \f{e_R} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & &
\ud{\nu_R} & \ud{u_R^r} & \ud{u_R^b} & \ud{u_R^g} \\
-\fr{1}{4} \f{e_R} \ph_+^* & -\fr{1}{4} \f{e_R} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B} &
\ud{e_R} & \ud{d_R^r} & \ud{d_R^b} & \ud{d_R^g} \\
& & & & i \f{B} & & & \\
& & & & & \fr{-i}{3} \f{B} \!+\! i \f{G^{3+8}} & i\f{G^1} \!-\! \f{G^2} & i\f{G^4} \!-\! \f{G^5} \\
& & & & & i\f{G^1} \!+\! \f{G^2} & \fr{-i}{3} \f{B} \!-\! i \f{G^{3+8}} & i\f{G^6} \!-\! \f{G^7} \\
& & & & & i\f{G^4} \!+\! \f{G^5} & i\f{G^6} \!+\! \f{G^7} & \fr{-i}{3} \f{B} \!-\!\! \fr{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
The $n=8$ dimensional [[Clifford algebra]], ''Cl(7,1)'', is built from 7 positive-norm and 1 negative-norm [[Clifford basis vectors]], $\Ga_\al$.
This algebra has many [[Clifford matrix representation]]s in real or complex $16 \times 16$ matrices. One particularly nice, [[chiral]], complex representation, related to the [[chiral representation|Dirac matrices]] of [[Cl(3,1)]], built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& \; \, \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_1 \\
\Ga_2 &=& \; \, \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& \; \, \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_3 \\
\Ga_4 &=& i \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
\Ga_5 &=& \; \, \si_2 \otimes \si_1 \otimes \si_1 \otimes \si_0 \\
\Ga_6 &=& \; \, \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_0 \\
\Ga_7 &=& \; \, \si_2 \otimes \si_1 \otimes \si_3 \otimes \si_0 \\
\Ga_8 &=& \; \, \si_2 \otimes \si_2 \otimes \si_0 \otimes \si_0 \\
\end{eqnarray}
A nice choice of 4 bivectors in a [[Cartan subalgra|Lie algebra structure]] is
$$
\begin{eqnarray}
\Ga_{12} &=& i \, \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \\
\Ga_{34} &=& \;\, \si_3 \otimes \si_3 \otimes \si_0 \otimes \si_3 \\
\Ga_{56} &=& i \, \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \\
\Ga_{78} &=& i \, \si_0 \otimes \si_3 \otimes \si_3 \otimes \si_0 \\
\end{eqnarray}
$$
Multiplying these together gives the psuedoscalar,
$$
\Ga = \Ga_{12345678} = - i \, \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0
$$
A real, chiral representation for $Cl(7,7)$ may be built by starting with a real, chiral [[Cl(8)]] rep and combining it with a [[Cl(0,6)]] rep, to build the ''real, chiral Cl(7,7) basis vectors'':
\begin{eqnarray}
\Ga_1 &=& \;\;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_0 \\
\Ga_2 &=& \;\;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_2 \otimes \si_0 \\
\Ga_3 &=& \;\;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_0 \\
&&\\
\Ga_4 &=& \;\;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_0 \otimes \si_0 \\
&&\\
\Ga_5 &=& \; \;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_1 \otimes \si_0 \otimes \si_2 \otimes \si_0 \\
\Ga_6 &=& \; \;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_0 \\
\Ga_7 &=& \; \;\;\; \, \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \otimes \si_2 \otimes \si_0 \\
&&\\
\Ga_8 &=& \; \;\, i \, \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
&&\\
\Ga_9 &=& \; \;\, i \, \si_1 \otimes \si_3 \otimes \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_1 \\
\Ga_{10} &=& -i \, \si_1 \otimes \si_0 \otimes \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \\
\Ga_{11} &=& -i \, \si_1 \otimes \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_1 \\
\Ga_{12} &=& \;\;\, i \, \si_1 \otimes \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \\
\Ga_{13} &=& \;\;\, i \, \si_1 \otimes \si_1 \otimes \si_2 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_1 \\
\Ga_{14} &=& -i \, \si_1 \otimes \si_2 \otimes \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \\
\end{eqnarray}
The basis elements of the usual [[Cartan subalgebra|Lie algebra structure]] of ''spin(7,7)'' are
\begin{eqnarray}
\Ga_{1,2} &=& \;\;\, i \, \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_0 \otimes \si_3 \otimes \si_0 \\
\Ga_{3,4} &=& -i \, \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_3 \otimes \si_0 \\
\Ga_{5,6} &=& \;\;\, i \, \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_3 \otimes \si_2 \otimes \si_0 \\
\Ga_{7,8} &=& \;\;\;\; \, \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \otimes \si_2 \otimes \si_2 \\
\Ga_{9,10} &=& -i \, \si_0 \otimes \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\Ga_{11,12} &=& -i \, \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\Ga_{13,14} &=& -i \, \si_0 \otimes \si_3 \otimes \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\end{eqnarray}
or this can be mixed up a bit to be better adapted to spacetime and the weak interaction, with
\begin{eqnarray}
\Ga_{3,8} &=& \;\;\;\; \, \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_2 \\
\Ga_{4,7} &=& \;\;\,i \, \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_1 \otimes \si_1 \otimes \si_2 \otimes \si_0 \\
\end{eqnarray}
Note that the maximal Cartan subalgebras are all noncompact. The pseudoscalar is
$$
\Ga = - \, \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
$$
The $n=8$ dimensional [[Clifford algebra]], $Cl(8,0)$, is built from 8 anti-commuting, positive norm [[Clifford basis vectors]], $\Ga_\al$. The full algebra has $2^8 = 256$ [[Clifford basis elements]],
| !Element(s) | !Grade | !Dimensions |!Names |
| $1$ | $0$ | $1$ |scalar |
| $\Ga_\al$ | $1$ | $8$ |vector |
| $\Ga_{\al \be}$ | $2$ | $28$ |bivector |
| $\Ga_{\al \be \ga}$ | $3$ | $56$ |trivector |
| $\Ga_{\al \be \ga \de}$ | $4$ | $70$ |4-vector |
| $\Ga_{\al \be \ga \de \ep}$ | $5$ | $56$ |5-vector |
| $\Ga_{\al \dots \be}$ | $6$ | $28$ |6-vector |
| $\Ga_{\al \dots \be}$ | $7$ | $8$ |7-vector |
| $\Ga_{\al \dots \be} = \ep_{\al \dots \be} \Ga$ | $8$ | $1$ |8-vector, psuedoscalar |
The [[pseudoscalar]], $\Ga$, satisfies $\Ga \Ga = 1$ and anti-commutes with odd-graded elements. This algebra has many [[Clifford matrix representation]]s in real or complex $16\times16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is
\begin{eqnarray}
\Ga_1 &=& \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_1 \\
\Ga_2 &=& \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_3 \\
\Ga_4 &=& \si_2 \otimes \si_2 \otimes \si_0 \otimes \si_0 \\
\Ga_5 &=& \si_2 \otimes \si_1 \otimes \si_1 \otimes \si_0 \\
\Ga_6 &=& \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_0 \\
\Ga_7 &=& \si_2 \otimes \si_1 \otimes \si_3 \otimes \si_0 \\
\Ga_0 = \Ga_8 &=& \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0
\end{eqnarray}
giving $\Ga = \si_3 \otimes \si_0 \otimes \si_0 \otimes \si_0$. These may all be expressed in a $16\times16$ matrix (using $2\times2$ sub-matrices) as
\begin{eqnarray}
& & \Ga_\pi + z \Ga_4 + a \Ga_5 + b \Ga_6 + c \Ga_7 + \Ga_8 = \\
& & \lb
\begin{array}{cccccccc}
& & & & 1-i\si^p_\pi & & -z-ic & -b-ia \\
& & & & & 1-i\si^p_\pi & b-ia & -z+ic \\
& & & & z-ic & -b-ia & 1+i\si^p_\pi & \\
& & & & b-ia & z+ic & & 1+i\si^p_\pi \\
1+i\si^p_\pi & & z+ic & b+ia & & & & \\
& 1+i\si^p_\pi & -b+ia & z-ic & & & & \\
-z+ic & b+ia & 1-i\si^p_\pi & & & & & \\
-b+ia & -z-ic & & 1-i\si^p_\pi & & & &
\end{array}
\rb
\end{eqnarray}
A nice chiral, real representation of $Cl(8,0)$ is
\begin{eqnarray}
\Ga_1 &=& \si_2 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_2 &=& \si_2 \otimes \si_1 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& \si_2 \otimes \si_2 \otimes \si_3 \otimes \si_0 \\
\Ga_4 &=& \si_2 \otimes \si_2 \otimes \si_1 \otimes \si_0 \\
\Ga_5 &=& \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_3 \\
\Ga_6 &=& \si_2 \otimes \si_0 \otimes \si_2 \otimes \si_1 \\
\Ga_7 &=& \si_2 \otimes \si_2 \otimes \si_2 \otimes \si_2 \\
\Ga_8 &=& \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0
\end{eqnarray}
And another one, matched to [[su(3)]], is
\begin{eqnarray}
\Ga_1 &=& \;\;\; \si_1 \otimes \si_0 \otimes \si_0 \otimes \si_0 \\
\Ga_2 &=& -\si_2 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\Ga_3 &=& \;\;\; \si_2 \otimes \si_3 \otimes \si_2 \otimes \si_1 \\
\Ga_4 &=& -\si_2 \otimes \si_0 \otimes \si_2 \otimes \si_3 \\
\Ga_5 &=& -\si_2 \otimes \si_2 \otimes \si_0 \otimes \si_1 \\
\Ga_6 &=& \;\;\; \si_2 \otimes \si_2 \otimes \si_3 \otimes \si_3 \\
\Ga_7 &=& \;\;\; \si_2 \otimes \si_1 \otimes \si_2 \otimes \si_1 \\
\Ga_8 &=& -\si_2 \otimes \si_2 \otimes \si_1 \otimes \si_3
\end{eqnarray}
There's also an [[octonionic representation of Cl(8)]].
The ''chirality operator for Cl(8)'' is $P^{\lp8\rp}_\pm = \ha \lp 1 \pm \Ga \rp$.
The 28 bivector [[Clifford basis elements]] of [[Cl(8,0)|Cl(8)]] may be represented by multiplying the 8 basis vectors, in the complex rep, to get:
\begin{eqnarray}
\Ga_{01} &=& i \si_3 \otimes \si_3 \otimes \si_0 \otimes \si_1 \\
\Ga_{02} &=& i \si_3 \otimes \si_3 \otimes \si_0 \otimes \si_2 \\
\Ga_{03} &=& i \si_3 \otimes \si_3 \otimes \si_0 \otimes \si_3 \\
\Ga_{12} &=& i \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_3 \\
\Ga_{13} &=& -i \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_2 \\
\Ga_{23} &=& i \si_0 \otimes \si_0 \otimes \si_0 \otimes \si_1 \\
&\,& \\
\Ga_{04} &=& i \si_3 \otimes \si_2 \otimes \si_0 \otimes \si_0 \\
\Ga_{05} &=& i \si_3 \otimes \si_1 \otimes \si_1 \otimes \si_1 \\
\Ga_{06} &=& i \si_3 \otimes \si_1 \otimes \si_2 \otimes \si_0 \\
\Ga_{07} &=& i \si_3 \otimes \si_1 \otimes \si_3 \otimes \si_0 \\
\Ga_{14} &=& -i \si_0 \otimes \si_1 \otimes \si_0 \otimes \si_1 \\
\Ga_{15} &=& i \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_1 \\
\Ga_{16} &=& i \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_1 \\
\Ga_{17} &=& i \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_1 \\
\Ga_{24} &=& -i \si_0 \otimes \si_1 \otimes \si_1 \otimes \si_2 \\
\Ga_{25} &=& i \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_2 \\
\Ga_{26} &=& i \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_2 \\
\Ga_{27} &=& i \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_2 \\
\Ga_{34} &=& -i \si_0 \otimes \si_1 \otimes \si_0 \otimes \si_3 \\
\Ga_{35} &=& i \si_0 \otimes \si_2 \otimes \si_1 \otimes \si_3 \\
\Ga_{36} &=& i \si_0 \otimes \si_2 \otimes \si_2 \otimes \si_3 \\
\Ga_{37} &=& i \si_0 \otimes \si_2 \otimes \si_3 \otimes \si_3 \\
&\,& \\
\Ga_{45} &=& -i \si_0 \otimes \si_3 \otimes \si_1 \otimes \si_0 \\
\Ga_{46} &=& -i \si_0 \otimes \si_3 \otimes \si_2 \otimes \si_0 \\
\Ga_{47} &=& -i \si_0 \otimes \si_3 \otimes \si_3 \otimes \si_0 \\
\Ga_{56} &=& i \si_0 \otimes \si_0 \otimes \si_3 \otimes \si_0 \\
\Ga_{57} &=& -i \si_0 \otimes \si_0 \otimes \si_2 \otimes \si_0 \\
\Ga_{67} &=& i \si_0 \otimes \si_0 \otimes \si_1 \otimes \si_0
\end{eqnarray}
A real, chiral representation for $Cl(8,8)$ may be built by starting with a real, chiral [[Cl(0,8)]] rep and picking out one of the vectors, such as $\Ga_8$, and using it to build the ''real, chiral Cl(8,8) basis vectors'':
$$
\begin{array}{rcl}
\ga^{\lp16\rp}_\al \!\!&\!\!=\!\!&\!\! \Ga_\al \otimes 1 \\
\ga^{\lp16\rp}_8 \!\!&\!\!=\!\!&\!\! \Ga_8 \otimes \Ga \\
\ga^{\lp16\rp}_{\al+8} \!\!&\!\!=\!\!&\!\! \Ga_8 \otimes \Ga_\al \\
\ga^{\lp16\rp}_{16} \!\!&\!\!=\!\!&\!\! \Ga_8 \otimes \Ga_8
\end{array}
$$
with $1 \le \al \le 7$. Or we can start with $\Ga_\al$ in [[Cl(7,7)]] and add
$$
\begin{array}{rcrl}
\ga_\al \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \Ga_\al \\
\ga_{15} \!\!&\!\!=\!\!&\!\! \!\!&\!\! \si_1 \otimes \Ga \\
\ga_{16} \!\!&\!\!=\!\!&\!\! -i \!\!&\!\! \si_2 \otimes 1 \\
\end{array}
$$
The ''Clifford adjoint'' transformation of a [[Clifford element]], $A$, by a [[Clifford group]] element, $U$, is
\[ A' = U A \, U^- \]
The ''Clifford inner [[automorphism]]'', a.k.a. //''similarity transformation''//, is the Clifford adjoint transformation of the [[Clifford basis vectors]],
\[ \ga'_\al = U \ga_\al U^- \]
This subsequently produces the Clifford adjoint transformation of all Clifford elements built from these basis vectors, since
$$
\ga'_{\al \dots \be} = \ga'_\al \dots \ga'_\be = U \ga_\al U^- \dots U \ga_\be U^- = U \ga_{\al \dots \be} U^-
$$
It is an automorphism because it is a map, specified by $U \in Cl^*$, from the [[Clifford algebra]] into itself.
The adjoint transformation does not necessarily preserve the [[grade|Clifford grade]] of elements. It does, however, preserve scalars, $\li A' B' \ri = \li UA \, U^- UB \, U^- \ri = \li A B \, \ri$. The [[fundamental Clifford identity|Clifford basis vectors]], $\ga_\al \cdot \ga_\be = \et_{\al \be}$, is preserved by the Clifford automorphism, $\ga'_\al \cdot \ga'_\be = \et_{\al \be}$, preserving the structure of the Clifford algebra and the decomposition of [[Clifford element]]s even though the transformed basis "vectors", $\ga'_\al$, may no longer be grade 1 with respect to the old basis.
For Clifford group elements near the identity, $U \simeq 1 + \ha C$, the Clifford adjoint is approximately
\[ A' = U A \, U^- \simeq \lp 1 + \ha C \rp A \lp 1 - \ha C \rp \simeq A + C \times A \]
with the "small" Clifford element, $C$, acting via the [[cross product|antisymmetric bracket]].
An "$n$ dimensional" ''Clifford algebra'', $Cl(p,q)$, is a $2^n$ dimensional algebra of [[Clifford element]]s consisting of coefficients multiplying [[Clifford basis elements]] constructed from $p$ positive norm and $q$ negative norm ($n=p+q$) [[Clifford basis vectors]], $\ga_\al$. The Clifford algebra product of any two Clifford elements, equivalent to the [[matrix product in a suitable representation|Clifford matrix representation]], is non-commutative and decomposes into ''symmetric product'' (''//dot product//'') and [[antisymmetric product|antisymmetric bracket]] (''//cross product//'') parts,
\begin{eqnarray}
AB &=& A \cdot B + A \times B\\
A \cdot B &=& \ha \lp AB+BA \rp\\
A \times B &=& \ha \lp AB-BA \rp
\end{eqnarray}
The product is associative and distributive,
\begin{eqnarray}
A \lp B C \rp = \lp A B \rp C\\
A \lp B + C \rp = A B + A C
\end{eqnarray}
And, just as for matrices, [[almost all elements|Clifford group]] have an [[inverse]], $AA^-=1$.
A Clifford algebra is a graded ''geometric algebra'' in that the elements of [[Clifford grade]] $0,1,2,3,\dots$ may be considered as scalars, vectors, areas, volumes, ... and the Clifford product as operations between them. For example, the product of two vectors is a scalar (their dot product) plus an area (their cross product). The antisymmetric product of three vectors is a 3-vector, or volume. The product and its decomposition are completely described by the [[Clifford basis identities]].
Any two [[Clifford basis elements]] are orthogonal under the [[scalar part]] operator. Taking the scalar part of two multiplied basis elements of [[grade|Clifford grade]] $r$ gives the orthogonality relation,
\[ \li \ga_{\al \dots \be} \ga^{\ga \dots \de} \ri = r! \de^\ga_{\lb \be \rd} \dots \de^\de_{\ld \al \rb} \]
in which the basis element indices have been raised with the [[Minkowski metric]]. The scalar part of any two multiplied basis elements of unequal grade is 0.
The orthogonality relation may be used to determine the ''scalar product'' of any two multivectors. For example, between a multivector, $A$, and bivector, $B$, the scalar product is
\[ \li A B \ri = \fr{1}{4} A^{\al \be} B_{\ga \de} \li \ga_{\al \be} \ga^{\ga \de} \ri = \fr{1}{4} A^{\al \be} B_{\ga \de} 2 \de^\ga_{\lb \be \rd} \de^\de_{\ld \al \rb} = \ha A^{\al \be} B_{\be \al} \]
These may be combined to express any [[Clifford element]] using a Clifford basis expansion,
$$
A = \li A \ri + \li A \ga^\al \ri \ga_\al + \fr{1}{2} \li A \ga^{\al \be} \ri \ga_{\al \be}
+ \fr{1}{3!} \li A \ga^{\al \be \ga} \ri \ga_{\al \be \ga}
+ ...
+ \fr{1}{n!} \li A \ga^{\al \be ... \de} \ri \ga_{\al \be ... \de}
$$
The scalar part operator, and the orthogonality relations, are equivalent to the matrix [[trace]] and [[Lie algebra]] generator orthogonality through the [[Killing form]].
The $2^n$ ''Clifford basis elements'' are formed by all possible products of the $n$ [[Clifford basis vectors]]. Because of the [[fundamental Clifford identity|Clifford basis vectors]], basis elements are antisymmetric under the exchange of indices, like [[coordinate basis forms]], and may be written via the [[antisymmetric bracket]]. Each basis element has a [[grade|Clifford grade]], $q$, corresponding to the number of constituent basis vectors, and a multiplicity, ${n \choose q} = \fr{n!}{q!(n-q)!}$, equal to the number of their ordered combinations,
| !Element(s) | !Grade | !Multiplicity |!Names |
| $1$ | $0$ | ${n \choose 0} = 1$ |scalar, real number |
| $\ga_\al$ | $1$ | ${n \choose 1} = n$ |vector, [[Clifford basis vectors]]|
| $\ga_{\al \be} = \ga_{\lb \al \be \rb} = \ga_\al \ga_\be = \lb \ga_\al,\ga_\be \rb_A$ | 2 | ${n \choose 2} = \ha n (n-1)$ |bivector, 2-vector |
| $\ga_{\al \be \ga} = \ga_{\lb \al \be \ga \rb} = \ga_\al \ga_\be \ga_\ga = \lb \ga_\al,\ga_\be,\ga_\ga \rb_A$ | 3 | ${n \choose 3} = \fr{1}{3!} n (n-1)(n-2)$ |trivector, 3-vector |
| $\vdots$ | $\vdots$ | $\vdots$ |$\vdots$ |
| $\ga_{\al \dots \be} = \ep_{\al \dots \be} \ga$ | $n$ | ${n \choose n} = 1$ |n-vector |
Each of these $\sum_{k=0}^n {n \choose k} =2^n$ Clifford basis elements is a [[Lie algebra]] generator, with structure coefficients corresponding to [[Clifford basis identities]]. The Clifford basis elements also satisfy [[Clifford basis element orthogonality]].
Using [[Clifford dual]]ity, it is often convenient to express high grade basis elements in terms of the Clifford [[pseudoscalar]],
$$\ga = \ga_0 \ga_1 \dots \ga_{n-1}$$
and the [[permutation symbol]]. In this way, the basis r-vectors can be written as
$$\ga_{\al \dots \be} = \fr{1}{\lp n-r \rp!} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \ga$$
For example, the $n$ basis (n-1)-vectors are
$$\ga_{\al \dots \be} = \fr{1}{\lp n-1 \rp!} \ep_{\al \dots \be \ga} \ga^\ga \ga$$
This reduces the number of indices necessary to represent high grade [[Clifford element]]s.
The $2^n$ basis elements can also be written via a generalized index as $\ga_A$, with $A$ enumerating
each different antisymmetric combination of the usual Clifford indices.
The ''Clifford basis identities'' are derived from the [[fundamental Clifford identity|Clifford basis vectors]] by splitting the product of two basis vectors into a scalar (a [[Minkowski metric]] component) plus a bivector,
\[ \ga_\al \ga_\be = \ga_\al \cdot \ga_\be + \ga_\al \times \ga_\be = \et_{\al \be} + \ga_{\al \be} \]
or going in reverse — rewriting a bivector as a scalar plus the product of two basis vectors. By selectively applying this rule, all [[Clifford element]]s can be written as sums of coefficients times [[Clifford basis elements]]. The structure coefficients characterizing the [[Clifford algebra]] as a [[Lie algebra]] can be read off the [[cross product|antisymmetric bracket]] identities,
\begin{eqnarray}
\ga_\al \times \ga_\be &=& \ga_{\al \be}\\
\ga_\al \times \ga_{\be \ga} &=& \et_{\al \be} \ga_{\ga} - \et_{\al \ga} \ga_{\be}\\
\ga_{\al \be} \times \ga_{\ga \de} &=& - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga}\\
\ga_\al \times \ga_{\be \ga \de} &=& \ga_{\al \be \ga \de} \\
&\vdots&
\end{eqnarray}
Equally useful identities arise for the symmetric product,
\begin{eqnarray}
\ga_\al \cdot \ga_\be &=& \et_{\al \be}\\
\ga_\al \cdot \ga_{\be \ga} &=& \ga_{\al \be \ga}\\
\ga_{\al \be} \cdot \ga_{\ga \de} &=& \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de}\\
\ga_\al \cdot \ga_{\be \ga \de} &=& \et_{\al\be} \ga_{\ga\de} - \et_{\al\ga} \ga_{\be\de} + \et_{\al\de} \ga_{\be\ga} \\
&\vdots&
\end{eqnarray}
Continuing the series, the product of two basis elements of [[grade|Clifford grade]]s $p$ and $q$, such as
\begin{eqnarray}
\ga_{\al \be} \ga_{\ga \de} &=& \ga_{\al \be} \cdot \ga_{\ga \de} + \ga_{\al \be} \times \ga_{\ga \de}\\
&=& \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \lp - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \rp + \ga_{\al \be \ga \de}
\end{eqnarray}
gives a result of mixed grades $|p-q|$ through $p+q \le n$ in steps of $2$. For example, a bivector times a 3-vector typically gives a vector plus a 3-vector plus a 5-vector if $n$ is at least 5, otherwise just a vector plus a 3-vector.
The products of even or odd graded elements are
| !grade of $A$ | !grade of $B$ | !grade of $AB$ |
| even | even | even |
| odd | odd | even |
| odd | even | odd |
Note that the cross product with a bivector is grade preserving.
A rest [[frame]] exists at each point in a curved [[spacetime]]. A sufficiently small surrounding region is described locally by a diagonal [[Minkowski metric]], $\eta_{\al \be}$, with $p$ positive and $q$ negative unit entries. This may be visualized by considering a set of $n$ ''orthonormal'' (orthogonal and unit length) geometric "vector" elements, the ''Clifford basis vectors'', or ''//Clifford algebra generators//'', $\ga_\al$. These Clifford basis vectors provide a means for invariantly describing local geometric objects.
Like [[coordinate basis 1-forms]], two unequal Clifford basis vectors anti-commute, and their product represents a geometric area, or ''bivector'', element such as $\ga_1 \ga_2 = - \ga_2 \ga_1$, representing a unit area element spanned by $\ga_1$ and $\ga_2$. The orthonormality of Clifford basis vectors is expressed by the ''fundamental Clifford identity'',
$$
\ga_\al \cdot \ga_\be = \ha \lp \ga_\al \ga_\be + \ga_\be \ga_\al \rp = \et_{\al \be}
$$
which gives a Clifford scalar (real number) as a result of the symmetric product of two Clifford vectors. The antisymmetric product of every combination of two unequal Clifford vectors gives the $\ha n (n-1)$ bivector [[Clifford basis elements]],
$$
\ga_\al \times \ga_\be = \ha \lb \ga_\al, \ga_\be \rb = \ha \lp \ga_\al \ga_\be - \ga_\be \ga_\al \rp = \ga_\al \ga_\be = \ga_{\lb \al \be \rb} = \ga_{\al \be}
$$
The ''Clifford bundle'', $Cl M$, with base [[manifold]] $M$ is an [[automorphism bundle]] and a [[vector bundle]] with $2^n$ fiber basis elements equal to the [[Clifford basis elements]], $\ga_{\al \dots \be}$. The fiber at each base manifold point, $p$, is the space of [[Clifford element]]s. The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford adjoint]]s,
$$
\ga_{\al \dots \be}^2 = U_{21} \ga_{\al \dots \be}^1 U_{21}^-
$$
which don't necessarily preserve [[Clifford grade]]. The structure group, $Aut(Cl)=Cl^*$, the automorphism group, is the [[Clifford group]] with adjoint action on the fiber. //(Is that true?)//
For a section, $C(x)$, transforming under the adjoint action [[gauge transformation]], $C \mapsto C'=U C U^-$, the [[covariant derivative]] is
$$
\f{\na} C = \f{d} C + \ha \f{A} C - \ha C \f{A} = \f{d} C + \f{A} \times C
$$
(defined with a $\ha$ in it to keep things pretty) with the [[Clifford connection]], $\f{A}$, applied using the [[cross product|Clifford algebra]].
Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \ha \int_0^t \f{A}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} U(t) = - \ha \ve{v} \f{A} U
$$
Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} C &=& \f{d} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp + \ha \f{A} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp + \ha \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp \f{A} \\
&=& \ha \lp \f{d} \f{A} \rp C - \ha \f{A} \f{d} C - \ha \lp \f{d} C \rp \f{A} - \ha C \f{d} \f{A}
+ \ha \f{A} \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp + \ha \lp \f{d} C + \ha \f{A} C - \ha C \f{A} \rp \f{A} \\
&=& \ff{F} \times C
\end{eqnarray}
gives the [[Clifford curvature|Clifford-Riemann curvature]],
$$
\ff{F} = \f{d} \f{A} + \ha \f{A} \times \f{A}
$$
a Clifford valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\ha$ in the path holonomy equation).
Under a gauge transformation, $C(x) \mapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& U \lp \f{\na} C \rp U^-\\
\f{d} \lp U C U^- \rp + \ha \f{A'} U C U^- - \ha U C U^- \f{A'} &=& U \lp \f{d} C \rp U^- + \ha U \f{A} C U^- - \ha U C \f{A} U^-
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = U \f{A} U^- - 2 \lp \f{d} U \rp U^- = U \f{A} U^- + 2 U \lp \f{d} U^- \rp
$$
An infinitesimal transformation, $U \simeq 1 + \ha C$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} C - \ha \f{A} C + \ha C \f{A} = \f{A} - \f{\na} C
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \ha \f{A'} \times \f{A'} = U \ff{F} U^- \simeq \ff{F} + C \times \ff{F}
$$
The covariant derivative acting on a [[Clifform]] such as the curvature, transforming under the adjoint action, $\ff{F'} = U \ff{F} U^-$, is still
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \times \ff{F}
$$
The [[graded Clifford bundle|Clifford vector bundle]] has the same fiber as the [[Clifford bundle]], but the transition functions (which for the graded Clifford bundle are grade preserving) are [[Clifford rotation]]s.
We can use two [[division algebra]]s, $\mathbb{D}$ and $\mathbb{D}'$, of dimension $n$ and $n'$ and signature $(p,q)$ and $(p',q')$, to construct chiral representations of [[Clifford algebra]]s $Cl(p+p',q+q')$ or $Cl(p+q',q+p')$, in a similar manner to the construction of [[Clifford division algebra representation]]s. In a ''Clifford compound division algebra representation'', Clifford vectors are expressed as
$$
v = v^\al \ga_\al =
\lb \ba{cc}
0 & v_- \\
v_+ & 0
\ea \rb
\s \s
v_- =
\lb \ba{cc}
x & \pm \os{y}{}' \\
y' & - \os{x}
\ea \rb
\s \s
v_+ =
\lb \ba{cc}
\os{x} & \pm \os{y}{}' \\
y' & - x
\ea \rb
$$
which may be understood as matrices of inter-commuting division algebra elements, $x \in \mathbb{D}$ and $y' \in \mathbb{D}'$, or as $\mathbb{R}(2(n\!+\!n'))$ matrices via their direct Clifford division algebra representation. The result of multiplying two vectors is
$$
u \, v = \lb \ba{cc}
u_- v_+ & 0 \\
0 & u_+ v_-
\ea \rb
\s \s
u_- v_+ =
\lb \ba{cc}
w & \pm \os{z}{}' \\
z' & - \os{w}
\ea \rb
\lb \ba{cc}
\os{x} & \pm \os{y}{}' \\
y' & - x
\ea \rb
=
\lb \ba{cc}
w \os{x} \pm \os{z}' y'
&
\pm w \os{y}' \mp \os{z}' x \\
z' \os{x} - \os{w} y'
&
\pm z' \os{y}' + \os{w} x
\ea \rb
$$
in which we see the result of squaring a vector is
$$
v \, v = \lb \ba{cc}
x \os{x} \pm \os{y}' y'
&
0 \\
0
&
\pm y' \os{y}' + \os{x} x
\ea \rb
$$
and so the represented Clifford algebra has signature $(p+p',q+q')$ or $(p+q',q+p')$ depending on the choice of $\pm$. The chiral bivector part of $u v$, an element of a [[spin Lie algebra]], $spin(p+p',q+q')$ or $spin(p+q',q+p')$, is
$$
u_- v_+ \; \in \;
\lb \ba{cc}
\mathbb{B} \oplus \mathbb{B}'
&
\mathbb{D} \otimes \mathbb{D}' \\
\mathbb{D} \otimes \mathbb{D}'
&
\mathbb{B} \oplus \mathbb{B}'
\ea \rb
$$
with the tensor product of the division algebras on the off diagonal, and the direct sum of bi-products on the diagonal.
Every Clifford algebra and spin Lie algebra representation relevant to the [[exceptional magic square]] can be computed this way.
As well as the usual [[Clifford matrix representation]] of [[Cl(2,4)]], there are [[Clifford compound division algebra representation]]s.
Using the [[bi-split-quaternion]]s, with $q \in \mathbb{D} = \mathbb{H}'$ and $z \in \mathbb{D}' = \mathbb{C}$, we have
$$
v = v^\al \ga_\al =
\lb \ba{cc}
0 & v_- \\
v_+ & 0
\ea \rb
\s \s
v_- =
\lb \ba{cc}
q & - z^* \\
z & - \os{q}
\ea \rb
\s \s
v_+ =
\lb \ba{cc}
\os{q} & - z^* \\
z & - q
\ea \rb
$$
with negative chiral bivector parts,
$$
v_{1-} v_{2+} =
\lb \ba{cc}
q_1 & - z^*_1 \\
z_1 & - \os{q}_1
\ea \rb
\lb \ba{cc}
\os{q}_2 & - z^*_2 \\
z_2 & - q_2
\ea \rb
=
\lb \ba{cc}
q_1 \os{q}_2 - z^*_1 z_2
&
- q_1 z^*_2 + z^*_1 q_2 \\
z_1 \os{q}_2 - \os{q}_1 z_2
&
- z_1 z^*_2 + \os{q}_1 q_2
\ea \rb
=
\lb \ba{cc}
q_3 - z_3
&
b \\
\os{b}^*
&
- z_3^* + q_4
\ea \rb
$$
Or, using the [[biquaternion]]s, with $z \in \mathbb{D} = \mathbb{C}$ and $q \in \mathbb{D}' = \mathbb{H}$, we have
$$
v = v^\al \ga_\al =
\lb \ba{cc}
0 & v_- \\
v_+ & 0
\ea \rb
\s \s
v_- =
\lb \ba{cc}
z & - \os{q} \\
q & - z^*
\ea \rb
\s \s
v_+ =
\lb \ba{cc}
z^* & - \os{q} \\
q & - z
\ea \rb
$$
Using the usual representation of biquaternions in terms of Pauli matrices, this gives
\begin{eqnarray}
\ga_\va &=& - i \, \si_1 \otimes \si_1 \otimes \si_\va \\
\ga_4 &=& - i \, \si_1 \otimes \si_2 \otimes \si_0 \\
\ga_5 &=& \;\;\;\; \, \si_1 \otimes \si_3 \otimes \si_0 \\
\ga_6 &=& \;\;\;\; \,\si_2 \otimes \si_0 \otimes \si_0 \\
\end{eqnarray}
and so
$$
v_\pm = \lb \ba{cc}
v^5 \pm i \, v^6 & - v^4 \si_0 - v^\ep i \si_\ep \\
v^4 \si_0 - v^\ep i \si_\ep & - v^5 \pm i \, v^6
\ea \rb
$$
The negative chiral bivector part is
$$
v_{1-} v_{2+} =
\lb \ba{cc}
z_1 & - \os{q}_1 \\
q_1 & - z^*_1
\ea \rb
\lb \ba{cc}
z^*_2 & - \os{q}_2 \\
q_2 & - z_2
\ea \rb
=
\lb \ba{cc}
z_1 z^*_2 - \os{q}_1 q_2
&
- z_1 \os{q}_2 + \os{q}_1 z_2 \\
q_1 z^*_2 - z_1^* q_2
&
- q_1 \os{q}_2 + z_1^* z_2
\ea \rb
=
\lb \ba{cc}
q_3 + z_3
&
b \\
\os{b}^*
&
z_3^* + q_4
\ea \rb
$$
which is a representative element of [[su(2,2)]].
//that might lead to some fun...//
One may carry out several unary operations on [[Clifford element]]s.
The [[inverse]] of a Clifford element, $A^-$, is most generally computed by working in a [[Clifford matrix representation]]. However, some cases may be handled easily, such as the inverse of a Clifford vector, $v^- = \fr{v}{v \cdot v}$.
The [[Clifford dual]] of an element, $A \ga^-$, is often a useful object.
The ''involution'' operator inverts the signs of all vectors in an element, producing a [[grade|Clifford grade]] dependent sign change for the parts of an element, $\hat{A^r} = \lp -1 \rp^r A^r$, also expressible as $\hat{A} = A^e - A^o$.
The ''reversion operator'', a.k.a. ''//reverse//'', reverses the order of all vectors multiplied in an element, producing $\tilde{A^r} = \lp -1 \rp^{\ha r(r-1)} A^r$.
''Clifford conjugation'' combines these last two, $\bar{A} = \tilde{\hat{A^r}} = \lp -1 \rp^{\ha r(r+1)} A^r$.
For a set of [[Dirac matrices]] in which $\ga_0$ is represented by a Hermitian matrix and all spatial [[Clifford basis vectors]] are represented by anti-Hermitian matrices, the [[Hermitian]] conjugate of a Clifford element is $A^\da = \ga_0 \tilde{A} \ga^0$. When written as a matrix, this gives the transpose of the complex conjugate, $A^\da = A^{*T}$. For the chiral representation of the $Cl(1,3)$ Dirac matrices, only $\ga_2$ is imaginary and the other basis vectors are real, so [[complex conjugates|complex structure]] of these represented Clifford elements are $A^* = \ga_2 A \ga_2$.
One often encounters the ''Dirac conjugate'', $\overline{A} = A^\da \ga_0 = \ga_0 \tilde{A}$, which shouldn't be confused with the Clifford conjugate.
The [[vector bundle connection]] for the [[Clifford bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on the [[Clifford basis vectors]] for the $Cl$ fiber. The structure group for the bundle is the [[Clifford group]], with group elements acting on the fiber through the [[Clifford adjoint]]. The covariant derivative may therefore be represented using a ''Clifford connection'', $\f{A} \in \f{Cl}$, acting on basis elements via the [[cross product|antisymmetric bracket]],
$$
\f{\na} \ga_\al = \f{A} \times \ga_\al
$$
which gives the ''Clifford covariant derivative'' acting on any Clifford valued field (Clifford bundle section),
$$
\f{\na} C = \f{d} C + \f{A} \times C
$$
Note that the covariant derivative for the Clifford bundle does not necessarily preserve [[Clifford grade]].
The ''Clifford covariant derivative'', $\na = \ve{e} \f{\na} = \ga^\mu (e_\mu)^i \na_i = \ga^\mu \na_\mu$, is a [[Clifford vector]] valued differential operator that comes from [[Cliffordizing|Cliffordization]] the [[Clifford vector bundle covariant derivative|Clifford vector bundle]]. Acting on [[Clifford basis vectors]], we have
$$
\na \ga_\mu = \ve{e} \lp \f{\om} \times \ga_\mu \rp
= \ga^\rh \ve{e}_\rh \lp \f{\om}^\nu{}_\mu \ga_\nu \rp
= \ga_{\rh \nu} \om^\rh{}^\nu{}_\mu + \om_\nu{}^\nu{}_\mu
$$
with the [[spin connection]], $\f{\om}$, applied using the [[cross product|Clifford algebra]], and noting the [[Clifford vector]] multiplication resulting in scalar and bivector parts. Acting on the associated tangent vectors and 1-forms we have
$$
\na \ve{e}{}_\mu = \ga_\rh \om^{\rh \nu}{}_\mu \ve{e}{}_\nu \s \na \f{e}^\mu = \ga_\rh \om{}^\rh{}_\nu{}^\mu \f{e}^\nu
$$
Combining these, we have
$$
\na \f{e} = \f{\na} \f{e}^\mu \ga_\mu = ( \na \f{e}^\mu ) \ga_\mu + \f{e}^\mu ( \na \ga_\mu )
= ( \ga_\rh \om{}^\rh{}_\nu{}^\mu \f{e}^\nu ) \ga_\mu + \f{e}^\mu ( \ga_{\rh \nu} \om^\rh{}^\nu{}_\mu + \et_{\rh \nu} \om^\rh{}^\nu{}_\mu) = \f{e}^\mu \mathrm{div}(\ve{e}_\mu)
$$
producing the ''frame [[divergence]]'', $\na \cdot \f{e} = \f{e}^\mu \mathrm{div}(\ve{e}_\mu)$, as well as $\na \times \f{e}=0$. This covariant derivative is the [[Clifford bundle]] analog of the [[Dirac operator]], which includes half of the [[Clifford connection]]. It also provides the Clifford bundle analog of both the [[exterior derivative]] and the [[codifferential]], raising or lowering the [[Clifford algebra]] grade using either the cross product or dot product, and can be used to help reformulate expressions involving forms to expressions using Clifford elements.
If $\f{A} = \f{e}^\mu A_\mu(x)$ is a [[1-form]] field, such as a [[connection]], then its associated Clifford field is $A = \ve{e} \f{A} = \ga^\mu A_\mu$. The Clifford antisymmetric covariant derivative of this is
\begin{eqnarray}
\na \times A &=& \ga^\mu \na_\mu \times \ga^\nu A_\nu = \ga^\mu \times ( \ga^\nu \pa_\mu A_\nu + \om_{\mu \rh}{}^\nu \ga^\rh A_\nu )
= \ga^{\mu\nu} ( \pa_\mu A_\nu + \om_{\mu \nu}{}^\rh A_\rh ) \\
&=& \ve{e} \f{\na} \times \ve{e} \f{A} = - \ve{e} \ve{e} \f{\na} \f{A} = - \ve{e} \ve{e} \ff{F} = F \\
\end{eqnarray}
in which $F$ is the Clifford bivector field, $F = \ha \ga^{\mu \nu} F_{\mu \nu}$, associated to the 2-form $\ff{F} = \ha \f{e}^\mu \f{e}^\nu F_{\mu\nu} = \f{\na} \f{A}$ ( $= \f{d} \f{A}$ if there's no [[torsion]]). The Clifford symmetric covariant derivative of a 1-form is a scalar, its [[divergence]],
$$
\na \cdot A = \ve{e} \f{\na} \cdot \ve{e} \f{A} = \ga^\mu \cdot \na_\mu \ga^\nu A_\nu
= \ga^\mu \cdot ( \ga^\nu \pa_\mu A_\nu + \om_{\mu \rh}{}^\nu \ga^\rh A_\nu )
= \pa^\mu A_\mu + \om_{\mu}{}^{\mu \nu} A_\nu
= \mathrm{div}(\f{A})
$$
equivalent to the [[codifferential]] of the 1-form field, $\ve{\de} \f{A}$, if there's no torsion. Combining these, we have $\na A = - \ve{e} \ve{e} ( \f{\na} \f{A} ) + \mathrm{div}(\f{A})$. If we wish to compute the equivalent of $\ve{\de} \ff{F} = \ve{\de} \f{d} \f{A}$, we can look at the Clifford antisymmetric derivative of a bivector, obtaining a vector,
\begin{eqnarray}
\na \times F &=& \ga^\rh \na_\rh \times ( \ha \ga^{\mu\nu} F_{\mu\nu})
= \ga^\rh \times ( \ha \ga^{\mu\nu} \pa_\rh F_{\mu\nu} + \ha (\om_\rh{}_\si{}^\mu \ga^{\si\nu} + \om_\rh{}_\si{}^\nu \ga^{\mu\si}) F_{\mu\nu} ) \\
&=& \ga^\mu ( \pa^\nu F_{\nu\mu} + \om_{\rh}{}^{\rh\nu} F_{\nu\mu} + \om^\nu{}_\mu{}^\rh F_{\nu\rh} ) \\
&=& \ve{e} (\ve{\na} \ff{F})
\end{eqnarray}
using [[Clifford basis identities]]. For completeness, we also have the trivector,
$$
\na \cdot F = \fr{1}{3!} \ga^{\rh\mu\nu} (3 \, \pa_\rh F_{\mu\nu} + 6 \, \om_{\rh\mu}{}^\si F_{\si\nu})
$$
associated to $\f{\na} \ff{F}$.
The ''Clifford curvature scalar'' is obtained by taking the [[scalar part]] of the [[frame]] contracted twice with the [[Clifford bundle]] curvature,
$$
R = \li \ve{e} \ve{e} \ff{F} \ri
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature scalar is then the result of the [[dot product|Clifford algebra]] of the frame with the [[Clifford-Ricci curvature]],
$$
R = \li \ve{e} \ve{e} \ff{R} \ri = \ve{e} \cdot \f{R} = \lp \ve{e} \times \ve{e} \rp \cdot \ff{R}
$$
and equals the [[curvature scalar]] written in terms of the [[spin connection]] and frame.
The Clifford curvature scalar also comes from the expression:
\begin{eqnarray}
\fr{2}{\lp n-2 \rp!} \li \lp \f{e} \rp^{n-2} \ff{R} \ga^- \ri
&=& \fr{2}{\lp n-2 \rp!} \f{e}^\al \dots \f{e}^\be \f{e}^\mu \f{e}^\nu \fr{1}{4} R_{\mu\nu}{}^{\rh\si} \li \ga_{\al \dots \be} \ga_{\rh \si} \ga^- \ri \\
&=& \fr{2}{\lp n-2 \rp!} \nf{e} \ep^{\al \dots \be \mu \nu} \fr{1}{4} R_{\mu\nu}{}^{\rh\si} \ep_{\al \dots \be \rh \si} \\
&=& \nf{e} \de_{\lb \rh \si \rb}^{\mu\nu} R_{\mu\nu}{}^{\rh \si}=\nf{e} R
\end{eqnarray}
using the [[volume form]] and [[permutation identities]].
We can use a [[division algebra]], $\mathbb{D}$, of signature $(p,q)$, to construct a [[Clifford algebra]] of the same or reversed signature, with [[chiral]] Clifford basis vectors expressed as
$$
\ga_c =
\lb \begin{array}{cc}
0 & \pm \os{e}_c \\
e_c & 0
\end{array} \rb
\sim
\lb \begin{array}{cc}
0 & \pm (\os{e}_c)^a{}_b \\
(e_c)^b{}_a & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \pm M_{\os{c}\os{b}}{}^a \\
M_{ca}{}^\os{b} & 0
\end{array} \rb
$$
a ''division algebra representation'', with $\os{e}_c$ the conjugate division algebra element, and the $\pm$ sign producing signatures $(p,q)$ or $(q,p)$. These can be understood as $\mathbb{D}(2)$ matrices, or as the $\mathbb{R}(2n)$ matrices of a real [[Clifford matrix representation]] using division algebra multiplication coefficients. For the [[octonion]]s, which are non-associative, multiplication is assumed to act to the right, and the specific matrices come from the action on a ''division algebra spinor'',
$$
\Ps = \lb \ba{c} \ps \\ \os{\ch} \ea \rb
= \lb \ba{c} \ps^a e_a \\ \ch^b \os{e}_b \ea \rb
\sim \lb \ba{c} \ps^a \\ \ch^b \ea \rb
$$
The representative basis vectors satisfy the fundamental Clifford identity by virtue of the division algebra metric,
$$
\ga_a \cdot \ga_b = \ha \lp \ga_a \ga_b + \ga_b \ga_a \rp =
\pm \ha
\lb \begin{array}{cc}
\os{e}_a e_b + \os{e}_b e_a & 0 \\
0 & e_a \os{e}_b + e_b \os{e}_a
\end{array} \rb
= \pm n_{ab}
$$
and these representative matrices are each other's transpose, $(\os{e}_c) = n_{cd} (e_d)^T$, and satisfy a ''cyclic identity'',
$$
M_{ca\os{b}} = M_{ab\os{c}} = M_{bc\os{a}}
= M_{\os{a}\os{c}b} = M_{\os{c}\os{b}a} = M_{\os{b}\os{a}c}
$$
related to [[triality]].
The representative Clifford bivector basis elements are
$$
\ga_{cd} = \lb \ba{cc} \pm \os{e}_c e_d & \\ & \pm e_c \os{e}_d \ea \rb
\sim \lb \ba{cc} \pm (\os{e}_c)^a{}_b (e_d)^b{}_e & \\ & \pm (e_c)^b{}_a (\os{e}_d)^a{}_f \ea \rb
= \lb \ba{cc} \pm M_{\os{c}\os{b}}{}^a M_{de}{}^\os{b} & \\ & \pm M_{ca}{}^\os{b} M_{\os{d}\os{f}}{}^a \ea \rb
$$
with it understood that, for example, $\os{e}_d$, multiplies to the right before $e_c$ multiplies the result. These are the [[chiral]] basis elements of the corresponding [[spin Lie algebra]], acting on division algebra spinors.
The ''Clifford dual'' of any [[Clifford element]], $A$, is obtained by right multiplying it by the inverse [[pseudoscalar]], $A \ga^-$. For a [[Clifford grade]] $r$ element, this gives a grade $(n-r)$ element,
\[ A^r \ga^- = \fr{1}{r!} A^{\al \dots \be} \ga_{\al \dots \be} \ga^- = \fr{1}{r! \lp n - r \rp !} A^{\al \dots \be} \li \ga_{\al \dots \be \ga \dots \de} \ga^- \ri \ga^{\ga \dots \de} = \fr{1}{r! \lp n - r \rp !} A^{\al \dots \be} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \]
in which $\ep_{\al \dots \be \ga \dots \de}$ is the [[permutation symbol]], and [[indices]] are raised with the [[Minkowski metric]].
The Clifford dual transformation is analogous to the [[Hodge dual]].
Thanks to [[Clifford basis element orthogonality]] all [[Clifford algebra]] elements may be written as a sum of $2^n$ real coefficients multiplying [[Clifford basis elements]], with multiplicative factors included to account for the redundant sums over [[indices]], in a ''Clifford basis expansion'',
\begin{eqnarray}
A &=& A^s + A^\al \ga_\al + \ha A^{\al \be} \ga_{\al \be} + \fr{1}{3!} A^{\al \be \ga} \ga_{\al \be \ga} + \dots + A^p \ga\\
&=& A^0 + A^1 + A^2 + A^3 + \dots + A^n
\end{eqnarray}
(Some people choose to limit the sums so they don't run over all index values — but this isn't done here.) Like the coefficients of [[differential form]]s, the Clifford element coefficients are [[antisymmetric|index bracket]] in their indices, $A^{\al \dots \be}=A^{\lb \al \dots \be \rb}$. Unlike differential forms, Clifford elements may be of mixed [[grade|Clifford grade]].
A clifford element has a geometric interpretation as a collection of variously sized scalar, vector, oriented area set, ..., and n-volume objects.
Clifford elements have a faithful [[matrix representation|Clifford matrix representation]].
The high grade terms of Clifford elements may be written with fewer indices by using the [[pseudoscalar]],
\begin{eqnarray}
A^r &=& \fr{1}{r!} A^{\al \dots \be} \ga_{\al \dots \be} = \fr{1}{r!} A^{\al \dots \be} \fr{1}{\lp n-r \rp!} \ep_{\al \dots \be \ga \dots \de} \ga^{\ga \dots \de} \ga \\
&=& \fr{1}{\lp n-r \rp!} \lp \fr{1}{r!} A^{\al \dots \be} \ep_{\al \dots \be \ga \dots \de} \rp \ga^{\ga \dots \de} \ga \\
&=& \fr{1}{\lp n-r \rp!} A^r_{\ga \dots \de} \ga^{\ga \dots \de} \ga \\
\end{eqnarray}
So, for example, the pseudoscalar (n-vector, grade $n$) part is
$$
A^n = \fr{1}{n!} A^{\al \dots \be} \ga_{\al \dots \be} = A^p \ga
$$
and the (n-1)-vector part is
$$
A^{n-1} = \fr{1}{\lp n-1 \rp!} A^{\al \dots \be} \ga_{\al \dots \be} = A^{n-1}_\al \ga^\al \ga
$$
A ''Clifford [[gauge transformation|vector bundle gauge transformation]]'' is a change of the fiber basis elements for a [[Clifford bundle]], [[Clifford vector bundle]], or any graded Clifford bundle. The change may be induced by the action of an arbitrary, position dependent element of the fiber bundle's structure group -- a subgroup of the [[Clifford group]] acting on the [[Clifford basis elements]] via the [[Clifford adjoint]],
$$
\ga'_\al = U \ga_\al U^-
$$
This gauge transformation is an active transformation of bundle elements, and transforms any Clifford valued field (section), $\Ph$, to
$$
\Ph' = U \Ph U^-
$$
By definition, the [[Clifford covariant derivative|Clifford connection]] of any Clifford valued field transforms under a gauge transformation such that,
$$
\f{\na'} \Phi' = \lp \f{\na} \Phi \rp'
$$
Writing out the covariant derivative operators in this equation using the [[Clifford connection]],
\begin{eqnarray}
\f{\na'} \lp U \Phi U^- \rp &=& U \lp \f{\na} \Phi \rp U^- \\
\lp \f{d} U \rp \Phi U^- + U \lp \f{d} \Phi \rp U^- + U \Phi \lp \f{d} U^- \rp + \f{A'} \times \lp U \Phi U^-\rp &=& U \lp \f{d} \Phi + \f{A} \times \Phi \rp U^- \\
U^- \lp \f{d} U \rp \Phi + \Phi \lp \f{d} U^- \rp U + \ha U^- \f{A'} U \Phi - \ha \Phi U^- \f{A'} U &=& \ha \f{A} \Phi - \ha \Phi \f{A}
\end{eqnarray}
gives the transformation law for the connection under a gauge transformation:
$$
\f{A'} = U \f{A} U^- - 2 \lp \f{d} U \rp U^-
$$
For an infinitesimal gauge transformation, $U \simeq 1 + \ha C$, the connection changes to
$$
\f{A'} \simeq \f{A} - \f{d} C - \ha \f{A} C + \ha C \f{A} = \f{A} - \f{\na} C
$$
giving the change $\de \f{A} = - \f{\na} C$.
The ''grade'' of a [[Clifford element]] corresponds to the number of [[Clifford basis vectors]] used in the [[Clifford basis elements]] needed to represent it. An element may be a single grade, $q$, in which case it is called a ''q-vector'', or it may be of mixed grade, and called a ''multivector''. For example,
\[ t = \fr{1}{3!} t^{\al \be \ga} \ga_{\al \be \ga} \]
is a 3-vector, or ''trivector'', while
\[ w = w^s + \ha w^{\al \be} \ga_{\al \be} \]
is a multivector of grades 0 and 2.
The ''grade operator'', $\li A \ri_q = A^q$, acts as a filter, passing only the grade $q$ parts of $A$. For example, the bivector part of $w$ is
\[ \li w \ri_2 = \ha w^{\al \be} \ga_{\al \be} \in \li Cl \ri_2 = Cl^2 \]
The grade operator may also be used to filter the even or odd graded parts of an element, such as $\li w\ri_e = \li w\ri_2$ and $\li w\ri_o = 0$. Of special interest is the grade 0 operator, $\li A\ri = \li A\ri_0 = A^0 = A^s$, or //''scalar part''// operator which gives the scalar part of $A$. This operator is proportional to the [[trace]] of an element in a [[Clifford matrix representation]]. It is useful since the grade 0 (scalar) part of a Clifford element is a real.
Combining [[Clifford basis identities]] with the [[grade|Clifford grade]] operator gives a ''Clifford graded [[commutation|commutator]]'' relationship for two Clifford elements of grades $r$ and $s$,
$$
\li A^r B^s \ri_q = \lp -1 \rp^\ep \li B^s A^r \ri_q
$$
with
$$
\ep = \ha \lp q^2 + r^2 + s^2 - q - r - s \rp
$$
This relation implies that any two Clifford elements commute inside the [[scalar part]] operator, $\li AB \ri = \li BA \ri$.
The Clifford product of two elements of grades $r$ and $s$ can produce elements of various grades,
$$
A^r B^s = \li A^r B^s \ri_{\ll r - s \rl} + \li A^r B^s \ri_{\ll r - s \rl + 2} + \dots + \li A^r B^s \ri_{r + s}
$$
The ''Clifford group'' consists of [[Clifford algebra]] elements having an inverse,
\[ Cl^* = \left\{ U \in Cl \mid \exists \;\; U^- \ni \, U \, U^- = 1 \right\} \]
The [[Clifford adjoint]] is the group action, and the [[Lie algebra]] corresponding to the Clifford group is the Clifford algebra.
Each [[Clifford algebra]] has a faithful representation in the real, quaternionic, or complex matrices, $GL(2^{[n/2]},\mathbb{C})$, with the Clifford product isomorphic to [[matrix|linear operator]] multiplication. This corresponds to the traditional definition of [[Pauli matrices]] and [[Dirac matrices]] as the $\gamma_{\alpha}$ for the purpose of using matrix algebra to do Clifford Algebra calculations, or simply for writing Clifford elements as matrices. Various unary operations on Clifford elements, the [[Clifford conjugate]]s, are equivalent to various matrix conjugates. The possible representations for any $Cl(p,q)$ are described by the [[table of Clifford matrix representations]].
A Clifford algebra is built by starting with the basis vectors and creating all possible multiples. For a seed example, we can build a representation for ''Cl(2,0)'' by starting with two Pauli matrices as the two [[Clifford basis vectors]],
$$
\begin{array}{cc}
\si_1 =
\left[\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right]
&
\si_2 = \left[\begin{array}{cc}
0 & -i\\
i & 0\end{array}\right]
\end{array}
$$
and multiplying to get the scalar and bivector,
$$
\begin{array}{cc}
1 = \si_1 \si_1 =
\left[\begin{array}{cc}
1 & 0\\
0 & 1\end{array}\right]
&
\si_{12} = \si_1 \si_2 =
\left[\begin{array}{cc}
i & 0\\
0 & -i
\end{array}\right]
= i \si_3
\end{array}
$$
completing the list of $Cl(2,0)$ [[Clifford basis elements]] represented as $2 \times 2$ complex matrices. To build larger Clifford algebras we can use the [[Kronecker product]] of smaller Clifford algebras -- ''Clifford periodicity''. For example, $Cl(2,2) = Cl(2,0) \otimes Cl(2,0)$, or [[Clifford representation doubling]], $Cl(p+1,q+1) = Cl(p,q) \otimes Cl(1,1)$. The tricky part is finding a set of orthogonal, anticommuting, ''Clifford basis vector matrix representatives'' after doing the product, such as picking out:
$$
\begin{array}{cc}
\ga_1 = \si_1 \otimes 1 =
\left[\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{array}\right]
&
\ga_2 = \si_2 \otimes \si_2 =
\left[\begin{array}{cccc}
0 & 0 & 0 & -1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0
\end{array}\right]
\\
\ga_3 = \si_2 \otimes \si_1 =
\left[\begin{array}{cccc}
0 & 0 & 0 & -i\\
0 & 0 & -i & 0\\
0 & i & 0 & 0\\
i & 0 & 0 & 0
\end{array}\right]
&
\ga_4 = \si_2 \otimes \si_{12} =
\left[\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & -1\\
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0
\end{array}\right]
\end{array}
$$
A matrix representation, such as above for $Cl(3,1)$, allows any element to be represented by a $4 \times 4$ complex matrix. For example,
$$
a \, \ga_{12} + b \, \ga_{34} = a \, \si_{12} \otimes \si_2 + b \, 1 \otimes \si_2 =
\left[\begin{array}{cccc}
0 & a - i b & 0 & 0\\
-a + i b & 0 & 0 & 0\\
0 & 0 & 0 & -a - i b\\
0 & 0 & a + i b & 0
\end{array}\right]
$$
To get a different signature we can multiply any basis vector representaive by $i$, such as multiplying $\ga_3$ above by $i$ to get a ''real representation'' of $Cl(2,2)$ -- in which all basis vectors, and hence all elements, are represented by real matrices. And to represent a Clifford algebra of one less dimension we can discard a vector.
Everything done with Clifford algebra can be identified with the corresponding matrix manipulation; however, it will almost always be more geometrically revealing to deal with the Clifford algebra elements directly.
Refs:
*http://en.wikipedia.org/wiki/Representations_of_Clifford_algebras
*Andrzej Trautman
**[[Clifford Algebras and their Representations|papers/Clifford Algebras and their Representations.pdf]]
***p20 describes construction of reps for arbitrarily high dimension
Iteration of the [[cross product|antisymmetric bracket]] produces the ''Clifford Jacobi identity'',
\[ A \times \lp B \times C \rp + B \times \lp C \times A \rp + C \times \lp A \times B \rp = 0 \]
and the ''cross product distributive rules'',
\begin{eqnarray}
A \times \lp B \times C \rp &=& \lp A \cdot B \rp \cdot C - B \cdot \lp A \cdot C \rp \\
A \times \lp B C \rp &=& \lp A \times B \rp C + B \lp A \times C \rp
\end{eqnarray}
A combination of [[Clifford algebra]] dot and cross products is
\[ A \cdot \lp B \times C \rp + A \times \lp B \cdot C \rp = \ha \lp ABC - CBA \rp = \lp A \cdot B \rp \times C + \lp A \times B \rp \cdot C \]
A string of cross products without parenthesis, $A \times B \times C$, is not well defined because $A \times \lp B \times C \rp \ne \lp A \times B \rp \times C$; but a string of dot products, $A \cdot B \cdot C$, or a string of Clifford products, $ABC$, is well defined. In general, parenthesis should always be used to group multiple operations when the cross product is employed.
To calculate Clifford products, it is best to use the [[Clifford basis identities]]
The [[inverse]] of a [[Clifford vector]], $u$, is $u^-= \fr{u}{u\cdot u}$. Using the [[Clifford algebra]] symmetric product, and [[Clifford basis identities]], any vector, $v = v_\parallel + v_\perp$, can be decomposed into its orthogonal parts, $v_\parallel = u (v \cdot u^-)$ and $v_\perp = v - v_\parallel$, parallel to and perpendicular to $u$. Since parallel Clifford vectors commute and orthogonal Clifford vectors anticommute, the ''reflection'' of a vector, $v$, along a vector $u$, is
$$
R_u \, v = - u v u^- = - u ( v_\perp + v_\parallel ) u^- = v_\perp - v_\parallel
$$
Reflection is not obviously a [[Clifford adjoint]]. However, in [[spacetime]] [[Cl(1,3)]] we can define $u' = u \ga$ using the [[pseudoscalar]], which gives $R_u \, v = u' v \, u'^-$. This extends to give reflections of any $Cl(1,3)$ element, $R_u \, A = (u \ga) A (u \ga)^-$, and reflections of [[Dirac spinor]]s, $R_u \, \Ps = (u \ga) \Ps$. The reflection element, $U = u \ga$, is an element of the [[pin group]], $Pin(1,3)$.
A [[Clifford rotation]] can be constructed by combining two Clifford reflections. Suppose one wishes to rotate a vector, $v_1$, to a vector of equal length, $v_2$ (so $v_1^2 = v_2^2$). First reflect along their unit-length bisector, $v_a = (v_1+v_2)/|v_1+v_2|$, to get
$$
-v_a v_1 v_a^- = - \fr{v_1 + v_2}{|v_1+v_2|} v_1 \fr{v_1 + v_2}{|v_1+v_2|} = - \fr{v_2(v_2^- v_1^3 + 2 v_1^2 + v_1 v_2)}{ v_1^2 + v_1 v_2 + v_2 v_1 +v_2^2 } = - v_2
$$
then reflect along $v_2$,
$$
v_2 v_a v_1 v_a^- v_2^- = v_2 v_2 v_2^- = v_2
$$
So we see that the rotation by a [[simple rotor|Clifford rotation]],
$$
U = v_2 v_a = \fr{v_2^2 + v_2 v_1}{|v_1+v_2|}
$$
takes $v_1$ to $U v_1 U^- = v_2$. If $v_1$ and $v_2$ are of unit length with an angle $\th$ between them then
$$
U = \fr{1 + \cos(\th) + v_2 \times v_1}{\sqrt{2+2 \cos(\th)}} = \cos(\fr{\th}{2}) + \fr{v_2 \times v_1}{2 \cos(\fr{\th}{2})}
$$
A [[Clifford algebra]] that is two dimensions larger, by a split signature, has the same structure,
$$
Cl(p+1,q+1) = Cl(p,q) \otimes Cl(1,1)
$$
Doubled representations can be constructed explicitly using [[Pauli matrices]] and [[pseudoscalar]]. For $n = p + q$ even, and $\Ga^2 = -1$,
\begin{eqnarray}
\Ga'_\mu &=& \;\;\, \; \, \si_1 \otimes \Ga_\mu \\
\Ga'_{\mu + 1} &=& \;\;\, i \, \si_1 \otimes \Ga \\
\Ga'_{\mu + 2} &=& - i \, \si_2 \otimes 1 \\
\end{eqnarray}
or the $i$'s can be removed. For $n = p + q$ odd, and $\Ga^2 = -1$,
\begin{eqnarray}
\Ga'_\mu &=& \, \, \Ga_\mu \otimes \si_3 \\
\Ga'_{\mu + 1} &=& \; i \, \Ga \otimes \si_1 \\
\Ga'_{\mu + 2} &=& \; \; \, \Ga \otimes \si_2 \\
\end{eqnarray}
or the $i$ can be swapped. We also have the doublings
\begin{eqnarray}
Cl(p+2,q) &=& Cl(p,q) \otimes Cl(2,0) \\
Cl(p,q+2) &=& Cl(p,q) \otimes Cl(0,2) \\
\end{eqnarray}
A bivector crossed with a vector gives a vector orthogonal to the original, in the plane (or planes) of the bivector. Using the [[Clifford basis identities]] and antisymmetry of bivector indices,
\begin{eqnarray}
B \times v &=& \ha B^{\al \be} v^\ga \ga_{\al \be} \times \ga_\ga = B^{\al \be} v^\ga \ga_{\lb \al \rd} \et_{\ld \be \rb \ga} = B^{\al \be} v_\be \ga_\al \\
v \cdot \lp B \times v \rp &=& v^\de B^{\al \be} v_\be \ga_\de \cdot \ga_\al = B^{\al \be} v_\al v_\be = 0
\end{eqnarray}
A small rotational transformation in the plane (or planes) of a bivector may be carried out by
$$
v' \simeq v + \fr{1}{N}B \times v \simeq \lp 1 + \fr{1}{2N} B \rp v \lp 1 - \fr{1}{2N} B \rp
$$
for a large parameter, $N$. A finite rotation thus comes from [[exponentiating|exponentiation]] the bivector,
$$
v' = \lp \lim_{N \to \infty} \lp 1 + \fr{1}{2N} B \rp^N \rp \, v \, \lp \lim_{N \to \infty} \lp 1 - \fr{1}{2N} B \rp^N \rp = e^{\ha B} v \, e^{- \ha B} = U \, v \, U^-
$$
in which $U = e^{\ha B}$ is called a ''rotor'', and is an element of the [[spin group]].
As an example, a [[spatial rotation]] of a [[spacetime]] vector, $v = v^\mu \ga_\mu$, by an angle of $\th$ clockwise in the $\ga_{12}$ plane gives
$$
\begin{array}{rcl}
v' \ae e^{-\fr{\th}{2} \ga_{12}} \, v \, e^{ \fr{\th}{2} \ga_{12}} = \lp \cos{\fr{\th}{2}} - \ga_{12} \sin{\fr{\th}{2}} \rp v \lp \cos{\fr{\th}{2}} + \ga_{12} \sin{\fr{\th}{2}} \rp \\
\ae v^0 \ga_0 + \lp v^1 \cos{\th} + v^2 \sin{\th} \rp \ga_1 + \lp v^2 \cos{\th} - v^1 \sin{\th} \rp \ga_2 + v^3 \ga_3
\end{array}
$$
Similarly, a [[Lorentz boost]] by $\ze$ along $\ga_3$ gives
$$
\begin{array}{rcl}
v' \!\!&\!\!=\!\!&\!\! e^{- \ha \ga_{03} \ze} \, v \, e^{ \ha \ga_{03} \ze} = \lp \cosh{\fr{\ze}{2}} - \ga_{03} \sinh{\fr{\ze}{2}} \rp v \lp \cosh{\fr{\ze}{2}} + \ga_{03} \sinh{\fr{\ze}{2}} \rp \\
\!\!&\!\!=\!\!&\!\! \lp v^0 \cosh{\ze} + v^3 \sinh{\ze} \rp \ga_0 + v^1 \ga_1 + v^2 \ga_2 + \lp v^3 \cosh{\ze} + v^0 \sinh{\ze} \rp \ga_3
\end{array}
$$
Since $U^- = e^{- \ha B} = \tilde{U}$ is the [[reverse|Clifford conjugate]] and the [[inverse]] of $U = e^{\ha B}$, Clifford rotation is a special case of the [[Clifford adjoint]]. Any [[Clifford element]] may be rotated, $A' = U A U^-$ -- preserving the [[Clifford grade]] of the element. A rotor also actively rotates a [[spinor]], $\Ps' = U \Ps$, with the spinor either a Clifford algebra ideal or in some [[Clifford matrix representation]]. The group of Clifford rotations, $\mbox{Spin}{}^+$, in any Lorentzian spacetime, is the identity component of a [[spin group]] -- a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. A Clifford rotation may be translated into tensor notation for the equivalent [[Lorentz rotation]] via
$$
\ga'_\al = \ga_\be L^\be{}_\al = U \ga_\al U^-
$$
A rotation may also be obtained from [[Clifford reflection]]s. A ''simple rotor'' is a rotor that can be written as the product of two vectors, $U_{s}=ab=e^{\frac{1}{2}i_{2}\theta}$, in which $i_{2}$ is a unit bivector of a rotation plane and $\theta$ is a rotation parameter. A rotor may always be factored into a product of $\leq\frac{n}{2}$ simple rotors. A standard decomposition uses the choice of a non-singular vector, $v$, to factor a rotor into $U=\pm U'U_{s}$, in which $U'$ is a rotor that leaves $v$ invariant, $U'v=vU'$, and $U_{s}$ is a simple rotor that rotates $v$. In a four-dimensional Lorentzian [[spacetime]], a ''spacetime rotor'' can thus be factored using the time-like frame vector, $\gamma_{0}$, into a [[spatial rotation]] and [[Lorentz boost]],
$$
U = U_a \, U_\nu = e^{-\frac{1}{2}\gamma\gamma_{0} a} \, e^{-\frac{1}{2}\gamma_{0} \nu}
$$
in which $a = \gamma_{\pi} a^{\pi}$ is the spatial rotation vector (rotating an angle of $|a|$ clockwise about the $a$ axis) and $\nu = \gamma_{\pi}\nu^{\pi}$ is the rapidity vector of the boost. Using the chiral representation of the [[Dirac matrices]], in which $\ga_0 = \ga_0^\da$ is Hermitian and $\ga_\pi = - \ga_\pi^\da$ are anti-Hermitian under the [[Hermitian]] conjugate, the [[Clifford conjugate]] of a spacetime rotor is $U^- = \tilde{U} = \ga^0 U^\da \ga_0$ -- so these rotors, $U$, as matrix operators, are not [[unitary]]. They do, however, satisfy $U_{L/R}^- = U_{R/L}^\da$, in a [[chiral Clifford rotation]].
A ''Clifford vector'' is a [[grade|Clifford grade]] 1 [[Clifford element]],
$$
v = v^\al \ga_\al
$$
in which $\ga_\al$ are [[Clifford basis vectors]].
The ''Clifford vector bundle'', $Cl^1 M$, with base [[manifold]] $M$ is a [[vector bundle]] with $n$ fiber basis elements equal to the [[Clifford basis vectors]], $\ga_\al$. The fiber at each base manifold point, $p$, is the space of grade 1 Clifford elements, $Cl^1 = \li Cl \ri_1$. Physically, each fiber corresponds to a [[rest frame]] at that point. The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford rotation]]s,
$$
\ga_\al^2 = U_{12} \ga_\al^1 U_{12}^- = \lp t^{12} \rp_\al{}^\be \ga_\be^1 = \lp L^{12} \rp^\be{}_\al \ga_\be^1
$$
By equating the transition functions, $L^\be{}_\al$, and using the [[frame]], $\ve{e}{}_\al \f{e} = \ga_\al$, the Clifford vector bundle is [[associated]], $\ga_\al \leftrightarrow \ve{e_\al}$, to the [[tangent bundle]], with a corresponding equivalence between all their geometric structures. The structure group of the Clifford vector bundle, $\mbox{Spin}{}^+$, is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. A Clifford vector field, $v = v(x) = v^\al(x) \ga_\al$, over the manifold is a section of the bundle, and gives a Clifford vector at each manifold point.
[[Clifford grade]] $p$ fields are sections of the ''Clifford p-vector bundle'', $Cl^p M$, which has the $\frac{n!}{\left(n-p\right)!p!}$ grade $p$ [[Clifford basis elements]], $\ga_{\la \dots \be}$, as basis. The combined collection of these Clifford vector product bundles is the ''graded Clifford bundle'', $Cl^g M = \bigoplus_{p=0}^{n} Cl^p M$, having dimension $2^{n}$. The transition functions for the graded Clifford bundle are also [[Clifford rotation]]s,
$$\ga_{\al \dots \be}^2 = U_{12} \ga_{\al \dots \be}^1 U_{12}^-$$
which preserve the grade of the basis elements. The graded Clifford bundle fiber, $Cl$, is the same as for the [[Clifford bundle]] — but the transition functions (which for the graded Clifford bundle are grade preserving) are in different groups for the two bundles -- the Clifford vector bundle is a Clifford bundle with a [[reduction of the structure group]].
A section, $C(x)$, transforms under the Clifford rotation [[gauge transformation]], $C \mapsto C'=U C U^-$, and the [[covariant derivative]] is thus
$$
\f{\na} C = \f{d} C + \ha \f{\om} C - \ha C \f{\om} = \f{d} C + \f{\om} \times C
$$
(defined with a $\ha$ in it to keep things pretty) with the [[spin connection]], $\f{\om}$, applied using the [[cross product|Clifford algebra]]. This ''Clifford vector bundle covariant derivative'' acts nontrivially on Clifford basis vectors, and on the associated tangent vectors and forms as the [[tangent bundle covariant derivative|tangent bundle connection]] and [[cotangent bundle covariant derivative|cotangent bundle connection]],
$$
\f{\na} \ga_\mu = \f{\om}^\nu{}_\mu \ga_\nu
\s
\f{\na} \ve{e}{}_\mu = \f{\om}^\nu{}_\mu \ve{e}{}_\nu
\s
\f{\na} \f{e}^\mu = \f{\om}{}_\nu{}^\mu{} \f{e}^\nu
$$
so we have
$$
\f{\na} \f{e} = \f{\na} \f{e}^\mu \ga_\mu = \lp \f{\na} \f{e}^\mu \rp \ga_\mu - \f{e}^\mu \lp \f{\na} \ga_\mu \rp
= \lp \f{\om}{}_\nu{}^\mu{} \f{e}^\nu \rp \ga_\mu - \f{e}^\mu \lp \f{\om}^\nu{}_\mu \ga_\nu \rp = 0
$$
and, similarly, $\f{\na} \ve{e}=0$. It also acts on other Clifford basis elements,
$$
\f{\na} \ga_{\mu \nu} = \f{\om} \times \ga_{\mu\nu} = \f{\om}^\rh{}_\mu \ga_{\rh\nu} + \f{\om}^\rh{}_\nu \ga_{\mu\rh}
$$
using [[Clifford basis identities]].
Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t)=U(t)CU^-$ along a path on the base by a parameter dependent Clifford element, the path holonomy, $U(t) = Pe^{- \ha \int_0^t \f{\om}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} U(t) = - \ha \ve{v} \f{\om} U
$$
Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} C &=& \f{d} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp + \ha \f{\om} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp + \ha \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp \f{\om} \\
&=& \ha \lp \f{d} \f{\om} \rp C - \ha \f{\om} \f{d} C - \ha \lp \f{d} C \rp \f{\om} - \ha C \f{d} \f{\om}
+ \ha \f{\om} \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp + \ha \lp \f{d} C + \ha \f{\om} C - \ha C \f{\om} \rp \f{\om} \\
&=& \ff{R} \times C
\end{eqnarray}
with the [[Clifford-Riemann curvature]],
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \times \f{\om}
$$
This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\ha$ in the path holonomy equation).
Under a gauge transformation, $C(x) \mapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& U \lp \f{\na} C \rp U^-\\
\f{d} \lp U C U^- \rp + \ha \f{\om'} U C U^- - \ha U C U^- \f{\om'} &=& U \lp \f{d} C \rp U^- + \ha U \f{\om} C U^- - \ha U C \f{\om} U^-
\end{eqnarray}
giving the transformation law for the spin connection,
$$
\f{\om'} = U \f{\om} U^- - 2 \lp \f{d} U \rp U^- = U \f{\om} U^- + 2 U \lp \f{d} U^- \rp
$$
An infinitesimal transformation, $U \simeq 1 + \ha B$, in which $B$ is a Clifford bivector, changes the spin connection to
$$
\f{\om'} \simeq \f{\om} - \f{d} B - \ha \f{\om} B + \ha B \f{\om} = \f{\om} - \f{\na} B
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{R'} = \f{d} \f{\om'} + \ha \f{\om'} \times \f{\om'} = U \ff{R} U^- \simeq \ff{R} + B \times \ff{R}
$$
These expressions equate to those for a [[tangent bundle gauge transformation]].
The covariant derivative acting on a [[Clifform]] such as the curvature, transforming under a Clifford rotation, $\ff{F'} = U \ff{F} U^-$, is still
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{\om} \times \ff{F}
$$
We can also define the [[Clifford covariant derivative]] by [[Cliffordization]], $\na = \ve{e} \f{\na}$, which can be used to help reformulate expressions involving forms to expressions using Clifford elements.
Clifford vector bundles or graded Clifford bundles may alternatively be defined as [[automorphism bundle]]s -- for which outer automorphisms may prove interesting.
The ''Clifford-Ricci curvature'' is a [[Clifform]] obtained by taking the [[cross product|Clifford algebra]] of the [[frame]] with the [[Clifford bundle]] curvature,
$$
\f{R} = \ve{e} \times \ff{F}
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford-Ricci curvature is then a Clifford vector valued 1-form,
$$
\f{R} = \f{dx^i} R_i{}^\al \ga_\al = \ve{e} \times \ff{R} = \ve{e} \times \lp \f{d} \f{\om} + \ha \f{\om} \times \f{\om} \rp
$$
with coefficients equal to those of the [[Ricci curvature]], $R_i{}^\al = \lp e_\be \rp^j R_{ji}{}^{\be \al} = \et^{\al \be} R_{i \be}$.
The ''Clifford curvature'' is a [[Clifform]] describing the [[curvature]] of a [[Clifford bundle]],
$$
\ff{F} = \f{d} \f{A} + \ha \f{A} \times \f{A}
$$
If, specifically, we are working with the [[Clifford vector bundle]], the Clifford curvature is then the ''Clifford-Riemann curvature'', a Clifford bivector valued 2-form calculated from the [[spin connection]],
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \times \f{\om} = \f{dx^i} \f{dx^j} \fr{1}{4} R_{ij}{}^{\al \be} \ga_{\al \be}
$$
$$
R_{ij}{}^{\al \be} = 2 \pa_{\lb i \rd} \om_{\ld j \rb}{}^{\al \be} + 2 \om_{\lb i \rd}{}^\al{}_\ga \om_{\ld j \rb}{}^{\ga \be}
$$
with coefficients equal to those of the [[Riemann curvature]], $R_{ij}{}^{\al\be}$, when the [[tangent bundle connection]] and spin connection coefficients are identified, $\f{w}^{\al\be}=\f{\om}^{\al\be}$.
Physically, at every [[manifold]] point, the [[frame]] encodes a map from [[tangent vector]]s to vectors in a [[rest frame]]. Using the [[Clifford vector]] valued coframe, $\f{e}$, any (unitless) tangent vector, $\ve{v}$, on the manifold may be ''Cliffordized'' to its associated (temporal, $[v]=T$) Clifford vector, $v = \ve{v} \f{e}$, via [[vector-form algebra]],
$$
\ve{v} = v^\al \ve{e_\al} \;\;\; \leftrightarrow \;\;\; v = \ve{v} \f{e} = v^i \ve{\pa_i} \f{dx^j} \lp e_j \rp^\al \ga_\al = v^i \lp e_i \rp^\al \ga_\al = v^\al \ga_\al
$$
Similarly, using the frame, $\ve{e}$, any [[differential form]], $\f{f}$, at a point on the manifold may be Cliffordized to its corresponding Clifford vector in a [[rest frame]],
$$
\f{f} = \f{e^\al} f_\al \;\;\; \leftrightarrow \;\;\; f = \ve{e} \f{f} = \ga^\al \lp e_\al \rp^i \ve{\pa_i} \f{dx^j} f_j = \ga^\al \lp e_\al \rp^i f_i = f_\al \ga^\al
$$
Also, differential p-forms may be Cliffordized to Clifford p-vectors, such as $b = \ve{e} \ve{e} \ff{b}$.
A field, $\ve{v}(x)$ or $\f{a}(x)$, that is a section of the [[tangent bundle]] or [[cotangent bundle]], can similarly be Cliffordized to a section, $v(x)$ or $a(x)$, of the associated [[Clifford vector bundle]].
A ''Clifform'' is a [[Clifford algebra]] valued [[differential form]], or, conversely, a [[Clifford element]] with form valued coefficients. A Clifform has a single form grade, $p$, but may consist of pieces with different Clifford grades. In terms of [[coordinate basis forms]] and [[Clifford basis elements]], an arbitrary Clifform may be written as
$$
\nf{A} = \f{dx^i} \dots \f{dx^k} \fr{1}{p!} \lp A_{i \dots k}{}^s + A_{i \dots k}{}^\al \ga_\al + \ha A_{i \dots k}{}^{\al \be} \ga_{\al \be} + \fr{1}{3!} A_{i \dots k}{}^{\al \be \ga} \ga_{\al \be \ga} + \dots + A_{i \dots k}{}^p \ga \rp
$$
For example, a bivector 2-form is written (using the coordinate or [[frame]] basis forms) as
$$
\ff{R} = \ff{R^2} = \f{dx^i} \f{dx^j} \fr{1}{4} R_{ij}{}^{\al \be} \ga_{\al \be}
= \f{e^\ga} \f{e^\de} \fr{1}{4} R_{\ga \de}{}^{\al \be} \ga_{\al \be}
$$
The form elements and Clifford elements act in different algebras. All scalar valued form elements commute with all Clifford basis elements. By convention, the form basis elements will be collected on the left and the Clifford basis elements on the right.
The product of Clifforms may be computed using [[Clifform algebra]]. A Clifform is a [[Lieform]] in which the [[Lie algebra]] generators are Clifford basis elements.
The algebra of [[Clifform]]s is the disjoint union of [[vector-form algebra]] and [[Clifford algebra]]. When performing calculations, it is best to move all [[coordinate basis 1-forms]] to the left of the expression (without commuting them) and all [[Clifford basis elements]] (and the operations between them) to the right. Then the basis contractions and products play out in their independent algebraic sandboxes. Clifford algebra operators like $\cdot$, $\times$, $[,]$, and $<>_q$ do not act on the forms, only on the Clifford basis elements. As an example, the dot product of a bivector (-2)-form and a bivector 2-form is a scalar plus a 4-vector,
\begin{eqnarray}
\vv{L} \cdot \ff{R} &=& \lp \ve{\pa_i} \ve{\pa_j} \fr{1}{4} L^{i j \al \be} \ga_{\al \be} \rp \cdot \lp \f{dx^k} \f{dx^m} \fr{1}{4} R_{km}{}^{\ga \de} \ga_{\ga \de} \rp\\
&=& \lp \ve{\pa_i} \ve{\pa_j} \lp \f{dx^k} \f{dx^m} \rp \rp \fr{1}{16} L^{i j \al \be} R_{km}{}^{\ga \de} \lp \ga_{\al \be} \cdot \ga_{\ga \de} \rp\\
&=& \lp - 2 \de_i^{\lb k \rd} \delta_j^{\ld m \rb} \rp \fr{1}{16} L^{i j \al \be} R_{km}{}^{\ga \de} \lp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de} \rp\\
&=& - \fr{1}{8} L^{i j \al \be} R_{ij}{}^{\ga \de} \lp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp + \ga_{\al \be \ga \de} \rp\\
&=& \fr{1}{4} L^{i j \al \be} R_{ij \al \be} - \fr{1}{8} L^{i j \al \be} R_{ij}{}^{\ga \de} \ga_{\al \be \ga \de}\\
\end{eqnarray}
using vector-form algebra and [[Clifford basis identities]].
Clifform product identities can be inferred from the identities of the two respective algebras. For example, since 1-forms anti-commute,
\[ \f{A} \ti \f{B} = \ha \lp \f{A} \f{B} + \f{B} \f{A} \rp = \f{B} \ti \f{A} \]
Some useful identities can be computed using the [[frame]]. For example, for any Clifford vector valued 2-form, $\ff{f}$,
\begin{eqnarray}
\ve{e} \ti \ff{f} & = & -\lp \ve{e} \ti \ve{e} \rp \lp \f{e} \cdot \ff{f} \rp + \ve{e} \ti \lp \lp \ve{e} \ti \ff{f} \rp \ti \f{e} \rp\\
\ff{f} & = & \lp \ve{e} \ti \ff{f} \rp \ti \f{e} - \ve{e} \cdot \lp \f{e} \cdot \ff{f} \rp\\
\lp n-2 \rp \ff{f} & = & \ve{e} \ti \lp \f{e} \ti \ff{f} \rp - \f{e} \cdot \lp \ve{e} \cdot \ff{f} \rp
\end{eqnarray}
(//add identities as needed//)
The [[Coleman-Mandula theorem|http://prola.aps.org/abstract/PR/v159/i5/p1251_1]] states:
<<<
Let G be a connected symmetry group of the S matrix, and let the following five conditions hold: (1) G contains a subgroup locally isomorphic to the Poincare group. (2) For any M>0, there are only a finite number of one-particle states with mass less than M. (3) Elastic scattering amplitudes are analytic functions of s and t, in some neighborhood of the physical region. (4) The S matrix is nontrivial in the sense that any two one-particle momentum eigenstates scatter (into something), except perhaps at isolated values of s. (5) The generators of G, written as integral operators in momentum space, have distributions for their kernels. Then, we show that G is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincare group.
<<<
The E8 theory proposed in [[An Exceptionally Simple Theory of Everything]] avoids condition (1) of this theorem because $G = E8$ does not containing a subgroup locally isomorphic to the Poincare group. The expected vacuum spacetime of E8 theory is [[de Sitter spacetime]], which has $SO(4,1)$ as symmetry group, which is nearly, but not, the Poincare group. At low energies the deviation from the Poincare group is infinitesimally small, and the Coleman-Mandula theorem applies to a good approximation, with gravity separate from the other symmetries.
Also, more fundamentally, before E8 symmetry breaking there is no spacetime. Only after symmetry breaking is there a [[frame]] and [[spacetime]], and no mixing with gauge groups, and thus the theorem is satisfied.
Ref:
*R. Percacci
**[[Mixing internal and spacetime transformations: some examples and counterexamples|http://arxiv.org/abs/0803.0303]]
*K. Cahill
**[[On the unification of the gravitational and electronuclear forces|papers/Cahill - On the unification of the gravitational and electronuclear forces.pdf]]
*** Phys. Rev. D 26, 1916 - 1922 (1982).
*T. Love
**The Geometry of Grand Unification
***Int. J. Th. Phys., 801 (1984).
*F. Nesti and R. Percacci
**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]
*S. Alexander
**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
/***
|Name|CollapseTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#CollapseTiddlersPlugin|
|Version|2.0.0|
|Author|Eric Shulman|
|OriginalAuthor|Bradley Meck - http://gensoft.revhost.net/Collapse.html|
|License|unknown|
|~CoreVersion|2.1|
|Type|plugin|
|Requires|CollapsedTemplate|
|Description|show/hide content of a tiddler while leaving tiddler title visible|
This plugin provides commands to quickly switch a rendered tiddler between its current ViewTemplate display and a minimal display (title and toolbar) defined by a separate CollapsedTemplate.
!!!Usage
<<<
In [[ToolbarCommands::ViewToolbar|ToolbarCommands]], add:
{{{
collapseTiddler collapseOthers
}}}
you can also embed the following macros in tiddler content:
*{{{<<collapseAll>>}}} - adds 'collapse all' command that applies CollapsedTemplate to each displayed tiddler
*{{{<<expandAll>>}}} - adds 'expand all' command that re-applies ViewTemplate (or equivalent custom template) to each displayed tiddler
*{{{<<foldFirst>>}}} - immediately apply CollapsedTemplate to a given tiddler, as soon as it is displayed.
<<<
!!!Revisions
<<<
2009.05.04 [2.0.0] standardized documentation and added version #
2008.10.05 collapseAll() and expandAll(): added "return false" to button handlers to prevent IE page transition
2008.03.06 refactored all code for size reduction, readability, and I18N/L10N-readiness. Also added 'folded' flag to tiddler elements (for use by other plugins that need to know if tiddler is folded (e.g., [[SinglePageModePlugin]]
2007.10.11 moved [[FoldFirst]] inline script and converted to {{{<<foldFirst>>}}} macro
2007.12.09 suspend/resume SinglePageMode (SPM/TPM/BPM) when folding/unfolding tiddlers
2007.05.06 add "return false" at the end of each command handler to prevent IE 'page transition' problem.
2007.03.30 add a shadow definition for CollapsedTemplate. Tweak ViewTemplate shadow so "fold/unfold" and "focus" toolbar items automatically appear when using default templates. Remove error check for "CollapsedTemplate" existence, since shadow version will now always work as a fallback.
2006.02.24 added fallback to "CollapsedTemplate" if "WebCollapsedTemplate" is not found
2006.02.06 added check for 'readOnly' flag to use alternative "WebCollapsedTemplate"
<<<
!!!Code
***/
//{{{
version.extensions.CollapseTiddlersPlugin= {major: 2, minor: 0, revision: 0, date: new Date(2009,5,4)};
config.commands.collapseTiddler = {
text: "-",
tooltip: "Collapse this note",
collapsedTemplate: "CollapsedTemplate",
webCollapsedTemplate: "WebCollapsedTemplate",
handler: function(event,src,title) {
var e = story.findContainingTiddler(src); if (!e) return false;
// don't fold tiddlers that are being edited!
if(story.isDirty(e.getAttribute("tiddler"))) return false;
var t=config.commands.collapseTiddler.getCollapsedTemplate();
config.commands.collapseTiddler.saveTemplate(e);
config.commands.collapseTiddler.display(title,t);
e.setAttribute("folded","true");
return false;
},
getCollapsedTemplate: function() {
if (readOnly&&store.tiddlerExists(this.webCollapsedTemplate))
return this.webCollapsedTemplate;
else
return this.collapsedTemplate
},
saveTemplate: function(e) {
if (e.getAttribute("savedTemplate")==undefined)
e.setAttribute("savedTemplate",e.getAttribute("template"));
},
// fold/unfold tiddler with suspend/resume of single/top/bottom-of-page mode
display: function(title,t) {
var opt=config.options;
var saveSPM=opt.chkSinglePageMode; opt.chkSinglePageMode=false;
var saveTPM=opt.chkTopOfPageMode; opt.chkTopOfPageMode=false;
var saveBPM=opt.chkBottomOfPageMode; opt.chkBottomOfPageMode=false;
story.displayTiddler(null,title,t);
opt.chkBottomOfPageMode=saveBPM;
opt.chkTopOfPageMode=saveTPM;
opt.chkSinglePageMode=saveSPM;
}
}
config.commands.expandTiddler = {
text: " | ",
tooltip: "Expand this note",
handler: function(event,src,title) {
var e = story.findContainingTiddler(src); if (!e) return false;
var t = e.getAttribute("savedTemplate");
config.commands.collapseTiddler.display(title,t);
e.setAttribute("folded","false");
return false;
}
}
config.macros.collapseAll = {
text: "-",
tooltip: "Collapse all notes",
handler: function(place,macroName,params,wikifier,paramString,tiddler){
createTiddlyButton(place,this.text,this.tooltip,function(){
story.forEachTiddler(function(title,tiddler){
if(story.isDirty(title)) return;
var t=config.commands.collapseTiddler.getCollapsedTemplate();
config.commands.collapseTiddler.saveTemplate(tiddler);
config.commands.collapseTiddler.display(title,t);
tiddler.folded=true;
});
return false;
})
}
}
config.macros.expandAll = {
text: " | ",
tooltip: "Expand all notes",
handler: function(place,macroName,params,wikifier,paramString,tiddler){
createTiddlyButton(place,this.text,this.tooltip,function(){
story.forEachTiddler(function(title,tiddler){
var t=config.commands.collapseTiddler.getCollapsedTemplate();
if(tiddler.getAttribute("template")!=t) return; // re-display only if collapsed
var t=tiddler.getAttribute("savedTemplate");
config.commands.collapseTiddler.display(title,t);
tiddler.folded=false;
});
return false;
})
}
}
config.commands.collapseOthers = {
text: "\xD8",
tooltip: "Expand this note and collapse all others",
handler: function(event,src,title) {
var e = story.findContainingTiddler(src); if (!e) return false;
story.forEachTiddler(function(title,tiddler) {
if(story.isDirty(title)) return;
var t=config.commands.collapseTiddler.getCollapsedTemplate();
if (e==tiddler) t=e.getAttribute("savedTemplate");
config.commands.collapseTiddler.saveTemplate(tiddler);
config.commands.collapseTiddler.display(title,t);
tiddler.folded=(e!=tiddler);
})
return false;
}
}
// {{{<<foldFirst>>}}} macro forces tiddler to be folded when *initially* displayed.
// Subsequent re-render does NOT re-fold tiddler, but closing/re-opening tiddler DOES cause it to fold first again.
config.macros.foldFirst = {
handler: function(place,macroName,params,wikifier,paramString,tiddler){
var e=story.findContainingTiddler(place);
if (e.getAttribute("foldedFirst")=="true") return; // already been folded once
var title=e.getAttribute("tiddler")
var t=config.commands.collapseTiddler.getCollapsedTemplate();
config.commands.collapseTiddler.saveTemplate(e);
config.commands.collapseTiddler.display(title,t);
e.setAttribute("folded","true");
e.setAttribute("foldedFirst","true"); // only when tiddler is first rendered
return false;
}
}
//}}}
<!--{{{-->
<div>
<span class='toolbar' macro='toolbar +editTiddler expandTiddler collapseOthers closeOthers -closeTiddler'></span>
<span class='title' macro='view title'></span>
</div>
<!--}}}-->
This site is powered by [[TiddlyWiki|https://classic.tiddlywiki.com/]] classic <<version>>
Except for the raw html tweakage, these can all be installed using [[ImportTiddlers]].
!Install these plugins:
*[[InlineJavascriptPlugin]]
**used for the [[DisplayControl]]
**and for [[HideTags]] (used for slides)
*[[MathJaxPlugin]]
**this processes LaTeX.
**insert custom LaTeX command abbreviations into plugin.
**install MathJax TeX otf fonts locally, to ~/Library/Fonts
*[[CollapsePlugin]]
**add symbols
**[[CollapsedTemplate]]
*[[RearrangeTiddlersPlugin]]
*[[ListTaggedPlugin]]
**used for folder/tag listings
*[[AllTagsExceptPlugin]]
**use advanced checkbox to see system tags
*[[CopyTiddlerPlugin]]
**add symbol
*[[DisableWikiLinksPlugin]]
**remove checkboxes and set to always disable
*[[FaviconPlugin]]
*[[ReferencesPlugin]]
*[[RecentPlugin]]
**set to show last 3
Check to make sure didn't install any<<tag plugin>>and forget to list it here. Try using the [[PluginManager]].
!Change these tiddlers to configure operation and appearance:
*These control the content of several boxes:
**[[SiteTitle]]
**[[SiteSubtitle]]
**[[WindowTitle]]
**[[SiteUrl]]
**[[DefaultTiddlers]]
**[[MainMenu]]
**[[SideBarOptions]]
**[[OptionsPanel]]
**[[SideBarTabs]]
***[[TabContents]]
***[[TabTags]]
**[[DisplayControl]]
*These are css layout templates:
**[[PageTemplate]]
**[[ViewTemplate]]
**[[EditTemplate]]
**[[CollapsedTemplate]]
*This is for slides
**[[HideTags]]
*And these change the system and css options:
**[[SystemConfig]]
**[[StyleSheet]]
***Trouble with [[MyColors]] conflicting with [[ColorPalette]]?
**[[StyleSheetPrint]]
The default config files are invisible and listed as [[ShadowTiddlers]]. These:
*[[StyleSheetLayout]]
*[[StyleSheetColors]]
are augmented and overriden by the [[StyleSheet]]. If they change in the future, with updates, the old version content will likely have to be added to the new [[StyleSheet]].
!Evil raw html/javascript TW source code tweakage
*edit cookie options, since setting them in [[SystemConfig]] overrides user cookies
*switch line order in {{{config.macros.search.handler}}} for search button after search field
*comment out a couple of displayMessage s?
*comment out tag prompt line in {{{config.macros.tags.handler}}}?
Put TiddlySaver.jar in dg directory and .java.policy in home directory so can save locally
http://arxiv.org/abs/gr-qc/0603062
*Concise treatment of Hamiltonian formulation of GR with a conformal factor.
*uses metric instead of frame
<<tiddler HideTags>><html>
<table class="gtable">
<tr>
<td>
<table class="gtable">
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td>
<img SRC="talks/Cate2010/Connection field.png" width=300>
</tr></td>
</table>
</td>
<td>
</td>
<td>
<table class="gtable">
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td>
<img SRC="talks/Cate2010/EM field.png" width=300>
</tr></td>
</table>
</td>
</tr>
</table>
</html>
[[Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity|papers/9403058.pdf]]
Authors: Sean M. Carroll, George B. Field
We discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments.
/***
|Name|CopyTiddlerPlugin|
|Source|http://www.TiddlyTools.com/#CopyTiddlerPlugin|
|Version|3.2.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.3|
|Type|plugin|
|Description|Quickly create a copy of any existing tiddler|
!!!Usage
<<<
The plugin automatically updates the default (shadow) ToolbarCommands definitions to insert the ''copyTiddler'' command, which will appear as ''copy'' when a tiddler is rendered. If you are already using customized toolbar definitions, you will need to manually add the ''copyTiddler'' toolbar command to your existing ToolbarCommands tiddler, e.g.:
{{{
|EditToolbar|... copyTiddler ... |
}}}
When the ''copy'' command is selected, a new tiddler is created containing an exact copy of the current text/tags/fields, using a title of "{{{TiddlerName (n)}}}", where ''(n)'' is the next available number (starting with 1, of course). If you copy while //editing// a tiddler, the current values displayed in the editor are used (including any changes you may have already made to those values), and the new tiddler is immediately opened for editing.
The plugin also provides a macro that allows you to embed a ''copy'' command directly in specific tiddler content:
{{{
<<copyTiddler TidderName label:"..." prompt:"...">>
}}}
where
* ''TiddlerName'' (optional)<br>specifies the //source// tiddler to be copied. If omitted, the current containing tiddler (if any) will be copied.
* ''label:"..."'' (optional)<br>specifies text to use for the embedded link (default="copy TiddlerName")
* ''prompt:"..."'' (optional)<br>specifies mouseover 'tooltip' help text for link
//Note: to use non-default label/prompt values with the current containing tiddler, use "" for the TiddlerName//
<<<
!!!Configuration
<<<
<<option chkCopyTiddlerDate>> use date/time from existing tiddler (otherwise, use current date/time)
{{{<<option chkCopyTiddlerDate>>}}}
<<<
!!!Revisions
<<<
2010.11.30 3.2.6 use story.getTiddler()
2009.06.08 3.2.5 added option to use timestamp from source tiddler
2009.03.09 3.2.4 fixed IE-specific syntax error
2009.03.02 3.2.3 refactored code (again) to restore use of config.commands.copyTiddler.* custom settings
2009.02.13 3.2.2 in click(), fix calls to displayTiddler() to use current tiddlerElem and use getTiddlerText() to permit copying of shadow tiddler content
2009.01.30 3.2.1 fixed handling for copying field values when in edit mode
2009.01.23 3.2.0 refactored code and added {{{<<copyTiddler TiddlerName>>}}} macro
2008.12.18 3.1.4 corrected code for finding next (n) value when 'sparse' handling is in effect
2008.11.14 3.1.3 added optional 'sparse' setting (avoids 'filling in' missing numbers that may have been previously deleted)
2008.11.14 3.1.2 added optional 'zeroPad' setting
2008.11.14 3.1.1 moved hard-coded '(n)' regex into 'suffixPattern' object property so it can be customized
2008.09.26 3.1.0 changed new title generation to use '(n)' suffix instead of 'Copy of' prefix
2008.05.20 3.0.3 in handler, when copying from VIEW mode, create duplicate array from existing tags array before saving new tiddler.
2007.12.19 3.0.2 in handler, when copying from VIEW mode, duplicate custom fields before saving new tiddler.
2007.09.26 3.0.1 in handler, use findContainingTiddler(src) to get tiddlerElem (and title). Allows 'copy' command to find correct tiddler when transcluded using {{{<<tiddler>>}}} macro or enhanced toolbar inclusion (see [[CoreTweaks]])
2007.06.28 3.0.0 complete re-write to handle custom fields and alternative view/edit templates
2007.05.17 2.1.2 use store.getTiddlerText() to retrieve tiddler content, so that SHADOW tiddlers can be copied correctly when in VIEW mode
2007.04.01 2.1.1 in copyTiddler.handler(), fix check for editor fields by ensuring that found field actually has edit=='text' attribute
2007.02.05 2.1.0 in copyTiddler.handler(), if editor fields (textfield and/or tagsfield) can't be found (i.e., tiddler is in VIEW mode, not EDIT mode), then get text/tags values from stored tiddler instead of active editor fields. Allows use of COPY toolbar directly from VIEW mode
2006.12.12 2.0.0 completely rewritten so plugin just creates a new tiddler EDITOR with a copy of the current tiddler EDITOR contents, instead of creating the new tiddler in the STORE by copying the current tiddler values from the STORE.
2005.xx.xx 1.0.0 original version by Tim Morgan
<<<
!!!Code
***/
//{{{
version.extensions.CopyTiddlerPlugin= {major: 3, minor: 2, revision: 6, date: new Date(2010,11,30)};
// automatically tweak shadow EditTemplate to add 'copyTiddler' toolbar command (following 'cancelTiddler')
config.shadowTiddlers.ToolbarCommands=config.shadowTiddlers.ToolbarCommands.replace(/cancelTiddler/,'cancelTiddler copyTiddler');
if (config.options.chkCopyTiddlerDate===undefined) config.options.chkCopyTiddlerDate=false;
config.commands.copyTiddler = {
text: '\xA9',
hideReadOnly: true,
tooltip: 'Make a copy of this note',
notitle: 'this note',
prefix: '',
suffixText: ' (%0)',
suffixPattern: / \(([0-9]+)\)$/,
zeroPad: 0,
sparse: false,
handler: function(event,src,title)
{ return config.commands.copyTiddler.click(src,event); },
click: function(here,ev) {
var tiddlerElem=story.findContainingTiddler(here);
var template=tiddlerElem?tiddlerElem.getAttribute('template'):null;
var title=here.getAttribute('from');
if (!title || !title.length) {
if (!tiddlerElem) return false;
else title=tiddlerElem.getAttribute('tiddler');
}
var root=title.replace(this.suffixPattern,''); // title without suffix
// find last matching title
var last=title;
if (this.sparse) { // don't fill-in holes... really find LAST matching title
var tids=store.getTiddlers('title','excludeLists');
for (var t=0; t<tids.length; t++) if (tids[t].title.startsWith(root)) last=tids[t].title;
}
// get next number (increment from last matching title)
var n=1; var match=this.suffixPattern.exec(last); if (match) n=parseInt(match[1])+1;
var newTitle=this.prefix+root+this.suffixText.format([String.zeroPad(n,this.zeroPad)]);
// if not sparse mode, find the next hole to fill in...
while (store.tiddlerExists(newTitle)||story.getTiddler(newTitle))
{ n++; newTitle=this.prefix+root+this.suffixText.format([String.zeroPad(n,this.zeroPad)]); }
if (!story.isDirty(title)) { // if tiddler is not being EDITED
// duplicate stored tiddler (if any)
var text=store.getTiddlerText(title,'');
var who=config.options.txtUserName;
var when=new Date();
var newtags=[]; var newfields={};
var tid=store.getTiddler(title); if (tid) {
if (config.options.chkCopyTiddlerDate) var when=tid.modified;
for (var t=0; t<tid.tags.length; t++) newtags.push(tid.tags[t]);
store.forEachField(tid,function(t,f,v){newfields[f]=v;},true);
}
store.saveTiddler(newTitle,newTitle,text,who,when,newtags,newfields,true);
story.displayTiddler(tiddlerElem,newTitle,template);
} else {
story.displayTiddler(tiddlerElem,newTitle,template);
var fields=config.commands.copyTiddler.gatherFields(tiddlerElem); // get current editor fields
var newTiddlerElem=story.getTiddler(newTitle);
for (var f=0; f<fields.length; f++) { // set fields in new editor
if (fields[f].name=='title') fields[f].value=newTitle; // rename title in new tiddler
var fieldElem=config.commands.copyTiddler.findField(newTiddlerElem,fields[f].name);
if (fieldElem) {
if (fieldElem.getAttribute('type')=='checkbox')
fieldElem.checked=fields[f].value;
else
fieldElem.value=fields[f].value;
}
}
}
story.focusTiddler(newTitle,'title');
return false;
},
findField: function(tiddlerElem,field) {
var inputs=tiddlerElem.getElementsByTagName('input');
for (var i=0; i<inputs.length; i++) {
if (inputs[i].getAttribute('type')=='checkbox' && inputs[i].field == field) return inputs[i];
if (inputs[i].getAttribute('type')=='text' && inputs[i].getAttribute('edit') == field) return inputs[i];
}
var tas=tiddlerElem.getElementsByTagName('textarea');
for (var i=0; i<tas.length; i++) if (tas[i].getAttribute('edit') == field) return tas[i];
var sels=tiddlerElem.getElementsByTagName('select');
for (var i=0; i<sels.length; i++) if (sels[i].getAttribute('edit') == field) return sels[i];
return null;
},
gatherFields: function(tiddlerElem) { // get field names and values from current tiddler editor
var fields=[];
// get checkboxes and edit fields
var inputs=tiddlerElem.getElementsByTagName('input');
for (var i=0; i<inputs.length; i++) {
if (inputs[i].getAttribute('type')=='checkbox')
if (inputs[i].field) fields.push({name:inputs[i].field,value:inputs[i].checked});
if (inputs[i].getAttribute('type')=='text')
if (inputs[i].getAttribute('edit')) fields.push({name:inputs[i].getAttribute('edit'),value:inputs[i].value});
}
// get textareas (multi-line edit fields)
var tas=tiddlerElem.getElementsByTagName('textarea');
for (var i=0; i<tas.length; i++)
if (tas[i].getAttribute('edit')) fields.push({name:tas[i].getAttribute('edit'),value:tas[i].value});
// get selection lists (droplist or listbox)
var sels=tiddlerElem.getElementsByTagName('select');
for (var i=0; i<sels.length; i++)
if (sels[i].getAttribute('edit')) fields.push({name:sels[i].getAttribute('edit'),value:sels[i].value});
return fields;
}
};
//}}}
// // MACRO DEFINITION
//{{{
config.macros.copyTiddler = {
label: 'copy',
prompt: 'Make a copy of %0',
handler: function(place,macroName,params,wikifier,paramString,tiddler) {
var title=params.shift();
params=paramString.parseParams('anon',null,true,false,false);
var label =getParam(params,'label',this.label+(title?' '+title:''));
var prompt =getParam(params,'prompt',this.prompt).format([title||this.notitle]);
var b=createTiddlyButton(place,label,prompt,
function(ev){return config.commands.copyTiddler.click(this,ev)});
b.setAttribute('from',title||'');
}
};
//}}}
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/TED08/images/Coral_reef_s620.JPG" width="827" height="620"></embed></center></html>@@
<<tiddler HideTags>>$$
\begin{array}{rcll}
\ff{F} \!\!&\!\!=\!\!&\!\! \f{d} \f{H} + \f{H} \f{H}
\s\s\; \f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W}
\\
\!\!&\!\!=\!\!&\!\! \Big( \ha ( \f{d} \f{\om} + \ha \f{\om} \f{\om} ) + \fr{1}{16} M^2 \f{e} \f{e} \Big)_{\p{(}}
\!&\!\! \leftarrow \text{spacetime} \; \ga_{\mu\nu} \\
&&\!\!\! + \Big( \fr{1}{4} \big( \f{d} \f{e} + \ha [ \f{\om}, \f{e} ] \big) \ph - \fr{1}{4} \f{e} \big( \f{d} \ph + [ \f{B} \!+\! \f{W}, \ph ] \big) \Big)_{\p{(}}
\!&\!\! \leftarrow \text{mixed} \; \ga_{\mu\ph} \\
&&\!\!\! + \Big( \f{d} \f{B} + \f{d} \f{W} + \f{W} \f{W} \Big)_{\p{\big(}}
\!&\!\! \leftarrow \text{higher} \; \ga_{\ph\ps} \\
\!\!&\!\!=\!\!&\!\! \ha \big( \ff{R} + \fr{1}{8} M^2 \f{e} \f{e} \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \big( \ff{F_B} + \ff{F_W} \big) \\
\!\!&\!\!=\!\!&\!\! \ff{F_s} + \ff{F_m} + \ff{F_h}
\end{array}
$$
Modified BF action over 4D base [[manifold]]:
\begin{eqnarray}
S &=& \int \big< \ff{B} \, \ff{F} + \Ph(\f{H},\ff{B}) \big>
= \int \big< \ff{B} \, \ff{F} - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga + \ff{B_m} \ff{*B_m} + \ff{B_h} \ff{*B_h} \big> \\
&=& \int \big< \ff{F_s} \, \ff{F_s} \ga^- + {\scriptsize \frac{1}{4}} \ff{F_m} \ff{*F_m} + {\scriptsize \frac{1}{4}} \ff{F_h} \ff{*F_h} \big>
\end{eqnarray}
New paper. How to go from a higher dimensional gauge theory, with Chern Simons or Born Infeld action, to Einstein gravity in 4D:
*[[D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction|papers/0703034.pdf]]
The kinetic [[Lagrangian density|action]] for a [[Dirac spinor]] field in curved [[spacetime]] is
$$
{\cal L}_{\Psi K} = i \bar{\Psi} \ve{e} \lp \f{d} + \ha \f{\om} \rp \Psi
$$
in which $\bar{\Psi}$ is the [[Dirac adjoint]]. //(that's not necessarily real...but maybe it is, up to a divergence term?)// Using the [[chiral]] representation for the [[Cl(1,3)]] [[Dirac matrices]], the [[spacetime frame]] and [[spacetime spin connection]] break up to give
\begin{eqnarray}
{\cal L}_{\Psi K} &=& i \lb \psi_L^\da \;\; \psi_R^\da \rb
\lb \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \rb
\lb \begin{array}{cc}
0 & \ve{e}_R \\
\ve{e}_L & 0
\end{array} \rb
\lp \f{\pa} +
\ha
\lb \begin{array}{cc}
\f{\om}{}_L & 0 \\
0 & \f{\om}{}_R
\end{array} \rb
\rp
\lb \begin{array}{c}
\psi_L \\
\psi_R
\end{array} \rb
\\
&=& i \psi_L^\da \ve{e}_L \lp \f{\pa} + \ha \f{\om}{}_L \rp \psi_L + i \psi_R^\da \ve{e}_R \lp \f{\pa} + \ha \f{\om}{}_R \rp \psi_R
\end{eqnarray}
in terms of the partnered [[Weyl spinor]]s. The full Dirac Lagrangian, including interaction with Higgs and gauge fields, can be written using the [[massive Dirac operator|Dirac operator]] as
$$
{\cal L}_{\Psi} = i \bar{\Psi} \not{\!\!D} \Psi
$$
The Lagrangian density corresponds to a 4-form over [[spacetime]],
$$
\nf{{\cal L}_{\Psi}} = \nf{e} \, {\cal L}_{\Psi}
$$
using the [[volume form]]. This 4-form in curved spacetime can also be expressed using volume form [[permutation identities]] and the [[massive covariant derivative|Dirac operator]] as
$$
\nf{{\cal L}_{\Psi}} = i \bar{\Psi} \nf{e} \ve{e} \f{D} \Psi
= \fr{i}{3!} \bar{\Psi} \, \ga^- \f{e} \f{e} \f{e} \f{D} \Psi
$$
or something like that.
Adding mass and [[gauge field|principal bundle]] interactions, the full Lagrangian density for a Dirac spinor field in curved spacetime is
$$
\ba{rcl}
{\cal L}_{\Psi} \ae i \bar{\Psi} \ve{e} \lp \f{d} + \ha \f{\om} + \fr{i}{4} \f{e} m + \f{A} \rp \Psi \\
\ae \bar{\Psi} \lp i \ga^\mu (e_\mu)^i \lp \pa_i + \ha \om_i + A_i \rp - m \rp \Psi \\
\ea
$$
In flat spacetime this is
\begin{eqnarray}
{\cal L}_{\Psi} &=& \bar{\Psi} \lp i \ga^\mu \pa_\mu + i \ga^\mu A_\mu - m \rp \Psi \\
&=& i \ps_L^\da \si^\mu (\pa_\mu + A_\mu) \ps_L + i \ps_R^\da \bar{\si}^\mu (\pa_\mu + A_\mu) \ps_R - m \lp \ps_R^\da \ps_L + \ps_L^\da \ps_R \rp
\end{eqnarray}
using [[Pauli matrices]].
The [[Dirac Lagrangian]] density in curved spacetime,
$$
{\cal L}_{\Ps} = i \bar{\Ps} \ve{e} \lp \f{d} + \ha \f{\om} + \fr{i}{4} \f{e} m + \f{A} \rp \Ps
$$
is invariant under any [[local Clifford rotation]]. This is evident by using the transformed fields to get
$$
{\cal L}_{\Ps} \to {\cal L}_{\Ps}' = i \bar{\Ps} U^- U \ve{e} \, U^- \lp \f{d} + \ha U \f{\om} \, U^- + U \f{d} \, U^- + \fr{i}{4} U \f{e} \, U^- m + \f{A} \rp U \Ps = {\cal L}_{\Ps}
$$
The [[Dirac Lagrangian]] density in curved spacetime,
$$
{\cal L}_{\Ps} = i \bar{\Ps} \ve{e} \lp \f{d} + \ha \f{\om} + \fr{i}{4} \f{e} m + \f{A} \rp \Ps
$$
is covariant under auto[[diffeomorphism]]s, $\ph : x^i \to x'^i = \ph^i(x)$,
$$
\ba{rcl}
{\cal L}_{\Ps}(x) \to {\cal L}'_{\Ps}(x) \ae
i \bar{\Ps}' \ve{e}' \lp \f{d} + \ha \f{\om}' + \fr{i}{4} \f{e}' m + \f{A}' \rp \Ps'(x) \\
\ae i \bar{\Ps}(x'(x)) \ve{e} \f{\ve{L}}^- \lp \f{\ve{L}} \f{d}' + \ha \f{\ve{L}} \f{\om} + \fr{i}{4} \f{\ve{L}} \f{e} m + \f{\ve{L}} \f{A} \rp \Ps(x'(x)) \\
\ae i \bar{\Ps} \ve{e} \lp \f{d}' + \ha \f{\om} + \fr{i}{4} \f{e} m + \f{A} \rp \Ps (x'(x)) = {\cal L}_{\Ps}(x'(x))
\ea
$$
with $\Ps' = \Ps'(x) = \Ps(x'(x))$ and $\f{\om}' = \f{\om}'(x) = \f{\ve{L}} \f{\om}(x'(x))$
<<tiddler HideTags>>
\begin{eqnarray}
L_D &=& \bar{\ps} \ga^\mu \lp e_\mu\rp^i \lp
\pa_i
+ \fr{1}{4} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
\rp \ps
+ \bar{\ps} \ph \ps
\end{eqnarray}
| $\; ( e_\mu )^i \; $ |gravitational [[frame]] components (//tetrad, vierbein//)|
| $\; \ga_\mu \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]] |
| $\; \ga_{\mu\nu} = \ga_\mu \ga_\nu \; $ |[[Clifford bivectors|Clifford basis elements]] |
| $\; \om_i^{\p{i}\nu\rh} \; $ |gravitational [[spin connection]] components |
| $\; T_A \; $ |[[Lie algebra]] generators |
| $\; G_i^{\p{i}A} \; $ |[[Yang-Mills gauge field|principal bundle]] components (//connection//) |
| $\; \ph \; $ |Higgs scalar field multiplet |
$$
\begin{array}{rcl}
{\rm Clifford \; algebra} \!\!&\!\! \longleftrightarrow \!\!&\!\! {\rm Lie \; algebra}^{\phantom{(}} \\
\searrow \!\!\!\!\!\! \nwarrow \!\!&\!\! \!\!&\!\! \swarrow \!\!\!\!\!\! \nearrow \\
& {\rm Matrices} &
\end{array}
$$
<<tiddler HideTags>>$$\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big( \pa_i + {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh} + A_i^{\p{i}B} T_B \big) \ps + \bar{\ps} \ph \ps \right\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \left\{ \bar{\ps} \ve{e} \big( \f{\pa} + {\small \frac{1}{2}} \f{\om} + \f{A} \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \right\}
\end{array}$$
| $\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;$ |$\in \f{Cl}^1(1,3)$ |gravitational [[frame]] (//tetrad, vierbein//) |
| $\; \ve{e} = \ga^\mu (e_\mu)^i \ve{\pa_i} \;$ |$\in \ve{Cl}{}^1(1,3)$ |inverse [[frame]] |
| $\; \f{\om} = \f{dx^i} \ha \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;$ |$\in \f{Cl}^2(1,3)$ |[[spin connection]] |
| $\; \f{A} = \f{dx^i}A_i^{\p{i}B} T_B \;$ |$\in G_{SM} = \f{su}(2)_L + \f{u}(1)_Y + \f{su}(3)$ |[[gauge fields|principal bundle]] |
| $\; \ph \; $ |$\in GL(N,\mathbb{C}) \leftarrow \mathbb{C}^2 = 2_L $ |Higgs scalar field multiplet |
| $\; \ud{\ps} \; $ |$\in 2 \!\times\! (2_L\!+\!2_R) \!\times\! (1\!+\!3) $ |Grassmann valued [[Dirac spinor]] field multiplet |
| $\; \bar{\ps} = \ud{\ps}^\da \ga_0 \; $ |$\in 2 \!\times\! (2_R\!+\!2_L) \!\times\! (1\!+\!\bar{3}) $ |conjugate spinor multiplet |
<<tiddler HideTags>>
$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ {\tiny \frac{1}{2}} \om_i^{\p{i}\nu\rh} {\tiny \frac{1}{2}} \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\}
$$
The ''Dirac adjoint'' of a [[Dirac spinor]] field, $\ud{\Ps}$, is the [[conjugate spinor]],
$$
\bar{\Psi} = \od{\Psi}^\da \ga^0
$$
in which $\ga^0$ is the temporal [[Dirac matrix|Dirac matrices]] and the [[Hermitian]] conjugate is the complex conjugate of the transpose, $\bar{\Psi} = \od{\Psi}^{T*}$.
A Dirac spinor contracted with its adjoint is a scalar,
$$
\bar{\Psi} \ud{\Psi} =
\lb \od{\psi}_L^\da \;\; \od{\psi}_R^\da \rb
\lb \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \rb
\lb \begin{array}{c}
\ud{\psi}_L \\
\ud{\psi}_R
\end{array} \rb
=
\od{\psi}_L^\da \ud{\psi}_R + \od{\psi}_R^\da \ud{\psi}_L
$$
using chiral Dirac matrices and partnered [[Weyl spinor]]s. If the fermion fields are [[Grassmann number]] fields this contraction gives a pure imaginary number, since
$$
\lp \od{\psi}_L^\da \ud{\psi}_R + \od{\psi}_R^\da \ud{\psi}_L \rp^\da = - \od{\psi}_R^\da \ud{\psi}_L - \od{\psi}_L^\da \ud{\psi}_R
$$
And this contraction is invariant under spacetime [[Clifford rotation]], due to how a Dirac spinor transforms and how the corresponding exponential of a [[Cl(1,3) bivector]] behaves under the adjoint,
$$
\bar{\Psi}' \ud{\Psi}' = \lp U \od{\Psi} \rp^\da \ga^0 U \ud{\Ps} = \od{\Psi}^\da U^\da \ga^0 U \ud{\Ps}
= \od{\Psi}^\da \ga^0 U^- \ga_0 \ga^0 U \ud{\Ps} = \bar{\Psi} \ud{\Psi}
$$
The ''Dirac delta function'', $\de(x)$, casually defined, is a singular distribution that multiplies an integrand to produce the value of that integrand at a point,
$$
\int{dx} \, f(x) \de(x) = f(0)
$$
This can be defined somewhat more precisely, using a [[Fourier transform]], as
$$
\de(x) = \int{\fr{dp}{2 \pi}} e^{i x p}
$$
in which we see that $\de(x)$ is the inverse Fourier transform of $\de'(p)=1$.
From the [[Dirac Lagrangian]], the ''Dirac equation'' for a [[Dirac spinor]] field in curved [[spacetime]] with background Higgs and gauge fields is
$$
0 = i \not{\!\!D} \Psi = i \ve{e} \, \f{D} \Psi = i \ga^\mu \lp e_\mu \rp^a \lp \pa_a + \fr{1}{4} \om_a{}^{\nu \rh} \ga_{\nu \rh} + G_a{}^B T_B \rp \Psi - \ph_0 \Psi
$$
In a [[rest frame]] in the absence of gauge fields, and with a Higgs background of $\ph_0 = m$, using chiral [[Dirac matrices]], this is
$$
0 = i \ga^\mu \pa_\mu \Psi - m \Psi =
\lb \begin{array}{cc}
-m & i \pa_0 + i \si_\va \pa_\va \\
i \pa_0 - i \si_\va \pa_\va & -m
\end{array} \rb
\lb \begin{array}{c}
\ps_L \\
\ps_R
\end{array} \rb
$$
which is satisfied by [[Dirac solutions]]. For the massless case, the left and right chiral ''Weyl equation''s for the [[Weyl spinor]] parts are
$$
0 = \lp i \pa_0 \mp i \si_\va \pa_\va \rp \psi_{L/R}
$$
or $0=i \si^\mu \pa_\mu \ps_L$ and $0=i \bar{\si}^\mu \pa_\mu \ps_R$ using extended [[Pauli matrices]], and are satisfied by [[Weyl solutions]].
<<tiddler HideTags>>
$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a \lp e_a \rp^\mu \lp \pa_\mu + {\textstyle \fr{1}{4}} \om_\mu^{\p{\mu}ab} \ga_{ab} \rp \ps + i m \, \ps \\[.5em]
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} \rp \ps + i m \, \ps
\end{array}
$$
| $\; \ga_a \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]], rep in $\mathbb{C}(4)$ |
| $\; \ga_{ab} = \ga_a \ga_b \;\;\; a \ne b \; $ |[[Clifford basis bivectors|Clifford basis elements]] of $Cl^2(1,3) = spin(1,3) \;\;$ |
| $\; ( e_a )^\mu \; $ |[[orthonormal basis vector|frame]] components (//vierbein//) |
| $\; \om_\mu^{\p{\mu}ab} \; $ |[[spin connection]] components |
| $\; \ps \; $ |[[Dirac spinor]], $4^\mathbb{C}_S$ |
| $\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;$ |$\in Cl^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^a (e_a)^\mu \ve{\pa_\mu} \;$ |$\in Cl^1(1,3)$ |inverse [[frame]], $\ve{e} \f{e} = 4 \;\;$ |
| $\; \f{\om} = \f{dx^\mu} \ha \om_\mu^{\p{\mu}ab} \ga_{ab} \;$ |$\in Cl^2(1,3) = spin(1,3)$ |[[spin connection]] |
<<tiddler HideTags>>
$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a \lp e_a \rp^\mu \lp \pa_\mu + {\textstyle \fr{1}{4}} \om_\mu^{\p{\mu}ab} \ga_{ab} + A_\mu^{\p{\mu}B} T_B \rp \ps + \ph \, \ps \\[.5em]
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} +\f{A} \rp \ps + \ph \, \ps
\end{array}
$$
| $\; T_B \; $ |[[Lie algebra]] basis elements (//generators//), $\;\; \in \; u(1) \oplus su(2) \oplus su(3) \;\;$ |
| $\; \f{A} = \f{dx^\mu} A_\mu^{\p{\mu}B} T_B \; $ |Yang-Mills [[gauge field|principal bundle]] (//connection//) |
| $\; \ps \; $ |spinor multiplet, $\;\; \in \; 2 \!\otimes\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \;\;\;\;\; (\otimes 3)$ |
| $\; \ph \; $ |Higgs scalar field multiplet (linear operator on $\ps$) |
$$
\begin{array}{rcl}
{\rm Clifford \; algebra} \!\!&\!\! \longleftrightarrow \!\!&\!\! {\rm Lie \; algebra}^{\phantom{(}} & \longleftrightarrow \;\; {\rm Lie \; group} \;\; \longleftrightarrow \;\; {\rm Geometry}\\
\searrow \!\!\!\!\!\! \nwarrow \!\!&\!\! \!\!&\!\! \swarrow \!\!\!\!\!\! \nearrow & \\
& {\rm Matrices} & &
\end{array}
$$
<<tiddler HideTags>>
$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} + \f{A} \rp \ps + \ph \, \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \rp \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \f{H} \rp \ps \\
\!\!&\!\!=\!\!&\!\! \ve{e} \f{D} \, \ps = D \!\!\!\! / \; \ps
\end{array}
$$
| $\; \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;$ |$\in Cl^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^a (e_a)^\mu \ve{\pa_\mu} \;$ |$\in Cl^1(1,3)$ |inverse [[frame]], $\ve{e} \f{e} = 4$ |
| $\; \f{\om} = \f{dx^\mu} \ha \om_\mu^{\p{\mu}ab} \ga_{ab} \;$ |$\in Cl^2(1,3) = spin(1,3)$ |[[spin connection]] |
| $\; \f{A} = \f{dx^\mu}A_\mu^{\p{\mu}B} T_B \;$ |$\in G_{SM} = su(2)_L \oplus u(1)_Y \oplus su(3) \;\;$ |[[gauge connection|principal bundle]] |
| $\; \ps \; $ |$\in 2 \!\otimes\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \;\;\;\;\; (\otimes 3)$ |spinor field multiplet |
| $\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \; $ |$\in \; ?$ |''unified bosonic connection'' |
| $\; \f{e} \ph \;$ |$\in \; ?$ |''frame-Higgs'' |
<<tiddler HideTags>>
$$
0 = \lp \ga^\mu \pa_\mu + i m \rp \ps
$$
$$
\ga^\mu \pa_\mu =
\lb \begin{array}{cccc}
0 & 0 & \pa_0+\pa_3 & \pa_1-i\pa_2 \\
0 & 0 & \pa_1+i\pa_2 & \pa_0-\pa_3 \\
\pa_0-\pa_3 & -\pa_1+i\pa_2 & 0 & 0 \\
-\pa_1-i\pa_2 & \pa_0+\pa_3 & 0 & 0
\end{array} \rb
$$
$$
\ps =
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\in 4^\mathbb{C}
$$
The ''Dirac matrices'' provide a $4\times4$ [[Clifford matrix representation]] of [[Cl(1,3)]] or [[Cl(3,1)]]. There are several standard choices, built from the [[Kronecker product]] of [[Pauli matrices]]:
The ([[chiral]]) ''Weyl representation'' of the Dirac matrices of Cl(1,3) is:
\begin{eqnarray}
\ga_0 &=& \;\;\;\, \si_1 \otimes \si_0 \\
\ga_1 &=& -i \si_2 \otimes \si_1 \\
\ga_2 &=& -i \si_2 \otimes \si_2 \\
\ga_3 &=& -i \si_2 \otimes \si_3
\end{eqnarray}
giving a complex rep for ''Cl(1,3) vectors'',
\begin{eqnarray}
v &=& v^\mu \ga_\mu =
\lb \begin{array}{cc}
0 & v_L \\
v_R & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & v^\mu \bar{\si}_\mu \\
v^\mu \si_\mu & 0
\end{array} \rb
\\
&=&
\lb \begin{array}{cccc}
0 & 0 & v^0-v^3 & -v^1+iv^2 \\
0 & 0 & -v^1-iv^2 & v^0+v^3 \\
v^0+v^3 & v^1-iv^2 & 0 & 0 \\
v^1+iv^2 & v^0-v^3 & 0 & 0
\end{array} \rb
\end{eqnarray}
in which $\bar{\si}_\mu$ are the conjugate [[Pauli matrices]] and the ''left and right chiral vector'' parts,
$$
v_{L/R} = v^0 \si_0 \mp v^\va \si_\va
$$
are $2\times2$ Hermitian matrices projected out by the [[left/right chirality projector]]. Note the curious fact that the [[determinant]] of a chiral vector is its norm, $\det v_{L/R} = v^\mu v_\mu = | v |^2$. The ''spacetime [[pseudoscalar]]'' is
$$
\ga = \ga_0 \ga_1 \ga_2 \ga_3 = - i \si_3 \otimes \si_0 = \lb \begin{array}{cc} - i & 0 \\ 0 & i \end{array} \rb
$$
and these Dirac matrices satisfy $\ga^*_\mu = \ga_2 \ga_\mu \ga_2$ and $\ga_\mu^\da = \ga_0 \ga_\mu \ga_0$.
Alternatively, there is a ''Dirac representation'' of CL(1,3),
\begin{eqnarray}
\ga_0 &=& \;\, - \si_3 \otimes \si_0 \\
\ga_\va &=& -i \, \si_2 \otimes \si_\va
\end{eqnarray}
And there is also a (real) ''Majorana representation'' of Cl(3,1),
\begin{eqnarray}
\ga_0 &=& \;\; i \, \si_1 \otimes \si_2 \\
\ga_1 &=& \;\; \;\; \si_0 \otimes \si_1 \\
\ga_2 &=& \;\; \;\; \si_2 \otimes \si_2 \\
\ga_3 &=& \;\; \;\; \si_0 \otimes \si_3 \\
\ga &=& - i \, \si_3 \otimes \si_2 \\
\end{eqnarray}
Multiplying these matrices by $i$ switches them between representations of Cl(1,3) and Cl(3,1).
The different matrix representations may be related by ''similarity transformation''s. For example, the Majorana rep is given in terms of the Weyl rep by $\ga^M_\mu = U \ga^W_\mu U^\da$, with (recalculate this)
$$
U = \ha
\lb \begin{array}{cccc}
1 & i & 1 & -i \\
i & 1 & i & -1 \\
1 & -i & -1 & -i \\
-i & 1 & i & 1
\end{array} \rb
$$
And the Weyl and Dirac equations are related by $U = e^{i \fr{\pi}{4} \ga^0 \ga}$.
Ref:
http://en.wikipedia.org/wiki/Dirac_matrices
If $\Psi$ is a [[spinor]] field and $\f{A} \in \f{\mathfrak{g}}$ a [[principal bundle]] connection in a representation matched to the spinor, the [[covariant derivative]] of the spinor field is
$$
\f{\na} \Psi = \lp \f{d} + \f{A} \rp \Psi
$$
Note that $\f{A}$ includes the [[spin connection]], $\f{\om}$ (as the connection for the [[Clifford vector bundle]] subbundle of the full principal bundle) and usually other parts, which will be written as $\f{G}$, so
$$
\f{A} = \ha \f{\om} + \f{G}
$$
If we write the [[frame]] over the base manifold as $\ve{e} = \ga^\mu \ve{e_\mu}$, the ''Dirac operator'', $\not{\!\!\na}=\ve{e} \, \f{\na}$ acting on the spinor is defined as
$$
\not{\!\!\na} \Psi = \ve{e} \, \f{\na} \Psi = \ga^\mu \lp e_\mu \rp^i \lp \pa_i + \fr{1}{4} \om_i{}^{\nu \rh} \ga_{\nu \rh} + G_i{}^B T_B \rp \Psi
$$
using the [[vector-form algebra]]. Note that the relation to the [[Clifford covariant derivative]] operator. If there is a background Higgs field, $\ph_0$, interacting with fermions to give them mass, then we can define the ''massive covariant derivative'' in curved spacetime as
$$
\f{D} \Psi = \lp \f{d} + \ha \f{\om} + \fr{1}{4} \f{e} \ph_0 + \f{G} \rp \Psi
$$
and the massive Dirac operator as
$$
\not{\!\!D} \Psi = \ve{e} \, \f{D} \Psi = \ga^\mu \lp e_\mu \rp^i \lp \pa_i + \fr{1}{4} \om_i{}^{\nu \rh} \ga_{\nu \rh} + G_i{}^B T_B \rp \Psi + \ph_0 \Psi
$$
[[Dirac solutions]] interrelate via several identities, related to the [[Pauli matrix|Pauli matrices]] identity, $\si_\va^* = \ep \, \si_\va \ep$, using the [[skew]] and chiral [[Dirac matrices]],
$$
i \ga_2 = \lb \ba{cc} 0 & \ep \\ -\ep & 0 \ea \rb
\s \s
i \ga_0 = \lb \ba{cc} 0 & i \\ i & 0 \ea \rb
\s \s
i \ga_0 \ga = \lb \ba{cc} 0 & -1 \\ 1 & 0 \ea \rb
\s \s
\ga_{13} = \lb \ba{cc} -\ep & 0 \\ 0 &-\ep \ea \rb
\s \s
\ga = \lb \ba{cc} -i & 0 \\ 0 & i \ea \rb
$$
corresponding to [[CPT symmetry]].
Mapping particles to antiparticles involves identities related to [[charge conjugation|charge conjugate]],
$$
i \ga_2 \, u_p^{\wedge/\vee \, *} = v_p^{\wedge/\vee}
\s \s
i \ga_2 \, v_p^{\wedge/\vee \, *} = u_p^{\wedge/\vee}
$$
Reversing the momentum involves identities related to [[parity conjugation|parity conjugate]],
$$
i \ga_0 \, u_{-p}^{\wedge/\vee} = + i \, u_p^{\wedge/\vee}
\s \s
i \ga_0 \, v_{-p}^{\wedge/\vee} = - i \, v_p^{\wedge/\vee}
$$
Reversing the momentum and spin involves identities related to [[time conjugation|time conjugate]],
$$
i \ga_0 \ga \, u_{-p}^{\wedge/\vee} = \pm v_p^{\vee/\wedge}
\s \s
i \ga_0 \ga \, v_{-p}^{\wedge/\vee} = \pm u_p^{\vee/\wedge}
$$
$$
\ga_{13} \, u_{-p}^{\wedge/\vee} = \mp \, u_p^{\vee/\wedge \, *}
\s \s
\ga_{13} \, v_{-p}^{\wedge/\vee} = \mp \, v_p^{\vee/\wedge \, *}
$$
$$
\ga_{13} \, u_{-p}^{\wedge/\vee \, *} = - \, v_p^{\wedge/\vee}
\s \s
\ga_{13} \, v_{-p}^{\wedge/\vee \, *} = + \, u_p^{\wedge/\vee}
$$
The positive and negative energy solutions of different spins can be related by combining all three of these, using the [[CPT group]], to get identities related to [[CPT symmetry]],
$$
\ga \, u_p^{\wedge/\vee} = \pm i v_p^{\vee/\wedge \, *}
\s \s
\ga \, v_p^{\wedge/\vee} = \mp i u_p^{\vee/\wedge \, *}
$$
Using the chiral [[Dirac matrices]], the [[Dirac equation]] has two positive energy, $\Ps = u_p^{\wedge/\vee} e^{- i p_\mu x^\mu}$, and two negative energy, $\Ps = v_p^{\wedge/\vee} e^{+ i p_\mu x^\mu}$, [[Dirac spinor]] basis solutions, satisfying
$$
0 =
\lb \begin{array}{cc}
m & - E + \si_\va p^\va \\
- E - \si_\va p^\va & m
\end{array} \rb
\lb \begin{array}{c}
u^{\wedge/\vee}_L \\
u^{\wedge/\vee}_R
\end{array} \rb
\s \s
0=
\lb \begin{array}{cc}
m & E - \si_\va p^\va \\
E + \si_\va p^\va & m
\end{array} \rb
\lb \begin{array}{c}
v^{\wedge/\vee}_L \\
v^{\wedge/\vee}_R
\end{array} \rb
$$
with solutions
$$
u'{}_p^{\wedge/\vee} =
\lb \begin{array}{c}
m \, \ch^{\wedge/\vee} \\
\lp E + p^\va \si_\va \rp \ch^{\wedge/\vee}
\end {array} \rb
\sim
\lb \begin{array}{c}
\lp E - p^\va \si_\va \rp \ch^{\wedge/\vee} \\
m \, \ch^{\wedge/\vee}
\end {array} \rb
\s\;\;\;
v'{}_p^{\wedge/\vee} =
\lb \begin{array}{c}
\lp E - p^\va \si_\va \rp \xi^{\wedge/\vee} \\
-m \, \xi^{\wedge/\vee}
\end {array} \rb
\sim
\lb \begin{array}{c}
m \, \xi^{\wedge/\vee} \\
- \lp E + p^\va \si_\va \rp \xi^{\wedge/\vee}
\end {array} \rb
$$
in which $p$ is the [[momentum]], $\si_\va$ are the three [[Pauli matrices]], and $\ch^{\wedge/\vee}$ and $\xi^{\wedge/\vee} = \ep \ch^{\wedge/\vee} = \pm \ch^{\vee/\wedge}$ are the up and down [[basis Pauli spinors|Weyl spinor]] and [[flipped spin]]ors. Note that we have chosen $u'{}_p^{\wedge/\vee}$ and $v'{}_p^{\wedge/\vee}$ such that they are oppositely signed eigenvectors of the [[spin operator]], $S_z = -\fr{i}{2} \ga_1\ga_2 = \ha \si_0 \otimes \si_3$, in the rest frame -- specifically, $S_z u'{}_0^{\wedge/\vee} = \pm \ha u'{}_0^{\wedge/\vee}$ and $S_z v'{}_0^{\wedge/\vee} = \mp \ha v'{}_0^{\wedge/\vee}$. And they are [[charge conjugate]]s,
$$
(u'{}_p^{\wedge/\vee})^C = i \ga_2 u'{}_p^{\wedge/\vee \, *} =
\lb \ba{cc} & \ep \\ -\ep & \ea \rb
\lb \ba{c} m \, \ch^{\wedge/\vee} \\ \lp E + p^\va \si_\va^* \rp \ch^{\wedge/\vee} \ea \rb
= \lb \ba{c} - \ep \lp E + p^\va \si_\va^* \rp \ep \ep \ch^{\wedge/\vee} \\ - m \, \ep \ch^{\wedge/\vee} \ea \rb
= \lb \ba{c} \lp E - p^\va \si_\va \rp \xi^{\wedge/\vee} \\ - m \, \xi^{\wedge/\vee} \ea \rb
= v'{}_p^{\wedge/\vee}
$$
These solutions are not normalized, but are well behaved at $m=0$. Normalized solutions, satisfying $\bar{u}u = u^\da \ga_0 u =1$ and $\bar{v} v=-1$ (using the [[Dirac adjoint]]), are [[Lorentz boost]]ed solutions of the rest frame solution, and can be written as
$$
\ba{rcl}
{u}_p^{\wedge/\vee} \!\!&\!\!=\!\!&\!\!
\frac{1}{\sqrt{2m (E \pm p^3)}} u'{}_p^{\wedge/\vee}
= \fr{1}{\sqrt{2m}}
\lb \begin{array}{c}
\sqrt{\lp E - p^\va \si_\va \rp} \, \ch^{\wedge/\vee} \\
\sqrt{\lp E + p^\va \si_\va \rp} \, \ch^{\wedge/\vee}
\end {array} \rb \\
\!\!&\!\!=\!\!&\!\!
U_p u_0^{\wedge/\vee} =
\lb \ba{cc} \cosh{\fr{\ze}{2}} - p_u \sinh{\fr{\ze}{2}} & 0 \\ 0 & \cosh{\fr{\ze}{2}} + p_u \sinh{\fr{\ze}{2}} \ea \rb
\fr{1}{\sqrt{2}} \lb \ba{c} \ch^{\wedge/\vee} \\ \ch^{\wedge/\vee} \ea \rb
\ea
\s \s
\ba{rcl}
{v}_p^{\wedge/\vee} \ae
\frac{1}{\sqrt{2m (E \pm p^3)}} v'{}_p^{\wedge/\vee}
= \fr{1}{\sqrt{2m}}
\lb \begin{array}{c}
\sqrt{\lp E - p^\va \si_\va \rp} \, \xi^{\wedge/\vee} \\
- \sqrt{\lp E + p^\va \si_\va \rp} \, \xi^{\wedge/\vee}
\end {array} \rb \\
\!\!&\!\!=\!\!&\!\!
U_p v_0^{\wedge/\vee} =
\lb \ba{cc} \cosh{\fr{\ze}{2}} - p_u \sinh{\fr{\ze}{2}} & 0 \\ 0 & \cosh{\fr{\ze}{2}} + p_u \sinh{\fr{\ze}{2}} \ea \rb
\fr{1}{\sqrt{2}} \lb \ba{c} \xi^{\wedge/\vee} \\ - \xi^{\wedge/\vee} \ea \rb
\ea
$$
using $m = \sqrt{(E-p)(E+p)}$ and $\{ \sinh{\ze} = \fr{\left| p \right|}{m}, \cosh{\ze} = \fr{E}{m}, \tanh{\ze} = \fr{\left| p \right|}{E} \}$. These solutions can be interrelated by several [[Dirac solution identities]].
A ''Dirac [[spinor]]'', $\Psi$, of the [[spacetime]] [[Cl(1,3)]] [[Clifford algebra]] may be written, using the [[Weyl representation|Dirac matrices]], as a sum of ''left-[[chiral]]'' and ''right-chiral'' parts,
$$
\Psi = \Psi_L + \Psi_R =
\lb \begin{array}{c}
\ps_L \\
0
\end {array} \rb
+
\lb \begin{array}{c}
0 \\
\ps_R
\end {array} \rb
=
\lb \begin{array}{c}
\ps_L \\
\ps_R
\end {array} \rb
=
\lb \begin{array}{c}
\ps_L^\wedge \\
\ps_L^\vee \\
\ps_R^\wedge \\
\ps_R^\vee
\end {array} \rb
$$
These parts are left and right handed [[Weyl spinor]]s, $\ps_L$ and $\ps_R$, which may be projected out,
$$
\Psi_{L/R} = P_{L/R} \Psi
$$
by the [[left/right chirality projector]],
$$
P_{L/R} = \ha \lp 1 \pm i \ga \rp
$$
(In this equation, the four component column with two zero entries is equated to a two component column.)
Under a [[Lorentz rotation]], such as a [[spatial rotation]] or [[Lorentz boost]], a Dirac spinor transforms via half of a [[Clifford rotation]], $\Ps'(x') = U \Ps = e^{\ha B} \Ps (x)$ -- a Dirac spinor is an element of the complex pinor [[representation space]] of the [[spacetime spin group]].
A ''Dirac spinor field'' over spacetime, $\ud{\Ps}(x)$, is a [[Grassmann number]] field or operator over spacetime, which we often conflate with a Dirac spinor.
/***
|Name|DisableWikiLinksPlugin|
|Source|http://www.TiddlyTools.com/#DisableWikiLinksPlugin|
|Version|1.6.0|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|selectively disable TiddlyWiki's automatic ~WikiWord linking behavior|
This plugin allows you to disable TiddlyWiki's automatic ~WikiWord linking behavior, so that WikiWords embedded in tiddler content will be rendered as regular text, instead of being automatically converted to tiddler links. To create a tiddler link when automatic linking is disabled, you must enclose the link text within {{{[[...]]}}}.
!!!!!Usage
<<<
You can block automatic WikiWord linking behavior for any specific tiddler by ''tagging it with<<tag excludeWikiWords>>'' (see configuration below) or, check a plugin option to disable automatic WikiWord links to non-existing tiddler titles, while still linking WikiWords that correspond to existing tiddlers titles or shadow tiddler titles. You can also block specific selected WikiWords from being automatically linked by listing them in [[DisableWikiLinksList]] (see configuration below), separated by whitespace. This tiddler is optional and, when present, causes the listed words to always be excluded, even if automatic linking of other WikiWords is being permitted.
Note: WikiWords contained in default ''shadow'' tiddlers will be automatically linked unless you select an additional checkbox option lets you disable these automatic links as well, though this is not recommended, since it can make it more difficult to access some TiddlyWiki standard default content (such as AdvancedOptions or SideBarTabs)
<<<
!!!!!Configuration
G disabled these so that WikiLinks would never happen.
<<<
<<option chkDisableWikiLinks>> Disable ALL automatic WikiWord tiddler links
<<option chkAllowLinksFromShadowTiddlers>> ... except for WikiWords //contained in// shadow tiddlers
<<option chkDisableNonExistingWikiLinks>> Disable automatic WikiWord links for non-existing tiddlers
Disable automatic WikiWord links for words listed in: <<option txtDisableWikiLinksList>>
Disable automatic WikiWord links for tiddlers tagged with: <<option txtDisableWikiLinksTag>>
<<<
!!!!!Revisions
<<<
2008.07.22 [1.6.0] hijack tiddler changed() method to filter disabled wiki words from internal links[] array (so they won't appear in the missing tiddlers list)
2007.06.09 [1.5.0] added configurable txtDisableWikiLinksTag (default value: "excludeWikiWords") to allows selective disabling of automatic WikiWord links for any tiddler tagged with that value.
2006.12.31 [1.4.0] in formatter, test for chkDisableNonExistingWikiLinks
2006.12.09 [1.3.0] in formatter, test for excluded wiki words specified in DisableWikiLinksList
2006.12.09 [1.2.2] fix logic in autoLinkWikiWords() (was allowing links TO shadow tiddlers, even when chkDisableWikiLinks is TRUE).
2006.12.09 [1.2.1] revised logic for handling links in shadow content
2006.12.08 [1.2.0] added hijack of Tiddler.prototype.autoLinkWikiWords so regular (non-bracketed) WikiWords won't be added to the missing list
2006.05.24 [1.1.0] added option to NOT bypass automatic wikiword links when displaying default shadow content (default is to auto-link shadow content)
2006.02.05 [1.0.1] wrapped wikifier hijack in init function to eliminate globals and avoid FireFox 1.5.0.1 crash bug when referencing globals
2005.12.09 [1.0.0] initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.DisableWikiLinksPlugin= {major: 1, minor: 6, revision: 0, date: new Date(2008,7,22)};
// G hard coded this
config.options.chkDisableNonExistingWikiLinks=true;
config.options.chkDisableWikiLinks=true;
config.options.chkAllowLinksFromShadowTiddlers=false;
if (config.options.txtDisableWikiLinksList==undefined) config.options.txtDisableWikiLinksList="DisableWikiLinksList";
if (config.options.txtDisableWikiLinksTag==undefined) config.options.txtDisableWikiLinksTag="excludeWikiWords";
// find the formatter for wikiLink and replace handler with 'pass-thru' rendering
initDisableWikiLinksFormatter();
function initDisableWikiLinksFormatter() {
for (var i=0; i<config.formatters.length && config.formatters[i].name!="wikiLink"; i++);
config.formatters[i].coreHandler=config.formatters[i].handler;
config.formatters[i].handler=function(w) {
// supress any leading "~" (if present)
var skip=(w.matchText.substr(0,1)==config.textPrimitives.unWikiLink)?1:0;
var title=w.matchText.substr(skip);
var exists=store.tiddlerExists(title);
var inShadow=w.tiddler && store.isShadowTiddler(w.tiddler.title);
// check for excluded Tiddler
if (w.tiddler && w.tiddler.isTagged(config.options.txtDisableWikiLinksTag))
{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
// check for specific excluded wiki words
var t=store.getTiddlerText(config.options.txtDisableWikiLinksList);
if (t && t.length && t.indexOf(w.matchText)!=-1)
{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
// if not disabling links from shadows (default setting)
if (config.options.chkAllowLinksFromShadowTiddlers && inShadow)
return this.coreHandler(w);
// check for non-existing non-shadow tiddler
if (config.options.chkDisableNonExistingWikiLinks && !exists)
{ w.outputText(w.output,w.matchStart+skip,w.nextMatch); return; }
// if not enabled, just do standard WikiWord link formatting
if (!config.options.chkDisableWikiLinks)
return this.coreHandler(w);
// just return text without linking
w.outputText(w.output,w.matchStart+skip,w.nextMatch)
}
}
Tiddler.prototype.coreAutoLinkWikiWords = Tiddler.prototype.autoLinkWikiWords;
Tiddler.prototype.autoLinkWikiWords = function()
{
// if all automatic links are not disabled, just return results from core function
if (!config.options.chkDisableWikiLinks)
return this.coreAutoLinkWikiWords.apply(this,arguments);
return false;
}
Tiddler.prototype.disableWikiLinks_changed = Tiddler.prototype.changed;
Tiddler.prototype.changed = function()
{
this.disableWikiLinks_changed.apply(this,arguments);
// remove excluded wiki words from links array
var t=store.getTiddlerText(config.options.txtDisableWikiLinksList,"").readBracketedList();
if (t.length) for (var i=0; i<t.length; i++)
if (this.links.contains(t[i]))
this.links.splice(this.links.indexOf(t[i]),1);
};
//}}}
<<tiddler HideTags>>What is done:
*All [[gauge fields|connection]], [[gravity|spacetime]], and Higgs in ''one'' [[connection]], with fermions as [[BRST ghosts|BRST technique]].
To do:
*Will particle assignments work with [[E8]]? (Get the CKMPMNS matrix?)
*Why is the action what it is? (How does symmetry breaking happen?)
*Is a four dimensional base [[manifold]] emergent?
*How does this theory get quantized? (LQG methods should apply.)
**Natural explanation for QM as a bonus?
What this theory will mean, if it all works:
*Gravitational [[frame]] and Higgs are intimately related.
*Naturally combines standard model with gravity -- so it's a [[T.O.E.|theory of everything]]
**(It's also a U.F.T., but I don't like to call it that.)
*Our universe is a very pretty shape!
@@display:block;text-align:center;Gar@Lisi.com
http://deferentialgeometry.org $\p{{}_{(}}$@@
<script label="O" title="toggle sidebar">
var sb=document.getElementById('sidebar');
var da=document.getElementById('displayArea');
if (sb.style.display == 'none') {
da.style.marginLeft = '18.5em';
sb.style.display = 'block';}
else {
da.style.marginLeft = '0em';
sb.style.display = 'none';}
</script> <script label="O" title="toggle title">
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if (h.style.height == '1.5em') {
h.style.height = '5.8em';}
else {
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[<img[images/png/dynkin g2.png]]A ''Dynkin diagram'' graphically summarizes the Cartan matrix of a [[root system]]. The $R$ nodes correspond to the $R$ simple roots, with one, two, or three lines connecting nodes corresponding to simple roots subtending an angle of $120^ \circ$, $135^\circ$, or $150^\circ$. The corresponding ratios of simple root lengths must be $1$, $\sqrt{2}$, and $\sqrt{3}$; in the diagram, the longer root is indicated by a $<$ symbol.
Use the e10 Kac-Moody algebra for unification, possibly including quantized states.
so(3,1)=d2 GR + so(10)=d5 GUT + so(6)=d3 gen in d10 in e10
or
so(5,1)=d3 = a3 GR + su(5) = a4 GUT + su(3) = a2 gen in a9 in e10
The [[Cartan matrix|root system]] for e10 is
$$
\left[\begin{array}{cccccccccc}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2\\
\end{array}\right]
$$
A set of [[simple roots|root system]] are, $\{ \al_{-1}, \al_{0}, \al_{1}, \al_{2}, \al_{3}, \al_{4}, \al_{5}, \al_{6}, \al_{7}, \al_{8} \}$, in 10D, using $i$ instead of Minkowski space,
$$
\left[\begin{array}{cccccccccc}
1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
-\ha & -\ha & -\ha & -\ha & -\ha & -\ha & -\ha & -\ha & -\ha & -i \ha\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0\\
\end{array}\right]
$$
Some roots are all 180 + 512 = 692 permutations of
$$
\begin{eqnarray}
& ( \pm 1, \pm 1, 0, 0, 0, 0, 0, 0, 0, 0 ) \textrm{ multiplying the last coordinate by } i \\
& ( \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm i \ha ) \textrm{ with even number plus}
\end{eqnarray}
$$
The E10 lattice is the integral span of the simple roots, ${II}_{9,1}$ -- the even self-dual Lorentzian lattice in 10D. (The algebrafication of this lattice, ${\frak g}_{{II}_{9,1}}$, is called a Borcherds algebra.) All ${II}_{9,1}$ lattice points with length less than or equal to $2$ are roots of E10. (Ref: Persson) Real roots have multiplicity $1$ (one Lie algebra generator per root). Real roots are "prime" -- they cannot be written as some multiple of another root. Imaginary roots, having zero or negative length, have lengths $\{ 0,-2,-4, ... \}$, and larger multiplicities. Roots of length zero are "null roots" and have multiplicity $8$, while "light-like roots" of negative length have higher multiplicities, such as $44$ for roots of length $-2$. Some imaginary roots, such as $n \de$ for any $n$, are not prime, but have the same multiplicities. Multiplicities for all imaginary roots are not currently known.
Some interesting imaginary roots can be found by inverting the Cartan matrix,
$$
\left[\begin{array}{cccccccccc}
0 & -1 & -2 & -3 & -4 & -5 & -6 & -4 & -2 & -3\\
-1 & -2 & -4 & -6 & -8 & -10 & -12 & -8 & -4 & -6\\
-2 & -4 & -6 & -9 & -12 & -15 & -18 & -12 & -6 & -9\\
-3 & -6 & -9 & -12 & -16 & -20 & -24 & -16 & -8 & -12\\
-4 & -8 & -12 & -16 & -20 & -25 & -30 & -20 & -10 & -15\\
-5 & -10 & -15 & -20 & -25 & -30 & -36 & -24 & -12 & -18\\
-6 & -12 & -18 & -24 & -30 & -36 & -42 & -28 & -14 & -21\\
-4 & -8 & -12 & -16 & -20 & -24 & -28 & -18 & -9 & -14\\
-2 & -4 & -6 & -8 & -10 & -12 & -14 & -9 & -4 & -7\\
-3 & -6 & -9 & -12 & -15 & -18 & -21 & -14 & -7 & -10\\
\end{array}\right]
$$
A null root, from reading off the top, is
$$
\de = 0 \al_{-1} - 1 \al_{0} - 2 \al_{1} - 3 \al_{2} - 4 \al_{3} - 5 \al_{4} - 6 \al_{5} - 4 \al_{6} - 2 \al_{7} - 3 \al_{8} = (1,0,0,0,0,0,0,0,0,i)
$$
The ''affine level'' of a root is $ - k_0 = < \al, \de >$. (For the [[e9]] subalgebra, all imaginary roots are multiples of this $\de$.) Some light-like roots of e10, of lengths $-2$ and $-4$, are
$$
-1 \al_{-1} -2 \al_{0} - 4 \al_{1} - 6 \al_{2} - 8 \al_{3} - 10 \al_{4} - 12 \al_{5} - 8 \al_{6} - 4 \al_{7} - 6 \al_{8} = (1,1,0,0,0,0,0,0,0,2i)
$$
and
$$
-2 \al_{-1} - 4 \al_{0} - 6 \al_{1} - 8 \al_{2} - 10 \al_{3} - 12 \al_{4} - 14 \al_{5} - 9 \al_{6} - 4 \al_{7} - 7 \al_{8} = (0,0,0,0,0,0,0,0,0,2i)
$$
One can nicely decompose e10 into finite a9 reps at each level with respect to $\al_8$.
Ref:
* Persson and Tabti, "Lectures on Kac-Moody Algebras with Applications in (Super-)Gravity", http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/KMModaveLectures2007.pdf
The rank $6$ exceptional group, ''E6'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $78$ dimensional [[Lie algebra]], [[e6]].
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E6.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E6 = spin(10) \,\oplus\, u(1)_{PQ} \,\,\oplus\,\, 16^\mathbb{C}_{S^+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E6.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E6 = spin(10) \,\oplus\, u(1)_{PQ} \,\,\oplus\,\, 16^\mathbb{C}_{S^+}</SPAN>
</td></tr>
</table>
</center></html>
The rank $8$ exceptional group, ''E8'', is the largest of the real, [[simple]], compact, connected [[Lie groups]] -- and is often regarded as the most beautiful. It may be described by [[exponentiating|exponentiation]] its $248$ dimensional [[Lie algebra]], [[e8]].
<<tiddler HideTags>>Build a real form of complex [[E8]] by using $Cl^2(1,7)=so(1,7)$ instead of $Cl^2(8)=so(8)$. Then ''E8 T.O.E. connection'' is:
$$
\udf{A} = \f{H} + \f{G} + \ud{\Ps}{}_I + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} =
$$
$$
\text{something like}_{\p{\big(}}
$$
$$
{\small
\begin{array}{c}
\!\!\! \lb \begin{array}{cccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!& \frac{1}{4} \f{e_R} \ph_+ \! \\
i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!& \frac{1}{4} \f{e_R} \ph_0 \! \\
-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \!\!\!\! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \!\! \!& & \! \\
\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \!\! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \!\!\!\!\!
\end{array} \rb
\!\!+\!\!
\lb \begin{array}{cccc}
i \f{B} \!\! & & & \\
&\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
&\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
&\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
\\
\; \\
+
\lb \begin{array}{cccc}
\ud{\nu}{}^e_L & \ud{u}{}_L^r & \ud{u}{}_L^g & \ud{u}_L^b \\
\ud{e}{}_L & \ud{d}{}_L^r & \ud{d}{}_L^g & \ud{d}{}_L^b \\
\ud{\nu}{}^e_R & \ud{u}{}_R^r & \ud{u}{}_R^g & \ud{u}{}_R^b \\
\ud{e}{}_R & \ud{d}{}_R^r & \ud{d}{}_R^g & \ud{d}{}_R^b
\end{array} \rb
\;+\;
\lb \begin{array}{cccc}
\ud{\nu}{}^\mu_L & \ud{c}{}_L^r & \ud{c}{}_L^g & \ud{c}_L^b \\
\ud{\mu}{}_L & \ud{s}{}_L^r & \ud{s}{}_L^g & \ud{s}{}_L^b \\
\ud{\nu}{}^\mu_R & \ud{c}{}_R^r & \ud{c}{}_R^g & \ud{c}{}_R^b \\
\ud{\mu}{}_R & \ud{s}{}_R^r & \ud{s}{}_R^g & \ud{s}{}_R^b
\end{array} \rb
\;+\;
\lb \begin{array}{cccc}
\ud{\nu}{}^\ta_L & \ud{t}{}_L^r & \ud{t}{}_L^g & \ud{t}_L^b \\
\ud{\ta}{}_L & \ud{b}{}_L^r & \ud{b}{}_L^g & \ud{b}{}_L^b \\
\ud{\nu}{}^\ta_R & \ud{t}{}_R^r & \ud{t}{}_R^g & \ud{t}{}_R^b \\
\ud{\ta}{}_R & \ud{b}{}_R^r & \ud{b}{}_R^g & \ud{b}{}_R^b
\end{array} \rb_{\p{(}}
\end{array}
}
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;\; \in Cl(3,1)^2 = spin(3,1)
\s\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\s\s\s \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big>
$$
<<tiddler HideTags>>@@display:block;text-align:center;
<html><center>
<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$
[[Garrett Lisi]] Joint Mathematics Meetings 1/15/10@@
*Quantization
**Coupling constants run.
***Large $\La$ compatible with UV fixed point.
**Just a connection -- amenable to LQG, spin foams, etc.
*Understand triality-generation relationship better
**Possible collapse or mixing to graviweak $SL(2,\mathbb{C})$.
**The role of $\f{w}+\f{x}\Ph$ and symmetry breaking.
**Getting the CKMPMNS matrix would be nice.
*Why is the action what it is?
**Pulling $\f{e}$ out and putting it into $\ff{F} \ff{*F}$ and $\fff{\od{B}}$ seems weird.
***Why $\f{e}\ph$ simple?
***Four dimensional base manifold emergent?
What this theory will mean, if it all works:
*Combines standard model with gravity -- with LQG, it's a T.o.E.
*Our universe is very pretty.
@@display:block;text-align:center; http://deferentialgeometry.org Garrett Lisi@@
<<tiddler HideTags>>
Everything in an $E8$ principal bundle connection,
$$
\udf{A} \in \udf{e8}
$$
Periodic table of interactions (Feynman vertices) from curvature,
$$
\udff{F} = \f{d} \udf{A} + {\scriptsize \frac{1}{2}} \big[ \udf{A}, \udf{A} \big]
$$
described by the $E8$ root polytope. Three generations through triality,
$$
T \, e = \mu \qquad T \, \mu = \ta \qquad T \, \ta = e
$$
Pati-Salam $SU(2)_L \times SU(2)_R \times SU(4)$ GUT and MM gravity together,
$$
S = \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi}{4}} \ff{B}{}_G \ff{B}{}_G \ga + \ff{B'} \ff{*B'} \big>
$$
No free parameters -- masses from Higgs VEV's,
$$
g_1 = \sqrt{\fr{3}{5}} \qquad g_2=1 \qquad g_3=1 \qquad \La=\fr{3}{4}\ph^2 \qquad \ph_0 , \ph_1, \Ph \dots
$$
Everything is pure geometry, and it's very beautiful.
<<tiddler HideTags>>
<<tiddler HideTags>>Superconnection:
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\! \big( \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \big) + \ud{\ps} \\
\!\!&\!\! \in \!\!&\!\! H + K \\
\!\!&\!\! \subset \!\!&\!\! \big( spin(3,1) + 4 \!\times\! (2 \!+\! \bar{2}) + su(2)_L + u(1)_Y + su(3) \big) + 2 \!\times\! (2_L\!+\!2_R) \!\times\! (1\!+\!3) \\
\!\!&\!\! \subset \!\!&\!\! \big( spin(3,1) + 4 \!\times\! 10 + spin(10) \big) + 2 \!\times\! 16_S^{+\mathbb{C}} \\
\!\!&\!\! \subset \!\!&\!\! spin(3,11) + 64_S^{+\mathbb{R}} \subset spin(4,12) + 128_S^{+\mathbb{R}} \subset E8
\end{array}
$$
Curvature:
$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F}^H + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}
$$
Action:
$$
S = \int \left< \big( \fff{\od{B}} + \ff{B} \big) \udff{F} + \nf{V}(B^H) \right>
$$
Generations?$\s\s\s\s\s\s\s\s\s\;\;
\mbox{Axions?} \;\;\;\; spin(1,1)_{PQ}, \;\; \th \ff{F} \ff{F},\;\; \big< \bar{\ps} \f{e} \th \f{e} \th \f{e} \th \ep \f{D} \ud{\ps} \big> \mbox{ ?}
$
Geometric interpretation of the superconnection?$\s\s\s\s\s\;
\mbox{BRST?} \;\;\;\; \ud{\de} \f{K} = - \f{D} \ud{\ps} \s \mbox{TQFT?}\vp{A_{\big(}}
$
Precise symmetry breaking mechanism?$\s\s\s\s\s\s\s\s\s\;\;
\nf{V}(B^H) = \ff{B} \ff{\vv{\Ph}} \ff{B} + \dots \mbox{ ?}
$
Quantization?$\s\s\s\s\s\s\s\s
\mbox{Asymptoticly safe R.G. flow of } \La, G, g, \dots \mbox{? Spinfoams?}
$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;\; \in Cl(3,1)^2 = spin(3,1)
\s\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\s\s\s \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big>
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\nu\rho} \ga_{\nu\rho} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\nu \ph^\ps \ga_{\nu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;\; \in Cl(3,1)^2 = spin(3,1)
\s\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\s\s\s \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) \;\; \subset spin(3,11) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^G \ff{*F}^G \big>
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \Ga^a (e_a)^\mu \big( \pa_\mu + \fr{1}{4} \om_\mu^{\p{\mu}bc} \Ga_{bc} + \ha A_\mu^{\p{\mu}xy} \Ga_{xy} + \fr{1}{4} (e_\mu)^b \ph^x \Ga_{bx} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^\mu} \, \om_\mu^{\p{\mu}ab} \ga_{ab} \;\; \in Cl^2(1,3) = spin(1,3)
\s\s \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;\; \in Cl^1(1,3) = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{A} \;\; \in \, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3)$
$\s\s\s \subset su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \otimes (2_L \oplus 2_R) \otimes (1 \oplus 3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^x \Ga_x \;\; \in Cl^1(4) = 4 \;$ or $\; Cl^1(10) = 10{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) \;\; \subset spin(11,3) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^A{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\, \subset spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} = E_{8(-24)}$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big>
$$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \Ga^a (e_a)^\mu \big( \pa_\mu + \fr{1}{4} \om_\mu^{\p{\mu}bc} \Ga_{bc} + \ha A_\mu^{\p{\mu}xy} \Ga_{xy} + \fr{1}{4} (e_\mu)^b \ph^x \Ga_{bx} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^\mu} \, \om_\mu^{\p{\mu}ab} \ga_{ab} \;\; \in Cl^2(1,3) = spin(1,3)
\s\s \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \;\; \in Cl^1(1,3) = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{A} \;\; \in \, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3)$
$\s\s\s \subset su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \otimes (2_L \oplus 2_R) \otimes (1 \oplus 3) = 32^\mathbb{C} = 64^\mathbb{R} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^x \Ga_x \;\; \in Cl^1(4) = 4 \;$ or $\; Cl^1(10) = 10{}^{\p{\big(}}$
Connection: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) \;\; \subset spin(11,3) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^A{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\, \subset spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} = E_{8(-24)}$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \big< \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \big> \sim \int \big< \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \f{D} \ph \fff{*D} \ph + {\textstyle \fr{1}{4g^2}} \ff{F}^A \ff{*F}^A \big>
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUT E8.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">E_{8(-24)} = spin(12,4) \,\,\oplus\,\, 128^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/E8.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math"> </SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center><iframe src="talks/Mindshare11/anim/E8toE8.html" width="540" height="540" frameborder="0"></iframe>
</center></html>
$$
\udf{A} = \f{H}{}_1 + \f{H}{}_2 + \ud{\Ps}{}_{I} + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} \quad \in \;\; \udf{e8} \vp{|_{\big(}}
$$
$$
\begin{array}{rclcl}
\f{H}{}_1 \!\!&\!\!=\!\!&\!\! {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph + \f{W} + \f{B}{}_1 & \in & \f{so}(7,1) \\[-.1em]
&& \f{\om} & \in & \f{so}(3,1) \\[-.1em]
&& \f{e} \ph = (\f{e}{}_1+\f{e}{}_2+\f{e}{}_3+\f{e}{}_4)\times(\ph_{+/0}+\ph_{-/1}) & \in & \f{4} \times (2+\bar{2}) \\
&& \f{W} + \f{B}{}_1 & \in & \f{su}(2) + \f{su}(2) \\
\f{H}{}_2 \!\!&\!\!=\!\!&\!\! \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} & \in & \f{so}(8) \\[-.2em]
&& \f{w} + \f{B}{}_2 & \in & \f{u}(1) + \f{u}(1) \\[-.3em]
&& \f{x} \Ph = (\f{x}{}_{1}+\f{x}{}_{2}+\f{x}{}_{3})\times(\Ph^{r/g/b} + {\Ph}{}^{\bar{r}/\bar{g}/\bar{b}}) & \in & \f{3} \times (3+\bar{3}) \\[-.1em]
&& \f{g} & \in & \f{su}(3) \\
\ud{\Psi}{}_{I} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_e + \ud{e} + \ud{u} + \ud{d} & \in & 8_{S+} \!\times 8_{S+} \\
\ud{\Psi}{}_{II} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_\mu + \ud{\mu} + \ud{c} + \ud{s} & \in & 8_{V} \times 8_{V} \\
\ud{\Psi}{}_{III} \!\!&\!\!=\!\!&\!\! \ud{\nu}{}_\ta + \ud{\ta} + \ud{t} + \ud{b} & \in & 8_{S-} \!\times 8_{S-} \\
\end{array}
$$
<<tiddler HideTags>>
$$
\udff{F} = \f{d} \udf{A} + \udf{A} \udf{A}
= \ff{F}{}_1+\ff{F}{}_2+ \f{D} \big( \ud{\Ps}{}_{I} + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} \big) \quad \in \;\; \udff{e8} \vp{|_{\Big(}}
$$
$$
\begin{array}{rlcl}
\ff{F}{}_1 \!\!\!\!&=
\ha \big( \ff{R} - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \big( \ff{F}{}_{B_1} + \ff{F}{}_W \big) & \in & \f{so}(7,1) \\[.1em]
& \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} & \in & \f{so}(3,1) \\[.1em]
& \ff{T} \ph \!-\! \f{e} \f{D} \ph = \big( \f{d} \f{e} \!+\! \ha [ \f{\om}, \f{e} ] \big) \ph - \f{e} \big( \f{d} \ph \!+\! [ \f{B}{}_1 \!+\! \f{W}, \ph ] \big) & \in & \f{4} \times (2+\bar{2}) \\[.2em]
& \ff{F}{}_{B_1} + \ff{F}{}_W = (\f{d} \f{B}{}_1 + \f{B}{}_1 \f{B}{}_1) + (\f{d} \f{W} + \f{W} \f{W}) & \in & \f{su}(2) \!+\! \f{su}(2) \\[.4em]
\ff{F}{}_2 \!\!\!\!&=
\big( \ff{F}{}_{w} + \ff{F}{}_{B_2} + \f{x}\Ph\f{x}\Ph \big)
+ \big( (\f{D} \f{x}) \Ph - \f{x} \f{D} \Ph \big)
+\ff{F}{}_{g}
& \in & \f{so}(8) \\[.1em]
& \ff{F}{}_{w} + \ff{F}{}_{B_2} = \f{d} \f{w} + \f{d} \f{B}{}_2 & \in & \f{u}(1) + \f{u}(1) \\[.1em]
& (\f{D} \f{x}) \Ph \!-\! \f{x} \f{D} \Ph \!=\!
\big( \f{d} \f{x} \!+\! [ \f{w} \!+\! \f{B}{}_2, \! \f{x} ] \big) \Ph \!-\! \f{x} \big( \f{d} \Ph \!+\! [ \f{g}, \! \Ph ] \big) \!\!\!
& \in & \f{3} \times (3+\bar{3}) \\[0em]
& \ff{F}{}_{g} = \f{d} \f{g} + \f{g} \f{g} & \in & \f{su}(3)
\end{array}
$$
$$
\f{D} \ud{\Psi} = \big( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph \big) \ud{\Ps}
+ \f{W} \ud{\Ps}{}_L + \f{B}{}_1 \ud{\Ps}{}_R - \ud{\Ps} \big( \f{w} + \f{B}{}_2 + \f{x} \Ph \big) - \ud{\Ps}{}_q \, \f{g}
\vp{|^{\Big(}}
$$
<<tiddler HideTags>>
<<tiddler HideTags>>Build new ${\rm Lie}(E8)$ generators from old ones:
$$
\begin{array}{rclclcll}
H_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\al\be} \!\!&\!\!=\!&\!\! \ga^{(8)+}_{\al\be} \otimes 1 \!\!&\!\!\in\!&\!\! so(8)^+ \otimes 1
\!\!&\!=\, so(8)^H \\
G_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\lp\al+8\rp\lp\be+8\rp} \!\!&\!\!=\!&\!\! P^{\lp8\rp}_+ \otimes \ga^{(8)}_{\al\be} \!\!&\!\!\in\!&\!\! 1 \otimes so(8)
\!\!&\!=\, so(8)^G \\
\Ps^I_{\al\be} \!\!&\!=\!&\!\! \ga^{\lp16\rp+}_{\al\lp\be+8\rp} \!\!&\!\!=\!&\! \ga^{(8)+}_\al \otimes \ga^{(8)}_\be \!\!&\!\!\in\!&\!\! v^{(8)+} \otimes v^{(8)}
\!\!&\!=\, S^I \\
\Ps^{II}_{ab} \!\!&\!=\!&\!\! Q^+_{16\lp a-1\rp+b} \!\!&\!\!=\!&\!\! q^+_a \otimes q^+_b \!\!&\!\!\in\!&\!\! S^{(8)+} \otimes S^{(8)+}
\!\!&\!=\, S^{II} \\
\Ps^{III}_{ab} \!\!&\!=\!&\!\! Q^+_{16\lp a-1\rp+b+8} \!\!&\!\!=\!&\!\! q^+_a \otimes q^-_b \!\!&\!\!\in\!&\!\! S^{(8)+} \otimes S^{(8)-}
\!\!&\!=\, S^{III}
\end{array}
$$
With these basis generators, the ${\rm Lie}(E8)$ elements are:
\begin{eqnarray}
E &=& H + G + \Ps_I + \Ps_{II} + \Ps_{III} \\
&=& \ha h^{\al\be} H_{\al\be} + \ha g^{\al\be} G_{\al\be} + \ps_I^{\al\be} \Ps^I_{\al\be} + \ps_{II}^{ab} \Ps^{II}_{ab} + \ps_{III}^{ab} \Ps^{III}_{ab} \\
&\in& so(8)^H + so(8)^G + S^I + S^{II} + S^{III}_{\p{(}}
\end{eqnarray}
<<tiddler HideTags>>@@display:block;text-align:center;[img[images/png/e8 periodic table.png]]@@
//"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."// -- Hermann Nicolai
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour.mov" width="602" height="602" controller="false" autoplay="false" loop="false"></embed>
<!-- <embed src="talks/Perimeter07/anim/e8tour (om up)/p1.png" width="608" height="609"></embed> -->
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p1.png" width="608" height="609"></embed>
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<embed src="talks/Perimeter07/anim/e8tour/p181.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p236.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p280.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p320.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p361.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p391.png" width="608" height="609"></embed>
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<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p410.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p422.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p430.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p482.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<embed src="talks/Perimeter07/anim/e8tour/p562.png" width="608" height="609"></embed>
</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
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<embed src="talks/Perimeter07/anim/e8tour/p642.png" width="608" height="609"></embed>
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</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@
<<tiddler HideTags>>@@display:block;text-align:center;${\rm Lie}(E8)$ has $(248-8)=240$ roots in 8D space -- vertices of $P4_{2,1}$:$\p{{}_{\big(}}$
<html><center><embed src="talks/FQXi07/video/e8anim.mov" width="510" height="510" controller="false" autoplay="false" loop="false"></embed></center></html>$E8$ T.O.E.: Each vertex corresponds to an elementary particle.$\p{{}{\Big(}^{(}}$@@
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Triality.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math"> </SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>The ${\rm Lie}(E8)$ brackets between elements in the various parts:
$$
\begin{array}{cc}
\begin{array}{rcl}
\big[ H_1, H_2 \big] \!\!&\!=\!&\!\! H_1 H_2 - H_2 H_1 \\
\big[ G_1, G_2 \big] \!\!&\!=\!&\!\! G_1 G_2 - G_2 G_1 \\
&&\\
\big[ H, \Ps_I \big] \!\!&\!=\!&\!\! H \, \Ps_I \\
\big[ H, \Ps_{II} \big] \!\!&\!=\!&\!\! H^+ \, \Ps_{II} \\
\big[ H, \Ps_{III} \big] \!\!&\!=\!&\!\! H^+ \, \Ps_{III} \\
&&\\
\big[ G, \Ps_I \big] \!\!&\!=\!&\!\! \Ps_I \, G \\
\big[ G, \Ps_{II} \big] \!\!&\!=\!&\!\! - \Ps_{II} \, G^+ \\
\big[ G, \Ps_{III} \big] \!\!&\!=\!&\!\! - \Ps_{III} \, G^-
\end{array}
&
\begin{array}{rcl}
\big[ \Ps^1_I, \Ps^2_I \big] \!\!&\!=\!&\!\! -2 \big( \Ps^1_I \, {\Ps^2_I}^T \big)_H \\
&& -2 \big( {\Ps^1_I}^T \Ps^2_I \big)_{G_{\p{(}}} \\
\\
\big[ \Ps^1_{II}, \Ps^2_{II} \big] \!\!&\!=\!&\!\! - \big( \Ps^1_{II} \Ga^+ {\Ps^2_{II}}^T \big)_H \\
&&\!\! - \big( {\Ps^1_{II}}^T \Ga^+ \Ps^2_{II} \big)_{G_{\p{(}}} \\
\big[ \Ps^1_{III}, \Ps^2_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps^1_{II} \Ga^+ {\Ps^2_{II}}^T \big)_H \\
&&\!\! - \big( {\Ps^1_{II}}^T \Ga^- \Ps^2_{II} \big)_G \\
&&\\
\big[ \Ps_I, \Ps_{II} \big] \!\!&\!=\!&\!\! - \big( \Ps_I \Ga^{++} \Ps_{II} \big)_{III} \\
\big[ \Ps_I, \Ps_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps_I \Ga^{+-} \Ps_{III} \big)_{II} \\
\big[ \Ps_{II}, \Ps_{III} \big] \!\!&\!=\!&\!\! - \big( \Ps_{II} \Ga^{++} \Ps_{III} \big)_I
\end{array}
\end{array}
$$
Note: $H$ acts on $\Ps$'s from the left and $G$ acts from the right.$^{\p{\big(}}_{\p{(}}$
<<tiddler HideTags>>$$\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \,\oplus\, 64^\mathbb{R}_{S-} \,\oplus\, 14_V \,\oplus\, 14_V \,\oplus\, spin(1,1) \\
\!\!&\!\!=\!\!&\!\! spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} \\
\!\!&\!\!=\!\!&\!\! E_{8(-24)} \\
\end{array}
$$
''$E_{8(-24)}$ structure''
$$\begin{array}{rcl}
\f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{A} = \ha \f{H}^{xy} \Ga'_{xy} \!\!&\!\! \in \!\!&\!\! spin(12,4) \\
\ud{\ps} = \ud{\ps}^\ph Q'_\ph \!\!&\!\! \in \!\!&\!\! 128^\mathbb{R}_{S+}
\end{array}
$$
$$
\begin{array}{rcl}
[\Ga'_{wx}, \Ga'_{yz}] \!\!&\!\!=\!\!&\!\! 2 \eta_{xy} \Ga'_{wz} - 2 \eta_{xz} \Ga'_{wy} + 2 \eta_{wz} \Ga'_{xy} - 2 \eta_{wy} \Ga'_{xz} \\
[\Ga'_{xy}, Q'_\ph] \!\!&\!\! = \!\!&\!\! - [Q'_\ph , \Ga'_{xy}] = \Ga'^+_{xy} Q'_\ph = Q'_\ps (\Ga'^+_{xy})^\ps_{\p{\ps} \ph} \\
[Q'_\ph, Q'_\ps] \!\!&\!\! = \!\!&\!\! - \Ga_{xy} (\Ga'^{+xy})_{\ps \ph} = - \Ga_{xy} \eta^{xw} \eta^{yz} (\Ga'^+_{wz})^\la_{\p{\la} \ph} g_{\la \ps} \\ [.5em]
g_{\la \ps} \!\!&\!\!=\!\!&\!\! (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+_{\la \ps}
\end{array}
$$
Spinors from E8 geometry,
$$\begin{array}{rcl}
\udff{F} \!\!&\!\! = \!\!&\!\! \ff{F^H} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps} \\[.5em]
\ve{e} \f{D} \ud{\ps} \!\!&\!\! = \!\!&\!\! \ve{e} \lp \f{d} \ud{\ps} + {\textstyle \ha} [ \f{H}, \ud{\ps} ] \rp
= \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps} = D \!\!\!\! / \ud{\ps}
\end{array}
$$
<<tiddler HideTags>>$$\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \rp \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+} \\
\!\!&\!\!\subset\!\!&\!\! spin(11,3) \,\oplus\, ( 64^\mathbb{R}_{S+} \,\oplus\, 64^\mathbb{R}_{S-}) \,\oplus\, (14_V \,\oplus\, 14_V) \,\oplus\, spin(1,1) \\
\!\!&\!\!=\!\!&\!\! spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+} = E_{8(-24)}
\end{array}
$$
''$E_{8(-24)}$ structure''
$$
\Ga'_{xy} \in spin(12,4) \s\;\; Q'_\ph \in 128^\mathbb{R}_{S+}
$$
$$\begin{array}{rcl}
[\Ga'_{wx}, \Ga'_{yz}] \!\!&\!\!=\!\!&\!\! 2 \eta_{xy} \Ga'_{wz} - 2 \eta_{xz} \Ga'_{wy} + 2 \eta_{wz} \Ga'_{xy} - 2 \eta_{wy} \Ga'_{xz} \\
[\Ga'_{xy}, Q'_\ph] \!\!&\!\! = \!\!&\!\! - [Q'_\ph , \Ga'_{xy}] = \Ga'^+_{xy} Q'_\ph = Q'_\ps (\Ga'^+_{xy})^\ps_{\p{\ps} \ph} \\
[Q'_\ph, Q'_\ps] \!\!&\!\! = \!\!&\!\! - \Ga_{xy} (\Ga'^{+xy})_{\ps \ph} = - \Ga_{xy} \eta^{xw} \eta^{yz} (\Ga'^+_{wz})^\la_{\p{\la} \ph} g_{\la \ps} \\ [.5em]
(Q'_\la, Q'_\ps) \!\!&\!\!=\!\!&\!\! g_{\la \ps} = (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+_{\la \ps}
\end{array}
$$
Spinors from E8 geometry,
$$
\f{H} = {\textstyle \ha} \f{H}{}^{xy} \Ga'_{xy} \s\;\; \ud{\ps} = \ud{\ps}{}^{\ph} Q'_\ph
$$
$$
\ve{e} \f{D} \ud{\ps} = \ve{e} \lp \f{d} \ud{\ps} + {\textstyle \ha} [ \f{H}, \ud{\ps} ] \rp = \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps}
$$
<<tiddler HideTags>>
Via the [[Pati-Salam]] GUT:
$$
\begin{array}{rcl}
\udf{A} = \f{H} + \ud{\ps} \;\; \!\!&\!\!\in\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \rp \!\otimes\! 2 \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, \lp su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) \rp \rp \,\oplus\, \lp 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! 4 \rp \!\otimes\! 2 \\[.5em]
\!\!&\!\!=\!\!&\!\! \lp spin(1,3) \,\oplus\, 4 \!\otimes\! 4 \,\oplus\, \lp spin(4) \,\oplus\, spin(6) \rp \rp \,\oplus\, \lp 2 \!\otimes\! 4 \!\otimes\! 4 \!\otimes\! 2 \rp \\[.5em]
\!\!&\!\!=\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(6) \rp \,\oplus\, \lp 8 \!\otimes\! 4 \!\otimes\! 2 \rp \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(7,1) \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \\[.5em]
\!\!&\!\!\subset\!\!&\!\! \lp spin(1,7) \,\oplus\, spin(7,1) \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \,\oplus\, \lp 8 \!\otimes\! 8 \rp \\[.5em]
\!\!&\!\!=\!\!&\!\! spin(8,8) \,\oplus\, 128^\mathbb{R}_{S+} \\[.5em]
\!\!&\!\!=\!\!&\!\! E_{8(8)}
\end{array}
$$
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The ''Ehresmann Cartan connection'', $\f{\ve{\cal C}}$, is an [[Ehresmann principal bundle connection]] over the total space, $E_G$, of an [[Ehresmann Cartan geometry]]. In coordinates ($x$ over $M$ patches, $x_s$ over $G/H$ patches, and $y$ over $H$ patches) adapted to the reference sections, the Ehresmann Cartan connection may be written locally as
\begin{eqnarray}
\f{\ve{\cal C}}(x, x_s, y) &=& \f{C^J}(x) \, \ve{\xi^L_J}(x_s, y) + \f{\ve{\cal I}} \\
&=& \f{C^J}(x) \, L^I{}_J(x_s, y) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}}
\end{eqnarray}
in which $\ve{\xi^L_J}$ and $\ve{\xi^R_J} \sim T_J$ are the [[left and right action vector fields|Lie group geometry]] for the fibers, $G_x$, the [[left-right rotator]] is
$$
L^I{}_J(x_s, y) = \ve{\xi^L_J} \f{\xi_R^I} = \lp T^I, g^-(x_s,y) \, T_J \, g(x_s,y) \rp
$$
the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers) is
$$
\f{\ve{\cal I}}(x_s,y) = \f{\xi_R^J} \ve{\xi^R_J} = \f{dx_s^a} \ve{\pa^s_a} + \f{dy^p} \ve{\pa_p} = \f{\ve{{\cal I}_{G/H}}} + \f{\ve{{\cal I}_H}}
$$
and $\f{C^J}(x)$ are the components of the [[Cartan connection|Cartan geometry]] over $M$. The Ehresmann Cartan connection is a projection, $\f{\ve{\cal C}} \f{\ve{\cal C}} = \f{\ve{\cal C}}$, is [[right invariant]], $R_g^*\f{\ve{\cal C}} = \f{\ve{\cal C}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a [[Lieform]] over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\f{\cal I}(x_s,y) = \f{\xi_R^J} T_J = g^- \f{d} g$, over the total space to get the ''Ehresmann Cartan connection form'',
\begin{eqnarray}
\f{\cal C}(x,x_s,y) &=& \f{\ve{\cal C}} \f{\cal I} = \f{C^J}(x) \ve{\xi^L_J} \f{\xi_R^I} T_I + \f{\ve{\cal I}} \f{\cal I} \\
&=& \lp \f{C^J} L^I{}_J(x_s,y) + \f{\xi_R^I} \rp T_I \\
&=& g^-(x_s,y) \, \f{C}(x) \, g(x_s,y) + g^-(x_s,y) \, \f{d} \, g(x_s,y)
\end{eqnarray}
This form [[pulls back|pullback]] along the canonical reference section, $\si_0^G$, to give the Cartan connection,
$$
\si_0^{G*} \f{\cal C} = \f{C}(x)
$$
and satisfies $R_g^* \f{\cal C} = g^- \f{\cal C} g$ under the right action.
The ''[[FuN curvature]] of the Ehresmann Cartan connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,x_s,y) &=& - \ha \lb \f{\ve{\cal C}}, \f{\ve{\cal C}} \rb_L \\
&=& \lp \f{d} \f{C^K} + \ha \f{C^I} \f{C^J} C_{IJ}{}^K \rp \ve{\xi^L_K}(x_s,y)
\end{eqnarray}
which is vector valued in the vertical subspace and right invariant, $R_g^* \ff{\ve{\cal F}} = \ff{\ve{\cal F}}$. The ''FuN curvature form of the Ehresmann Cartan connection'' is a ${\frak g}$ valued 2-form over $E_G$,
\begin{eqnarray}
\ff{\cal F} &=& \ff{\ve{\cal F}} \f{\cal I} = \lp \f{d} \f{C^K} + \ha \f{C^I} \f{C^J} C_{IJ}{}^K \rp g^-(x_s,y) \, T_K \, g(x_s,y) \\
&=& g^-(x_s,y) \lp \f{d} \f{C} + \ha \lb \f{C}, \f{C} \rb \rp g(x_s,y) \\
&=& g^-(x_s,y) \lp \f{d} \f{C} + \f{C} \f{C} \rp g(x_s,y)
\end{eqnarray}
This form pulls back along the canonical reference section to give the [[Cartan geometry]] curvature,
$$
\si_0^{G*} \ff{\cal F} = \f{d} \f{C} + \f{C} \f{C} = \ff{F}(x)
$$
and satisfies $R_g^* \ff{\cal F} = g^- \ff{\cal F} g$ under the right action.
When $H$ is [[reductive]] in $G$ (which is usually assumed) the [[Cartan connection|Cartan geometry]] splits as
$$
\f{C}(x) = \f{e}(x) + \f{A}(x) = \f{e^A} K_A + \f{A^P} H_P
$$
the [[Ehresmann Cartan connection]] can be made to follow this split,
$$
\begin{eqnarray}
\f{\ve{\cal C}}(x,x_s,y) &=& \f{C^J}(x) \, L^I{}_J(x_s, y) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}} \\
&=& \f{\ve{\cal E}} + \f{\ve{\cal A}}
\end{eqnarray}
$$
with the ''Ehresmann Cartan frame'' and ''Ehresmann Cartan H-connection'' defined over patches of the total space, $E_G$, of the [[Ehresmann Cartan geometry]] as:
$$
\begin{eqnarray}
\f{\ve{\cal E}}(x, x_s, y) &=& \f{e^A}(x) \, (L^h)^I{}_K(y) \, (L^r)^K{}_A(x_s) \, \ve{\xi^R_I}(x_s, y) \\
\f{\ve{\cal A}}(x, x_s, y) &=& \f{A^P}(x) \, (L^h)^I{}_K(y) \, (L^r)^K{}_P(x_s) \, \ve{\xi^R_I}(x_s, y) + \f{\ve{\cal I}}
\end{eqnarray}
$$
with the [[left-right rotator]] and [[Killing vector fields|Lie group geometry]] split over the [[reductive Lie group geometry]]. The [[Ehresmann Cartan connection form|Ehresmann Cartan connection]], $\f{\cal C} = \f{\ve{\cal C}} \f{\cal I}$, also splits,
$$
\begin{eqnarray}
\f{\cal C}(x,x_s,y) &=& g^-(x_s,y) \, \f{C}(x) \, g(x_s,y) + g^-(x_s,y) \, \f{d} \, g(x_s,y) \\
&=& \f{\cal E} + \f{\cal A}
\end{eqnarray}
$$
with the ''Ehresmann Cartan frame form'' and ''Ehresmann Cartan H-connection form'' defined over $E_G$ as:
$$
\begin{eqnarray}
\f{\cal E}(x, x_s, y) &=& \f{\ve{\cal E}} \f{\cal I} = \f{{\cal E}^I} T_I = g^- \, \f{e} \, g(x_s,y) \in \f{\mathfrak{g}} \\
\f{\cal A}(x, x_s, y) &=& \f{\ve{\cal A}} \f{\cal I} = \f{{\cal A}^I} T_I = g^- \, \f{A} \, g(x_s,y) + g^- \, \f{d} \, g(x_s,y) \in \f{\mathfrak{g}}
\end{eqnarray}
$$
with $g(x_s,y) = r(x_s) \, h(y)$.
This splitting is not a natural thing to do for the Ehresmann Cartan connection or connection form, for which the gauge group is $G$; however, it makes more sense when the Ehresmann Cartan conenction form is pulled back to the [[Cartan homogeneous space bundle]] or [[Cartan H-bundle]].
An ''Ehresmann [[Cartan geometry]]'' modeled on an $n_K$ dimensional [[homogeneous space]], $S=G/H$, is described by an [[Ehresmann principal bundle connection]] (the [[Ehresmann Cartan connection]]), $\f{\ve{\cal A}} = \f{\ve{\cal C}}$, over a $(n_M + n_G)$ dimensional total space, $E_G \sim M \times G$, built from an $n_M = n_S$ dimensional base, $M$, and $n_G$ dimensional fiber, $F = G$. This fiber of an ''Ehresmann Cartan geometry'' has a subgroup, $H \subset G$, so the bundle produces two [[associated]] bundles, the [[Cartan H-bundle]], $E_H \sim M \times H$, and the [[Cartan homogeneous space bundle]], $E_S \sim M \times S$. The Ehresmann Cartan connection gives the ''Ehresmann Cartan connection form'', $\f{\cal C} = \f{\ve{\cal C}} \f{\cal I} \in \f{\mathfrak{g}}$, which gives the associated [[Cartan H-bundle connection form|Cartan H-bundle]], $\f{{\cal C}_H}$, over $E_H$ and [[Cartan homogeneous space connection form|Cartan homogeneous space bundle]], $\f{{\cal C}_S}$, over $E_S$. A ''generalized Ehresmann Cartan geometry'' has $n_M \neq n_S$.
There is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, cover patches of the base manifold, $M$, the $n_H$ coordinates, $y^p$, correspond to elements $h(y) \in H \subset G$, and the remaining $n_S$ homogeneous space coordinates, $x_s^a$, correspond to $x_s \in S$. So the combined coordinates, $(x_s, y)$, cover patches of $G$ and the total combined coordinates, $(x,x_s,y)$, cover patches of $E_G$ -- so a point of $E_G$ may be written as
$$
p \sim (x,x_s,y) \sim (x,x_s,h(y)) \sim (x,g(x_s,y))
$$
The chosen [[coset representative section|homogeneous space]], $r : S \to G$, allows points of $G$ to be specified in terms of points of $G/H$ and $H$ via the right action,
$$
g(x_s, y) = R_{h(y)} r(x_s) = r(x_s) \, h(y)
$$
The ''Cartan geometry [[reference section|Ehresmann gauge transformation]]'', $\si^G : M \to E_G$, is then determined by the reference section, $\si^H : M \to E_H$, of the Cartan H-bundle and the reference section, $\si^S : M \to E_S$, of the Cartan homogeneous space bundle. With $\si^H(x) = {\big (} x,h(y_\si(x)) {\big )}$ and $\si^S(x) = (x, x_{s\si}(x))$ we have:
$$
\si^G(x) = {\big (} x, r(x_{s\si}(x)) \, h(y_\si(x)) {\big )}
$$
The [[canonical reference section|Ehresmann principal bundle connection]], $\si_0^G(x) = (x,1) \sim (x,0,0)$, of $E_G$ corresponds to the canonical reference section, $\si_0^H(x) = (x,1) \sim (x,0)$, of $E_H$ and the zero point reference section, $\si_0^S(x) = (x, 0)$, of $E_S$.
The Ehresmann Cartan geometry total space, $E_G$, is not only a bundle over $M$ -- it is also a bundle over $E_H$ and over $E_S$. The fundamental bundle maps, $\pi^G_H : E_G \to E_H$ and $\pi^G_S : E_G \to E_S$, are given by $\pi^G_H(x,x_s,y)=(x,y)$ and $\pi^G_S(x,x_s,y) = (x,x_s)$. There are also reference sections, $\si'^S : E_H \to E_G$ and $\si'^H : E_S \to E_G$, over these bases, determined by the reference sections over their partner bundle, $\si'^S(x,y)=(x,x_{s\si}(x),y)$ and $\si'^H(x,x_s)=(x,x_s,y_\si(x))$. The complete web of bundle maps is summarized by:
$$
\begin{array}{ccc}
E_G & \matrix{\lower8mu {\overset{\si'^S}{\longleftarrow}}\\ \raise8mu {\underset{\pi^G_H}{\longrightarrow}}} & E_H\\
{}^{\pi^G_S} \! {\big \downarrow} {\big \uparrow} \! {}_{\si'^H} & {}_{\pi_G} \! \! \! \! \searrow \! \! \nwarrow \! \! \! \! {}^{\si^G} & {}^{\pi_H} \! {\big \downarrow} {\big \uparrow} \! {}_{\si^H}\\
E_S & \matrix{\lower8mu {\overset{\si^S}{\longleftarrow}}\\ \raise8mu {\underset{\pi_S}{\longrightarrow}}} & M
\end{array}
$$
and we have
$$
\begin{eqnarray}
\pi_G &=& \pi_H \circ \pi^G_H = \pi_S \circ \pi^G_S \\
\si^G &=& \si'^H \circ \si^S = \si'^S \circ \si^H
\end{eqnarray}
$$
The geometry of an Ehresmann Cartan geometry and its associated bundles is described by the [[Ehresmann Cartan connection]] and its curvature.
The geometry of a [[fiber bundle]] may be described via a [[connection]], $\f{A}$, and [[covariant derivative]], $\f{\na}$, defined over the base manifold, $M$, or alternatively via an ''Ehresmann connection'', $\f{\ve{\cal A}}$, defined over the total space, $E$, of the bundle. This [[vector valued form]] is a [[vector projection]], $\f{\ve{\cal A}}\f{\ve{\cal A}}=\f{\ve{\cal A}}$, that succinctly describes the geometric structure of the bundle, including the [[Lie group]] symmetry. As a projection, it splits the tangent vector space at each point, $p$, of $E$, into range and kernel subspaces,
$$
T_p E = V_r + V_0
$$
The range subspace of $\f{\ve{\cal A}}$ is the ''vertical subspace'', $V_r=V_V$, and the collection of these vertical vector fields over $E$ is an involutive [[distribution]], $\ve{\De_V}=\ve{\De_r}$, of vectors tangent to the fibers of the bundle, $\pi_* \ve{\De_V} = 0$. In this way, the Ehresmann connection determines the fibers of the fiber bundle. The vector fields, $\ve{\xi_A} \sim T_A$, in $\ve{\De_V}$ are the flow fields of the group action on the fibers. They are in involution since
$$
\lb \ve{\xi_A},\ve{\xi_B} \rb_L = C_{AB}{}^C \ve{\xi_C}
$$
The kernel subspace of $\f{\ve{\cal A}}$ is the ''horizontal subspace'', $V_0=V_H$, and the collection of these horizontal vector fields over $E$ form a ''horizontal distribution'', $\ve{\De_H}=\ve{\De_0}$, that may or may not be in involution (more on that further down).
The Ehresmann connection respects the symmetry of the structure group. If we take the group action on manifold points to be a right action $p \mapsto R_g p = pg$, the Ehresmann connection at different points along fibers are related by the [[pullback]],
$$
R_g^* \f{\ve{\cal A}} (pg) = \f{\ve{\cal A}} (p)
$$
In this way the Ehresmann connection is related to the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] on the fiber. Note that the 1-form and vector parts of the Ehresmann connection are being pulled back -- in coordinates this could well be written as
$$
\lp R_g^* \f{dz^i} \rp {\cal A}_i{}^j (pg) \ve{\pa_j} = \f{dz^i} {\cal A}_i{}^j (p) \lp R_{g*} \ve{\pa_j} \rp
$$
The vertical and horizontal distributions satisfy
\begin{eqnarray}
\ve{\De_V} \f{\ve{\cal A}} &=& \ve{\De_V} \\
\ve{\De_H} \f{\ve{\cal A}} &=& 0
\end{eqnarray}
and so must also respect the symmetry of the structure group,
\begin{eqnarray}
\ve{\De_V} (p g) &=& R_{g*} \ve{\De_V} (p) \\
\ve{\De_H} (p g) &=& R_{g*} \ve{\De_H} (p)
\end{eqnarray}
so knowing the distributions at any point $p$ in $E$ implies the distributions at any other point on the fiber containing $p$.
Iff the [[FuN curvature]] of the Ehresmann connection vanishes,
$$
\ff{\ve{{\cal F}}} = - \ha \lb \f{\ve{\cal A}},\f{\ve{\cal A}} \rb_L = 0
$$
the horizontal distribution is also in involution and may be integrated to get ''horizontal section''s.
Refs:
*http://philsci-archive.pitt.edu/archive/00002133/01/geometrie.pdf
*http://www.mat.univie.ac.at/~michor/gaubook.pdf
*http://www.mat.univie.ac.at/~michor/listpubl.html
*http://www.emis.de/monographs/KSM/index.html
For any [[vector valued form]] field, $\nf{\ve{\cal K}}$, on the total space of a [[fiber bundle]], a natural grade 1 [[derivation]] is provided by the [[FuN derivative]] with respect to the [[Ehresmann connection]], defining the ''Ehresmann covariant derivative'',
$$
\f{\cal D} \nf{\ve{\cal K}} = - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
$$
Once a choice of [[gauge|Ehresmann gauge transformation]] is made, the Ehresmann connection may be expressed in local coordinates as
$$
\f{\ve{\cal A}}(x,y) = \f{A^B}(x) \ve{\xi_B}(y) + \f{\ve{W}}(y)
$$
If the vector valued form field is right invariant over the total space and may be written as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi_B}(y)
$$
then, using a couple of [[FuN identities]], its Ehresmann covariant derivative is
\begin{eqnarray}
\f{\cal D} \nf{\ve{\cal K}} &=& - \lb \f{\ve{\cal A}}, \nf{\ve{\cal K}} \rb_L = - \lb \f{A^B}(x) \ve{\xi_B}(y), \nf{K^C}(x) \ve{\xi_C}(y) \rb_L - \lb \f{\ve{W}}(y), \nf{K^C}(x) \ve{\xi_C}(y) \rb_L \\
&=& - \f{A^B} \nf{K^C} \lb \ve{\xi_B}, \ve{\xi_C} \rb_L - \f{A^B} \lp {\cal L}_{\ve{\xi_B}} \nf{K^C} \rp \ve{\xi_C}
+ \lp {\cal L}_{\ve{\xi_C}} \f{A^B} \rp \nf{K^C} \ve{\xi_B}
+ \lp \f{d} \f{A^B} \rp \ve{\xi^B} \nf{K^C} \ve{\xi_C}
+ \lp \ve{\xi^C} \f{A^B} \rp \lp \f{d} \nf{K^C} \rp \ve{\xi_B} \\
&-& \lp \f{\ve{W}} \f{\pa} \rp \nf{K^C} \ve{\xi_C} + \lp -1 \rp^k \lp \nf{K^C} \ve{\xi_C} \f{\pa} \rp \f{\ve{W}} + \lp \f{\pa} \f{\ve{W}} \rp \nf{K^C} \ve{\xi_C} + \lp \f{\pa} \nf{K^C} \ve{\xi_C} \rp \f{\ve{W}} \\
&=& \lp \f{d} \nf{K^C} \rp \ve{\xi_C} - \f{A^B} \nf{K^C} \lb \ve{\xi_B}, \ve{\xi_C} \rb_L
\end{eqnarray}
An [[Ehresmann connection]] may be described in local coordinates by choosing a ''reference [[section|fiber bundle]]'', $\si_0$, that maps from some base manifold, $M$, to the total space, $E$. If coordinates $x^a$ are used in a local patch over $M$, and coordinates $y^p$ are used in patches over a typical fiber, these coordinates can be chosen so $y=0$ on the reference section, and the Ehresmann connection can be written locally over $E$ as
$$
\f{\ve{\cal A}}(x,y) = \f{dx^a} A_a{}^B(x) \xi_B{}^p(y) \ve{\pa_p} + \f{dy^p} \ve{\pa_p} = \f{A^B} \ve{\xi_B} + \f{\ve{\cal I}}
$$
In which $\ve{\xi_B}$ are the [[right (or left) invariant vector fields|Lie group geometry]] on the fibers. Another section ([[gauge|gauge transformation]]), $\si:M \rightarrow E$, may be chosen by flowing the original section along a [[diffeomorphism]] along the fibers, $\ph(x,y) = (x,y_\ph(x,y))$, to $\si = \ph \circ \si_0$. The new section is described in the original coordinates by $y^p_\si(x)$. Since the Ehresmann connection is valued in $TE$ it can't be [[pulled back|pullback]] along the section; however, the [[vector projection onto a section]],
$$
\f{\ve{P_\si}} = \f{dx^a} \ve{\pa_a} + \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} \ve{\pa_p}
$$
can be used to project to the TE valued 1-form on the section,
$$
\f{\ve{{\cal A}_\si}} = \f{\ve{P_\si}} \f{\ve{\cal A}} = \f{dx^a} \lp A_a{}^B(x) \xi_B{}^p(y_\si) + \fr{\pa y_\si^p}{\pa x^a} \rp \ve{\pa_p}
$$
The Ehresmann connection everywhere in the total space is determined by the connection components, $A_a{}^B(x)$, on a chosen section. Changing to the connection on a different section is called a passive [[gauge transformation]].
An alternative way of effecting a gauge transformation is to flow the Ehresmann connection by the diffeomorphism, $\f{\ve{\cal A'}} = \phi^*\f{\ve{\cal A}}$, along the fibers while projecting it onto the original section, $\si_0$. This is called an ''active gauge transformation'', and gives the same result,
$$
\f{\ve{{\cal A'}_{\si_0}}} = \f{\ve{P_{\si_0}}} \f{\ve{\cal A'}} = \f{\ve{P_{\si_0}}} \ph^* \f{\ve{\cal A}} = \f{\ve{P_{\ph \circ \si_0}}} \f{\ve{\cal A}} = \f{\ve{P_\si}} \f{\ve{\cal A}} = \f{\ve{\cal A_\si}}
$$
Ref:
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
It was enlightening to consider the [[Ehresmann principal bundle connection]] as a construction in the entire space, $E$, of a [[principal bundle]] with base space, $M=S$. It is equally enlightening to consider the ''Ehresmann homogeneous space geometry'' as the analogous construction in the [[Lie group geometry]], $G$, with a base space that is a [[homogeneous space geometry]], $S = G/H$.
Points of the [[homogeneous space]], $x = [r(x)] \in M = G/H$, are mapped to $G$, by the homogeneous reference section, $r: S \to G$, and any point (element) in $G$, as a function of coordinates $x$ of $M$ and $y$ of $H$, may be specified by
$$
g(z) = g(x,y) = r(x) h(y) \in G
$$
in which $h(y) \in H$ operates on $r(x) \in G$ via the [[right action|group]]. In these coordinates, adapted to the reference section, the reference section is the [[submanifold]] corresponding to $y=0$. The [[Maurer-Cartan form]], $\f{\cal I} = \f{{\cal I}^J} T_J$, over $G$, is
$$
\begin{eqnarray}
\f{\cal I}(z) &=& g^- \f{d} g = h^- \lp r^- \f{d} r \rp h + h^- \f{d} h \\
&=& h^- \f{I} h + h^- \f{d} h \\
&=& \f{{\cal E}_S} + \f{{\cal A}_S}
\end{eqnarray}
$$
When $H$ is [[reductive]] in $G$ (which is assumed) the [[Maurer-Cartan frame|homogeneous space geometry]],
$$
\f{I}(x) = r^- \f{d} r = \f{e_S} + \f{A_S}
$$
splits into the homogeneous space frame, $\f{e_S}(x) = \f{e_S^A} K_A \in \f{\mathfrak{g}/\mathfrak{h}}$, and homogeneous H-connection, $\f{A_S}(x) = \f{A_S^P} H_P \in \f{\mathfrak{h}}$. These correspond to the ''Ehresmann homogeneous space frame form'' and ''Ehresmann homogeneous H-connection form'' over $G$,
$$
\begin{eqnarray}
\f{{\cal E}_S} = \f{{\cal E}_S^A} K_A &=& h^- \f{e_S} h \in \f{\mathfrak{g}/\mathfrak{h}} \\
\f{{\cal A}_S} = \f{{\cal A}_S^P} H_P &=& h^- \f{A_S} h + h^- \f{d} h \in \f{\mathfrak{h}}
\end{eqnarray}
$$
Their 1-form components are computed using the [[Killing form]],
$$
\begin{eqnarray}
\f{{\cal E}_S^A}(z) &=& \f{e_S^B} \lp K^A, h^- K_B h \rp = \lp L^h \rp^A{}_B \, \f{e_S^B}(x) \\
\f{{\cal A}_S^P}(z) &=& \f{A_S^Q} \lp H^P, h^- H_Q h \rp + \lp H^P, h^- \f{d} h \rp = \lp L^h \rp^P{}_Q \, \f{A_S^Q}(x) + \f{{\cal I}_H^P}(y)
\end{eqnarray}
$$
with the appearance of the [[left-right rotator]]s, $\lp L^h \rp^I{}_J(y)$, and the Maurer-Cartan form, $\f{{\cal I}_H}$, for $H$. These are the same as the frame components, $\f{e^A}(z) = \f{{\cal E}_S^A}(z)$ and $\f{e^P}(z) = \f{{\cal A}_S^P}(z)$, for a [[reductive Lie group geometry]]. Using the correspondence between the Lie algebra generators and the left invariant vector fields of the Lie group geometry, $T_I \sim \ve{\xi^R_I}$, allows us to write the [[Ehresmann-Maurer-Cartan vector valued form|Maurer-Cartan form]] as
$$
\begin{eqnarray}
\f{\ve{\cal I}} &=& \f{\ve{{\cal E}_S}} + \f{\ve{{\cal A}_S}} \\
&=& \f{{\cal E}_S^A} \ve{\xi^R_A} + \f{{\cal A}_S^P} \ve{\xi^R_P} \\
&=& \f{e_S^B} \lp L^h \rp^A{}_B \, \ve{\xi^R_A} + \f{A_S^Q} \lp L^h \rp^P{}_Q \, \ve{\xi^R_P} + \f{\ve{{\cal I}_H}} \\
&=& \f{I^J} \lp L^r \rp_J{}^K \, \ve{\xi^L_K} + \f{\ve{{\cal I}_H}}
\end{eqnarray}
$$
with the ''Ehresmann homogeneous space frame'', $\f{\ve{{\cal E}_S}}$, and ''Ehresmann homogeneous H-connection'', $\f{\ve{{\cal A}_S}}$, satisfying $\f{\ve{{\cal E}_S}} \f{\cal I} = \f{{\cal E}_S}$ and $\f{\ve{{\cal A}_S}} \f{\cal I} = \f{{\cal A}_S}$ using the Maurer-Cartan form, $\f{\cal I} = \f{\xi_R^J}(z) T_J$, over $G$. The Ehresmann homogeneous H-connection, $\f{\ve{{\cal A}_S}} = \f{{\cal A}_S^P} \ve{\xi^R_P}(z)$, is a [[Ehresmann principal bundle connection]] for an H-bundle and satisfies $\f{\ve{{\cal A}_S}} \f{\cal I_H} = \f{{\cal A}_S}$, since $\ve{\xi^R_P}(z) = \ve{\xi^{HR}_P}(y)$ in a reductive Lie group geometry. Another way of looking at an Ehresmann homogeneous space geometry is as a [[Cartan H-bundle]] with $\f{C} = \f{I}$.
Since the Ehresmann-Maurer-Cartan VVF is the identity projection, its [[FuN curvature]] vanishes, $\ff{\ve{\cal F}} = -\ha \lb \f{\ve{\cal I}}, \f{\ve{\cal I}} \rb_L = 0$.
from [[Ehresmann connection]]
equivalent to [[parallel transport]]
[<img[images/png/fiber bundle.png]]A [[principal bundle]] consists of a total space, $E$, built locally from the direct product of a [[Lie group geometry]] (the typical fiber, $F=G$) over a base manifold, $M$. The same [[Lie group]], $G$, is the structure group, acting on the fibers, and hence on the total space, via left action. This group also acts on the fibers, and the total space, via right action. A [[connection]], $\f{A}(x)=\f{A^B}T_B=\f{dx^a}A_a{}^BT_B$, a [[Lieform]] over the base space, describes principal bundle geometry. By choosing a [[reference section|Ehresmann gauge transformation]], $\si_0:M \to E$, this connection may be related to an [[Ehresmann connection]], $\f{\ve{\cal A}}$, over the total space.
There is a convenient set of local coordinates for the total space. The $n_M$ coordinates, $x^a$, with [[spacetime]] [[indices]], cover patches of the base manifold and the $n_G$ coordinates, $y^p$, are from the typical fiber. So a point of $E$ may be described by $p=(x,y)$ or equivalently by $p=(x,g)$ -- where $g(y)$ is the Lie group (fiber) element parameterized by $y$. The fiber bundle projection is then simply $\pi(x,y)=x$. The coordinates are chosen so that $y^p=0$, and hence $g=1$, on the ''canonical reference section'', $\si_0(x)=(x,y_0(x))=(x,0) \sim (x,1)$, which provides the ''canonical local trivialization'', $(x,g) \in E$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The $y$ [[coordinate basis vectors]] are in the vertical subspace, $\ve{\pa_p} \in \ve{\De_V}$, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\ve{\pa_a} \notin \ve{\De_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the ''Ehresmann principal bundle connection'' (a [[vector projection]]) over the total space is
$$
\f{\ve{\cal A}}(x,y) = \f{A}^B(x) \ve{\xi}{}^L_B(y) + \f{\ve{\cal I}}
$$
in which $\ve{\xi^L_B}$ are the [[left action vector fields|Lie group geometry]] for the Lie group geometry, defined by
$$
\ve{\xi^L_B} \f{\pa} g(y) = T_B g
$$
and
$$
\f{\ve{\cal I}} = \f{\xi_L^B}(y) \ve{\xi^L_B}(y) = \f{\xi_R^B}(y) \ve{\xi^R_B}(y) = \f{dy^p} \ve{\pa_p}
$$
is the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers). The Ehresmann connection is a projection, $\f{\ve{\cal A}} \f{\ve{\cal A}} = \f{\ve{\cal A}}$, is [[right invariant]], $R_h^*\f{\ve{\cal A}} = \f{\ve{\cal A}}$, and has a natural [[spectral decomposition|spectral decomposition of the Ehresmann principal bundle connection]]. It may also be written as a Lie algebra valued 1-form over the total space by [[contracting|vector-form algebra]] it with the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B}(y)T_B = g^- \f{d} g$, over the total space to get the ''Ehresmann connection form'',
\begin{eqnarray}
\f{\cal A}(x,y) &=& \f{\ve{\cal A}} \f{\cal I} = \f{A^B}(x) \ve{\xi^L_B}(y) \f{\xi_R^C}(y) T_C + \f{\ve{\cal I}} \f{\cal I} \\
&=& \lp \f{A^B} L^C{}_B(y) + \f{\xi_R^C} \rp T_C \\
&=& g^-(y) \f{A}(x) g(y) + g^-(y) \f{d^y} g(y)
\end{eqnarray}
using the defining equation for the [[left-right rotator]],
$$
L^C{}_B(y) = \ve{\xi^L_B}(y) \f{\xi_R^C}(y) = \lp T^C, g^-(y) T_B g(y) \rp
$$
This form pulls back along the canonical reference section ($y=0$) to give the principal bundle connection,
$$
\si_0^*\f{\cal A} = \f{A}(x)
$$
and satisfies $R_h^* \f{\cal A} = h^- \f{\cal A} h$ under the right action.
The ''[[FuN curvature]] of the Ehresmann principal bundle connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,y) &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L = \lp 1 - \f{\ve{\cal A}} \rp \lp \f{\pa} \f{\ve{\cal A}} \rp = \lp 1 - \f{\ve{\cal A}} \rp \lp \lp \f{\pa^x} + \f{\pa^y} \rp \f{\ve{\cal A}} \rp \\
&=& \lp 1 - \f{A^B}(x) \ve{\xi^L_B}(y) - \f{dy^p} \ve{\pa_p} \rp
\lp \lp \f{d^x} \f{A^C} \rp \ve{\xi^L_C}(y) - \f{A^C} \f{\pa^y} \ve{\xi^L_C} \rp \\
&=& \lp \f{d^x} \f{A^C} \rp \ve{\xi^L_C} - \f{A^B} \f{A^C} \ve{\xi^L_B} \f{\pa^y} \ve{\xi^L_C} \\
&=& \lp \f{d^x} \f{A^D} + \ha \f{A^B} \f{A^C} C_{BC}{}^D \rp \ve{\xi^L_D}(y)
\end{eqnarray}
using the [[Lie bracket for left action vector fields|Lie group geometry]],
$$
\ve{\xi^L_{\lb B \rd}} \lp \f{\pa} \ve{\xi^L_{\ld C \rb}} \rp = \ha \lb \ve{\xi^L_B} , \ve{\xi^L_C} \rb_L = - \ha C_{BC}{}^D \ve{\xi^L_D}
$$
This curvature is vector valued in the vertical subspace, and is right invariant, $R_h^* \ff{\ve{\cal F}} = \ff{\ve{\cal F}}$. The ''FuN curvature form of the Ehresmann connection'' is a ${\frak g}$ valued 2-form over $E$,
\begin{eqnarray}
\ff{\cal F} &=& \ff{\ve{\cal F}} \f{\cal I} = \lp \f{d^x} \f{A^D} + \ha \f{A^B} \f{A^C} C_{BC}{}^D \rp g^-(y) T_D g(y) \\
&=& g^-(y) \lp \f{d^x} \f{A} + \ha \lb \f{A}, \f{A} \rb \rp g(y) \\
&=& g^-(y) \lp \f{d^x} \f{A} + \f{A} \f{A} \rp g(y)
\end{eqnarray}
This form pulls back along the canonical reference section to give the [[principal bundle]] curvature,
$$
\si_0^*\ff{\cal F} = \f{d} \f{A} + \f{A} \f{A} = \ff{F}(x)
$$
and satisfies $R_h^* \ff{\cal F} = h^- \ff{\cal F} h$ under the right action.
The [[Ehresmann covariant derivative]] of a [[vector valued form]] field, $\nf{\ve{\cal K}}$, over the total space of a [[principal bundle]] using an [[Ehresmann principal bundle connection]] is
$$
\f{\cal D} \nf{\ve{\cal K}} = - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
$$
using the [[FuN derivative]]. The VVF will usually be right invariant and expressible as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi^L_B}(y)
$$
corresponding to the [[Lieform]],
$$
\nf{\cal K} = \nf{\ve{\cal K}} \f{\cal I} = \nf{K^B}(x) g^-(y) T_B g(y)
$$
obtained with the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B} T_B$, and the [[left-right rotator]]. The [[pullback]] of this form along a section, $\si_1=(x,g_1(x))$, gives the Lieform over the base,
$$
\nf{K_1}(x) = \si_1^* \nf{\cal K} = \nf{K^B}(x) g^-_1(x) T_B g_1(x) = g^-_1(x) \nf{K}(x) g_1(x)
$$
in which the form pulled back along the reference section is $\nf{K}(x) = \nf{K^B}(x) T_B$. The Ehresmann covariant derivative of the VVF,
$$
\f{\cal D} \nf{\ve{\cal K}} = \lp \f{d} \nf{K^C} \rp \ve{\xi^L_C} - \f{A^B} \nf{K^C} \lb \ve{\xi^L_B}, \ve{\xi^L_C} \rb_L
= \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp \ve{\xi^L_D}
$$
gives a definition for the ''Ehresmann covariant derivative of a Lieform'',
$$
\f{\cal D} \nf{\cal K} = \lp \f{\cal D} \nf{\ve{\cal K}} \rp \f{\cal I} = \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp g^-(y) T_D g(y)
$$
which pulls back along any chosen section to give the [[covariant derivative|principal bundle]] of $\nf{K}$ on the base,
\begin{eqnarray}
\lp \f{D_1} \nf{K_1} \rp(x) &=& \si_1^* \lp \f{\cal D} \nf{\cal K} \rp
= \lp \f{d} \nf{K^D} + \f{A^B} \nf{K^C} C_{BC}{}^D \rp g^-_1(x) T_D g_1(x) \\
&=& g^-_1(x) \lp \f{d} \nf{K} + \lb \f{A}, \nf{K} \rb \rp g_1(x)
= g^-_1(x) \lp \f{\na} \nf{K} \rp g_1(x)
\end{eqnarray}
(This should be equivalent to the usual definition of the Ehresmann covariant derivative of Lieform via
$$
\ve{v_1} \ve{v_2} \dots \ve{v_{k+1}} \f{\cal D} \nf{\cal K} = \ve{v^H_1} \ve{v^H_2} \dots \ve{v^H_{k+1}} \f{d} \nf{\cal K}(x,y)
$$
in which the vectors on the right are horizontal projections of the ones on the left, $\ve{v^H} = \ve{v}(1-\f{\ve{\cal A}})$. //That definition needs to be checked.//)
Similarly, if a VVF is left invariant and may be expressed as
$$
\nf{\ve{\cal K}} = \nf{K^B}(x) \ve{\xi^R_B}(y)
$$
corresponding to the Lieform,
$$
\nf{\cal K} = \nf{\ve{\cal K}} \f{\cal I} = \nf{K^B}(x) T_B
$$
The [[pullback]] of this form along any section, $\si_1=(x,g_1(x))$, gives the same Lieform over the base,
$$
\nf{K_1}(x) = \si_1^* \nf{\cal K} = \nf{K^B}(x) T_B = \nf{K}(x)
$$
The Ehresmann covariant derivative of this VVF,
$$
\f{\cal D} \nf{\ve{\cal K}} = \lp \f{d} \nf{K^C} \rp \ve{\xi^R_C} - \f{A^B} \nf{K^C} \lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L
= \lp \f{d} \nf{K^C} \rp \ve{\xi^R_C}
$$
gives the Lieform,
$$
\f{\cal D} \nf{\cal K} = \lp \f{\cal D} \nf{\ve{\cal K}} \rp \f{\cal I} = \lp \f{d} \nf{K^C} \rp T_C = \f{d} \nf{\cal K}
$$
which pulls back along any chosen section to give the [[exterior derivative]] of $\nf{K}$ on the base,
$$
\lp \f{D_1} \nf{K_1} \rp = \si_1^* \lp \f{\cal D} \nf{\cal K} \rp
= \f{d} \nf{K}
$$
A passive [[Ehresmann gauge transformation]] for an [[Ehresmann principal bundle connection]] corresponds to changing to a different choice of section along which to pull back the Ehresmann connection form. The choice of reference section is equivalent to the choice of a local trivialization for a [[fiber bundle]]. Once a reference section, $\si_0:M \to E$, and principal bundle connection, $\f{A}$, have been used to build the principal bundle Ehresmann connection, $\f{\ve{\cal A}}$, a different section, $\si_1$, can be introduced and used to pull back a different principal bundle connection, $\f{A'}$ -- this is a [[gauge transformation]]. Using coordinates adapted to the reference section, the Ehresmann connection is
$$
\f{\ve{\cal A}}(x,y) = \f{A^B} \ve{\xi^L_B}(y) + \f{\ve{\cal I}}
$$
and the Ehresmann connection form is
$$
\f{\cal A} = \f{\ve{\cal A}} \f{\cal I} = g^-(y) \f{A}(x) g(y) + g^- \f{d^y} g(y)
$$
using the [[Maurer-Cartan form]], $\f{\cal I}(y) = \f{\xi_R^B} T_B$, and [[left-right rotator]]. The new section, $\si_1 = \ph \circ \si_0$, may be obtained by flowing the reference section by an equivariant vertical [[diffeomorphism]], $\ph(x,y) = (x,y_\ph(x,y))$, satisfying $\ph(ph)=\ph(p)h$ and giving $\si_1(x) = (x,y_1(x)) = (x,y_\ph(x,0))$. This is equivalent to transforming the original section by right (//?//) action by an element of $G$ to $\si_1(x) = \si_0(x) \, g(y_1(x)) = \si_0 \, g_1(x)$. The [[vector projection onto a section]], $\si_1$, is
$$
\f{\ve{P_1}} = \f{dx^a} \ve{\pa_a} + \f{dx^a} \fr{\pa y_1^p}{\pa x^a} \ve{\pa_p}
$$
and is used to project the Ehresmann connection to
$$
\f{\ve{{\cal A}_1}} = \f{\ve{P_1}} \f{\ve{\cal A}} = \f{A^B}(x) \ve{\xi^L_B}(y_1) + \f{dx^a} \fr{\pa y_1^p}{\pa x^a} \ve{\pa_p}
$$
and the Ehresmann connection form to
$$
\f{{\cal A}_1} = \f{\ve{P_1}} \f{\cal A} = \f{\ve{{\cal A}_1}} \f{\cal I} = \f{\ve{P_1}} \f{\ve{\cal A}} \f{\cal I} = g^-(y_1(x)) \f{A}(x) g(y_1(x)) + g^-(y_1(x)) \f{d^x} g(y_1(x))
$$
on the section. This, the gauge transformed connection, gives the [[pullback]] of the Ehresmann connection form along the section,
$$
\f{A'}(x) = \si_1^* \f{\cal A} = \si_1^* \f{{\cal A}_1} = g^-_1(x) \f{A} g_1(x) + g^-_1(x) \f{d^x} g_1(x)
$$
identified as the [[principal bundle gauge transformation|principal bundle]] with $g_1(x)=g^-(x)$.
An alternative way of effecting a gauge transformation is to flow the Ehresmann connection form by the diffeomorphism, $\f{\cal A'} = \ph^*\f{\cal A}$, then pull it back along the original section, $\si_0$. This ''active gauge transformation'' gives the same result,
$$
\si_0^* \f{\cal A'} = \si_0^* \lp \ph^* \f{\cal A} \rp
= \lp \ph \circ \si_0 \rp^* \f{\cal A}
= \si_1^* \f{\cal A} = \f{A'}
$$
Ref:
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
[[Ehresmann lift]] for an [[Ehresmann principal bundle connection]].
A [[vector bundle]] consists of a total space, $E$, built locally from the direct product of a [[vector space]] (the typical fiber, consisting of elements $v = v^\al b_\al \in V = F$) and a base manifold, $M$. A [[vector bundle connection]], $\f{A}{}_\al{}^\be(x) = \f{dx^i} A_{i \al}{}^\be(x)$, a 1-form over the base space valued in some subgroup of the general linear group, describes the geometry of the vector bundle. By choosing a ''reference [[section|fiber bundle]]'', $\si_0:M \to E$, this connection may be related to an [[Ehresmann connection]], $\f{\ve{\cal A}}$, over the total space.
There is a convenient set of local coordinates for the total space. The $n$ coordinates, $x^a$, with [[spacetime]] [[indices]], are from the base manifold and the $K$ coordinates, $v^\al$, are from the typical fiber (vector space). So a point of $E$ (above some patch of $M$) may be described by $p = (x,v)$. The fiber bundle projection is then simply $\pi(x,v) = x$. The coordinates are chosen so that $v^\al = 0$ on the reference section, a ''canonical local trivialization'', $\si_0(x) = (x,v_0(x)) = (x,0)$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The [[coordinate basis vectors]] for the $v^\al$ coordinates, $\ve{\pa_\al} \in \ve{\De_V}$, are in the vertical subspace, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\ve{\pa_a} \notin \ve{\De_H}$. We will abuse the use of the same label, $x^a$, for the coordinates on the base and some on the total space. Using these coordinates, the linear ''Ehresmann vector bundle connection'' over the total space is
$$
\f{\ve{\cal A}}(x,v) = \lp \f{dx^i} A_{i \al}{}^\be(x) v^\al + \f{dv^\be} \rp \ve{\pa_\be}
$$
In analogy with the [[Maurer-Cartan form]], we build a vector ($V$) valued 1-form,
$$
\f{\cal I} = \f{dv^\be} b_\be
$$
and use it to define the ''Ehresmann vector bundle connection form'',
$$
\f{\cal A} = \f{\ve{\cal A}} \f{\cal I} = \lp \f{A}{}_\al{}^\be v^\al + \f{dv^\be} \rp b_\be
$$
This allows us to define the [[vector bundle covariant derivative|vector bundle connection]] of any section, $\si_1 : M \mapsto E$, $\si_1(x) = (x, v_1(x))$, as the [[pullback]] of $\f{\cal A}$ along the section to M,
$$
\si_1^* \f{\cal A} = \f{A}{}_\al{}^\be v_1^\al(x) b_\be + \f{dx^a} \fr{\pa v_1^\be}{\pa x^a} b_\be = \f{\na} v_1(x)
$$
The ''[[FuN curvature]] of the Ehresmann vector bundle connection'' is
\begin{eqnarray}
\ff{\ve{\cal F}}(x,y) &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L = \lp 1 - \f{\ve{\cal A}} \rp \lp \f{\pa} \f{\ve{\cal A}} \rp \\
&=& \lp 1 - \f{A}{}_\al{}^\be v^\al \ve{\pa_\be} - \f{dv^\be} \ve{\pa_\be} \rp
\lp \lp \f{d^x} \f{A}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de} - \f{A}{}_\ga{}^\de \f{dv^\ga} \ve{\pa_\de} \rp \\
&=& \lp \f{d^x} \f{A}{}_\al{}^\de - \f{A}{}_\al{}^\be \f{A}{}_\be{}^\de \rp v^\al \ve{\pa_\de} \\
&=& \ff{F}{}_\al{}^\de v^\al \ve{\pa_\de}
\end{eqnarray}
in which the [[vector bundle curvature]], $\ff{F}{}_\al{}^\de$, appears.
The [[Ehresmann covariant derivative]] of any [[vector valued form]] over the total space (such as the curvature above) that can be written as
$$
\nf{\ve{\cal K}} = \nf{K}{}_\ga{}^\de(x) v^\ga \ve{\pa_\de}
$$
is defined using the [[FuN derivative]] as
\begin{eqnarray}
\f{\cal D} \nf{\ve{\cal K}} &=& - {\cal L}_{\f{\ve{\cal A}}} \nf{\ve{\cal K}}
= - \f{\ve{\cal A}} \lp \f{\pa} \nf{\ve{\cal K}} \rp + \lp -1 \rp^k \nf{\ve{\cal K}} \lp \f{\pa} \f{\ve{\cal A}} \rp + \f{\pa} \lp \nf{\ve{\cal K}} \f{\ve{\cal A}} \rp \\
&=& - \lp \f{A}{}_\al{}^\be(x) v^\al + \f{dv^\be} \rp \ve{\pa_\be} \lp \lp \f{d^x} \nf{K}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de} + \lp -1 \rp^k \nf{K}{}_\ga{}^\de \f{dv^\ga} \ve{\pa_\de} \rp \\
& & + \lp -1 \rp^k \nf{K}{}_\ga{}^\de v^\ga \ve{\pa_\de} \lp \lp \f{d^x} \f{A}{}_\la{}^\be \rp v^\al \ve{\pa_\be} - \f{A}{}_\al{}^\be \f{dv^\al} \ve{\pa_\be} \rp
+ \f{\pa} \lp \nf{K}{}_\ga{}^\de v^\ga \ve{\pa_\de} \rp \\
&=& \lp \f{d^x} \nf{K}{}_\ga{}^\de - \f{A}{}_\ga{}^\be \nf{K}{}_\be{}^\de + \f{A}{}_\be{}^\de \nf{K}{}_\ga{}^\be \rp v^\ga \ve{\pa_\de} \\
&=& \lp \f{\na} \nf{K}{}_\ga{}^\de \rp v^\ga \ve{\pa_\de}
\end{eqnarray}
An [[Ehresmann gauge transformation]] corresponding to a change in local trivialization, $b_\be \mapsto b'_\be = g_\be{}^\al(x) b_\al$, gives changes in $\f{A}{}_\al{}^\be$ and other coefficients corresponding to a [[vector bundle gauge transformation]].
In a [[spacetime]], equivalent to a [[Cl(1,3)]] or [[Cl(3,1)]] [[Clifford vector bundle]], the [[Clifford-Ricci curvature]], $\f{R}$, [[Clifford curvature scalar]], $R$, [[frame]], $\f{e}$, ''cosmological constant'', $\La$, and ''Clifford energy-momentum tensor'', $\f{T}$, for matter are dynamically related by ''Einstein's equation'',
$$
\f{R} - \ha R \f{e} = \et_{00} ( \La \f{e} - 8 \pi G \f{T} )
$$
in which $\et_{00}$ specifies the [[Minkowski metric]] sign convention. In a vacuum, $\f{T} = 0$, Einstein's equation contracted with the coframe, $\ve{e}$, gives
$$
\ve{e} \cdot \lp \f{R} - \ha R \f{e} \rp = R - \ha R n = \et_{00} \La n
$$
requiring the curvature scalar to be constant, $R = - \fr{2n}{n-2} \et_{00} \La = - 4 \et_{00} \La$ (with $n=4$), and giving the ''vacuum Einsten's equation'',
$$
\f{R} = - \fr{2}{n-2} \et_{00} \La \f{e} = - \et_{00} \La \f{e}
$$
Any spacetime that satisfies $\f{R} = \al \f{e}$ for some constant, $\al$, is an ''Einstein space''.
Einstein's equation is derived by extremizing the ''Einstein-Hilbert [[action]]'',
$$
S = \int \nf{e} \lp \fr{1}{16 \pi G} \lp R + 2 \et_{00} \La \rp + L_M \rp
$$
with respect to $\ve{e}$, in natural [[units]].
http://arxiv.org/abs/gr-qc/0606062
*looks to be a good reveiw of ECT
In ECT...
The curvature picks up a contribution from torsion, and the Ricci curvature is no longer guaranteed to be symmetric in its indices. This change in the equation of motion allows matter with a spin component to couple to the angular momentum of the gravitational field.
In teleparallel theories of gravity the spin connection is purely torsional ($\f{\nu}=0$, $\f{d} \f{e}=0$, $\f{\ka} \neq 0$) and the [[spacetime]] is, in that sense, flat, with the gravitational field a force represented solely by torsion.
Ref:
[[Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity|papers/Huang - Cosmological Solutions with Torsion in a Model of de Sitter Gauge Theory of Gravity.pdf]]
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<td><div class="math">
\begin{array}{c}
W =
\lb \begin{array}{cc}
{\small \frac{i}{2}} W^3 & W^+ \\
W^- & {\small - \! \frac{i}{2}} W^3
\end{array} \rb \vp{|_{\Big(}}
\quad
B_1 =
\lb \begin{array}{cc}
{\small \frac{i}{2}} B_1^3 & B_1^+ \\
B_1^- & {\small - \! \frac{i}{2}} B_1^3
\end{array} \rb \vp{|_{\Big(}}
\\
\big[
\lb \begin{array}{cc}
W & \\
& B_1
\end{array} \rb
,
\lb \begin{array}{cc}
& \ph_B \\
\ph_W &
\end{array} \rb
\big] \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad \quad
\ph_{W/B} =
\lb \begin{array}{cc}
- \ph_{0/1} & \ph_+ \\
\ph_- & \ph_{1/0}
\end{array} \rb \vp{|_{\Big(}}
\\
\lb \begin{array}{cc}
W & \\
& B_1
\end{array} \rb
\quad
\lb \begin{array}{c}
\nu_{eL} \\ e_L \\ \nu_{eR} \\ e_R
\end{array} \rb
\quad
\lb \begin{array}{c}
u_L \\ d_L \\ u_R \\ d_R
\end{array} \rb \\[.5em]
\big( \fr{\sqrt{3}}{\sqrt{5}} B_1^3 - \fr{\sqrt{2}}{\sqrt{5}} B_2 \big)
= (\fr{\sqrt{3}}{\sqrt{5}}) \ha Y
\;\to\; g_1=\fr{\sqrt{3}}{\sqrt{5}}
\end{array}
</div></td>
<td> </td>
<td border=none>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(4)</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">W^3</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">B_1^3</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">\fr{\sqrt{2}}{\sqrt{3}} B_2</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">\ha Y</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">Q</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W^+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W^-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">B_1^+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">B_1^-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_+</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_-</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mdia{#4D4D4D}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph_1</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#B2B200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_{eL}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_{eR}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\btri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#BF6000}}{\mtri{#668000}}{\mtri{#8F00B2}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#F77C00}}{\mtri{#99BF00}}{\mtri{#AD00F7}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d_L</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#990000}}{\mtri{#009900}}{\mtri{#0000B2}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\fr{2}{3}</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\trip{\mtri{#D90000}}{\mtri{#00BF00}}{\mtri{#0000F7}}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d_R</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{6}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
</table>
</td>
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<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
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<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
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<SPAN class="math">su(2)_L \oplus u(1)_Y \oplus (2_L+2_R) \otimes (1+1)</SPAN>
</td></tr>
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<img src="talks/RW12/images/Electroweak breaking.png" width="480" height="480">
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<SPAN class="math">Q = Y \cos(\th_W) + W \sin(\th_W)</SPAN>
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<img src="images/png/ewhiggs.png" height="300">
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<SPAN class="math">\;</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\;</SPAN>
</td></tr>
<tr><td>
A fiber twisting around maximal torus inside <SPAN class="math">SU(2)_L \otimes U(1)_Y</SPAN>
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<center>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Forces (bosons)</SPAN></th>
<th><SPAN class="math">\;\;\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Matter (fermions)</SPAN></th>
<th><SPAN class="math">\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Second generation</SPAN></th>
<th><SPAN class="math">\;\;</SPAN></th>
<th ALIGN=CENTER COLSPAN="3"><SPAN>Third generation</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#000000} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{(}_{\big(}} \ga \p{{\Big(}^{(}_{\big(}} </SPAN></td>
<td ALIGN=CENTER><SPAN>electromagnetism</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200} \btri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e</SPAN></td>
<td ALIGN=CENTER><SPAN>electron</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mtri{#F2F200} \mtri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\mu</SPAN></td>
<td ALIGN=CENTER><SPAN>muon</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\stri{#F2F200} \stri{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\tau</SPAN></td>
<td ALIGN=CENTER><SPAN>tau</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">W,Z</SPAN></td>
<td ALIGN=CENTER><SPAN>weak</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\butr{#F2F200} \butr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{e}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{\Big(}}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mutr{#F2F200} \mutr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\mu}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\sutr{#F2F200} \sutr{#D9D9D9}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\tau}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">g</SPAN></td>
<td ALIGN=CENTER><SPAN>strong</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\btri{#B2B200} \btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_e</SPAN></td>
<td ALIGN=CENTER><SPAN>electron <br> neutrino</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mtri{#B2B200} \mtri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_\mu</SPAN></td>
<td ALIGN=CENTER><SPAN>muon <br> neutrino</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\stri{#B2B200} \stri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\nu_\tau</SPAN></td>
<td ALIGN=CENTER><SPAN>tau <br> neutrino</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}^{\big(}_{\big(}} \om \p{{\Big(}^{\big(}_{\big(}}</SPAN></td>
<td ALIGN=CENTER><SPAN>gravity</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\butr{#B2B200} \butr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_e</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\p{{\Big(}_(}</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\mutr{#B2B200} \mutr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_\mu</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\sutr{#B2B200} \sutr{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{\nu}{}_\tau</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mdia{#F2F200} \, \mdia{#BF6000} \\[-.5em]
\msqu{#B2B200} \, \msqu{#F77C00}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ph</SPAN></td>
<td ALIGN=CENTER><SPAN>Higgs</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\btri{#BF6000} \btri{#990000} \\[-.8em]
\btri{#668000} \btri{#009900} \\[-.8em]
\btri{#8F00B2} \btri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">u</SPAN></td>
<td ALIGN=CENTER><SPAN>up <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mtri{#BF6000} \mtri{#990000} \\[-.8em]
\mtri{#668000} \mtri{#009900} \\[-.8em]
\mtri{#8F00B2} \mtri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">c</SPAN></td>
<td ALIGN=CENTER><SPAN>charm <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\stri{#BF6000} \stri{#990000} \\[-.8em]
\stri{#668000} \stri{#009900} \\[-.8em]
\stri{#8F00B2} \stri{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">t</SPAN></td>
<td ALIGN=CENTER><SPAN>top <br> quark</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\butr{#BF6000} \butr{#990000} \\[-.8em]
\butr{#668000} \butr{#009900} \\[-.8em]
\butr{#8F00B2} \butr{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{u}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mutr{#BF6000} \mutr{#990000} \\[-.8em]
\mutr{#668000} \mutr{#009900} \\[-.8em]
\mutr{#8F00B2} \mutr{#0000B2}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{c}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\sutr{#BF6000} \sutr{#990000} \\[-.8em]
\sutr{#668000} \sutr{#009900} \\[-.8em]
\sutr{#8F00B2} \sutr{#0000B2}
\end{array}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{t}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\btri{#F77C00} \btri{#D90000} \\[-.8em]
\btri{#99BF00} \btri{#00BF00} \\[-.8em]
\btri{#AD00F7} \btri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">d</SPAN></td>
<td ALIGN=CENTER><SPAN>down <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mtri{#F77C00} \mtri{#D90000} \\[-.8em]
\mtri{#99BF00} \mtri{#00BF00} \\[-.8em]
\mtri{#AD00F7} \mtri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">s</SPAN></td>
<td ALIGN=CENTER><SPAN>strange <br> quark</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\stri{#F77C00} \stri{#D90000} \\[-.8em]
\stri{#99BF00} \stri{#00BF00} \\[-.8em]
\stri{#AD00F7} \stri{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">b</SPAN></td>
<td ALIGN=CENTER><SPAN>bottom <br> quark</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN class="math"></SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\butr{#F77C00} \butr{#D90000} \\[-.8em]
\butr{#99BF00} \butr{#00BF00} \\[-.8em]
\butr{#AD00F7} \butr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{d}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\mutr{#F77C00} \mutr{#D90000} \\[-.8em]
\mutr{#99BF00} \mutr{#00BF00} \\[-.8em]
\mutr{#AD00F7} \mutr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{s}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">
\begin{array}{c}
\sutr{#F77C00} \sutr{#D90000} \\[-.8em]
\sutr{#99BF00} \sutr{#00BF00} \\[-.8em]
\sutr{#AD00F7} \sutr{#0000F7}
\end{array}
</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\bar{b}</SPAN></td>
<td ALIGN=CENTER><SPAN></SPAN></td>
</tr>
</table>
</center>
</html>
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}}}
Decent intros using MaxEnt:
http://arxiv.org/abs/cond-mat/0507388
http://arxiv.org/abs/physics/9805024
some Q.A.Wang papers
*http://arxiv.org/abs/cond-mat/0312329
**I don't like his use of ergodicity in defining the long time average equal to the Bayesian expectation value.
**nice: uses fixed average action and MaxEnt to get partition function with action per path instead of energy per state
***I might like using Rovelli's Hamiltonian constraint dynamics better.
*http://arxiv.org/abs/cond-mat/0407515
**seems inferior to previous paper
Good Hawking paper on it:
http://arxiv.org/abs/gr-qc/9501014
*boundary term in depth
*Hamiltonian formulation
*relationship between partition functions for static spacetimes (timelike Killing vector).
related discussion on time and Tomita flow in Rovelli book
Nice new paper including possible action for BF gravity:
*http://arxiv.org/pdf/1103.2971v1
Each [[Cl(4)]] Weylish spinor in a ''Euclidean [[twistor]]'', $\{ \ps_L,\ps_R \}$, can be represented as a $\mathbb{R}^4$ column or as a $\mathbb{C}^2$ column using a [[complexification|complex structure]], $\ps_L = M_L \, \ps$ and $\ps_R = M_R \, \ch$,
$$
\ps_L = \fr{1}{\sqrt{2}} \lb \begin{array}{c} i \, \ps^0 + \ps^3 \\ \ps^1 + i \, \ps^2 \end{array}\rb \leftrightarrow \ps = \ps^a e_a
\s
\ps_R = \fr{1}{\sqrt{2}} \lb \begin{array}{c} i \, \ch^0 - \ch^3 \\ - \ch^1 - i \, \ch^2 \end{array}\rb \leftrightarrow \ch = \ch^a e_a
$$
mapping between real [[quaternion]]s and complex Weylish spinors. Explicitly, these are
$$
M_L = \fr{1}{\sqrt{2}} \lb \begin{array}{cccc} i & 0 & 0 & 1 \\ 0 & 1 & i & 0 \end{array} \rb
\s
M_R = \fr{1}{\sqrt{2}} \lb \begin{array}{cccc} i & 0 & 0 & -1 \\ 0 & -1 & -i & 0 \end{array} \rb
$$
This map is based on the identification of a [[complex structure]] on the quaternions,
$$
J_{L/R} = \pm \lb \ba{cccc} & & & 1 \\ & & -1 & \\ & 1 & & \\ -1 & & & \ea \rb
$$
differing in sign for each [[chiral]]ity, satisfying $-i M_{L/R} J_{L/R} = M_{L/R}$. There is also a map between the right chiral vector, $v_R$, and a quaternion, $v$,
$$
v_R = v_R^0 \si_0 - v_R^\va i \, \si_\va \leftrightarrow v = v^a e_a = v^0 e_0 + v^\va e_\va = v^0 \si_0 - v^\va i \, \si_\va
$$
written using [[Pauli matrices]], with the components directly related by $v_L^0 = v^0$ and $v_L^\va = v^\va$. Via [[division algebra confusion]], the quaternionic [[triality]],
$$
T(v,\ps,\ch) = \lp \tilde{\ch}, v \, \ps \rp = v^c \ps^a \ch^b \Ga_{cba} \; \in \; \mathbb{R}
$$
corresponds directly to ''Euclidean twistor triality'', using the [[Hermitian form]],
\begin{eqnarray}
T(v_R,\ps_L,\ps_R) &=& \langle \ps_R | v_R \, \ps_L \rangle = \ps_R^\da \, v_R \, \ps_L + \ps_L^\da \, v_R^\da \, \ps_R \\
&=& \ch^T M_R^\da \, v_R \, M_L \ps + \ps^T M_L^\da \, v_R^\da \, M_R \, \ch \\
&=& \ps^a \ch^b \, 2\, \text{Re} \lp M_R^\da (v^0 \si_0 - v^\va i \, \si_\va) M_L \rp_{ba} \\
&=& v^c \ps^a \ch^b \Ga_{cba} \; \in \; \mathbb{R}
\end{eqnarray}
in which $\Ga_{cba}$ is the quaternion multiplication table and the Clifford vector matrix for the [[quaternionic representation of Cl(4)]].
Going the other way, we can write $\ps = M_L^- \ps_L$ and $\ch = M_R^- \ps_R$, with
$$
M^-_L = \fr{1}{\sqrt{2}} \lb \ba{cc} \fr{1}{i}(1-K) & 0 \\ 0 & (1+K) \\ 0 & \fr{1}{i}(1-K) \\ (1+K) & 0 \ea \rb
\s
M^-_R = \fr{1}{\sqrt{2}} \lb \ba{cc} \fr{1}{i}(1-K) & 0 \\ 0 & -(1+K) \\ 0 & -\fr{1}{i}(1-K) \\ -(1+K) & 0 \ea \rb
$$
in which $K$ is the complex conjugation operator. This allows us to write
$$
T(v_R,\ps_L,\ps_R) = v^c \ps^a \ch^b \Ga_{cba} = \ch^T (v^c \Ga_c) \ps = \ps_R^T M_R^{-T} (v^c \Ga_c) M_L^- \ps_L
= [ \ba{cc} \ps_R^T & \ps_R^\da \ea ] (v^c_L \Ga_c^{\mathbb{C}}) \lb \ba{c} \ps_L \\ \ps_L^* \ea \rb
$$
The ''Euclidean twistor incidence relation'',
$$
\ps_R = \frac{1}{\langle v_R \ps_L | v_R \ps_L \rangle} v_R \, \ps_L
$$
like the quaternion incidence relation, $\tilde{\ch} = v \ps / |v \ps|^2$, comes from requiring $T(v_R,\ps_L,\ps_R) = 1$. Our Hermitian form works for vectors as well,
$$
\langle v_1 | v_2 \rangle = v_1^\da v_2 + v_2^\da v_1 = v_1^a v_2^b \de_{ab}
$$
so Euclidean twistor triality also gives us
$$
v_R = \frac{1}{\langle \ps_R \ps_L^\da | \ps_R \ps_L^\da \rangle} \ps_R \ps_L^\da
\s \text{and} \s
\ps_L = \frac{1}{\langle v_R^\da \ps_R | v_R^\da \ps_R \rangle} v_R^\da \ps_R
$$
Euclidean twistor triality is sort of invariant under permutation,
$$
T(v_R,\ps_L,\ps_R) = \langle \ps_R | v_R \ps_L \rangle = \langle v_R \ps_L | \ps_R \rangle = \langle \ps_L | v_R^\da \ps_R \rangle
= T(v_R^\da,\ps_R,\ps_L)
$$
The identification between Euclidean twistors and the quaternions works pretty well, but things don't work as well for twistors of signature $(1,3)$. For one thing, we might have $\langle v_R | v_R \rangle = 0$ for a null vector. We could fudge by letting $V^0$ be imaginary, but that's not very satisfying. Instead, we might be better to consider the conformal algebra, [[spin(2,4)]], and its action on twistors, and then see that a better [[twistor confusion]] exists, based on the [[split-octonion]]s.
Equations of motion for a dynamical system can be obtained by insisting that an [[action]] is at an extrema under variations,
$$
0 = \de S = \int \nf{e} \, \de {\cal L} (\Ph,\f{\pa} \Ph) = \int \nf{e} \, \left\{ \de \Ph \fr{\pa {\cal L}}{\pa \Ph} + \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \f{\pa} \de \Ph \right\}
$$
Presuming the variations, $\de \Ph$, to vanish at the [[integration]] boundary, and obtaining the rule for integration by parts involving the [[divergence]],
$$
\begin{eqnarray}
0 &=& \int_\pa \de \Ph \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \nf{e}
= \int \f{d} \lp \de \Ph \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \nf{e} \rp \\
&=& \int \nf{e} \, \mathrm{div} \lp \de \Ph \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \rp
= \int \nf{e} \left\{ \de \Ph \, \mathrm{div} \lp \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \rp + \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \f{\pa} \de \Ph \right\}
\end{eqnarray}
$$
this can be used in the expression for the variation to obtain a ''Euler-Lagrange equation'',
$$
0 = \fr{\pa {\cal L}}{\pa \Ph} - \mathrm{div} \lp \fr{\pa {\cal L}}{\pa (\f{\pa} \Ph)} \rp
$$
or, in components, using the [[covariant derivative|Christoffel symbols]],
$$
0 = \fr{\pa {\cal L}}{\pa \Ph} - D_i \lp \fr{\pa {\cal L}}{\pa (\pa_i \Ph)} \rp
$$
<<tiddler HideTags>>$$
\begin{array}{llcl}
1918, \!\!&\!\! {\rm Weyl} \!\!&\!\! : & \f{A} \in \f{\rm Lie}(G) \p{{}_{\big(}} \\
1954, \!\!&\!\! {\rm Y.M.} \!\!&\!\! : & \f{A} = \f{B} + \f{W} + \f{G}
\;\; \in \; \f{\rm Lie}(G) = \f{su}(1) + \f{su}(2) + \f{su}(3) \p{{}_{\big(}} \\
1967, \!\!&\!\! {\rm F.P.} \!\!&\!\! : & \udf{A} = \f{A} + \ud{g} \p{{}_{\big(}}
\;\; \in \; \udf{\rm Lie}(G) \\
1977, \!\!&\!\! {\rm M.M.} \!\!&\!\! : & \f{A} = \f{\om} + \f{e}
\;\; \in \; \f{\rm Lie}(G) = \f{so}(1,4) \p{{}_{\Big(_(}} \\
2002, \!\!&\!\! {\rm Y.T.} \!\!&\!\! : & \ud{\ps} = \ud{g} \p{{}_{\big(}} \\
2005, \!\!&\!\! {\rm Y.T.} \!\!&\!\! : & \udf{A} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d} \\
& & & \;\;\;\,\, \in \; \f{\rm Lie}(G) = \f{Cl}(1,7) \p{{}_{\big(}} \\
{\rm now}, \!&\! {\rm Y.T.} \!\!&\!\! : & \udf{A} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d} \\
& & & \s\;\; + \ud{\nu^\mu} + \ud{\mu} + \ud{c} + \ud{s}
+ \ud{\nu^\ta} + \ud{\ta} + \ud{t} + \ud{b} \\
& & & \;\;\;\,\, \in \; \f{\rm Lie}(G) = \f{e8} ? \p{{}_{\big(_(}}
\end{array}
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUTSM.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\oplus\, 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/GraviGUTSM.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(1,3) \,\oplus\, 4 \!\otimes\! (2 \!\,\oplus\,\! \bar{2}) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\oplus\, 2 \!\otimes\! (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>The $14$ Lie algebra elements of the smallest exceptional Lie group, $G2$:
$$
\begin{array}{rcccccccl}
g2 \!\!&\!\!=\!\!&\!\! su(3) \!\!&\!\! + \!\!&\!\! 3 \!\!&\!\! + \!\!&\!\! \bar{3} \!\!&& \\
&&\!\! \f{g} \!\!&\!\! + \!\!&\!\! \ud{q} \!\!&\!\! + \!\!&\!\! \ud{\bar{q}} \!\!&\! \in \! &\! \udf{g2}
\end{array}
$$
Structure of $G2$ implies Lie bracket equivalent to fundamental action,
$$
[ g,q ] = \big[ g^A T_A,q^B T_B \big] = g \, q =
\lb
\matrix{
\! \fr{i}{2} g^3 \!+\! {\scriptsize \frac{i}{2\sqrt{3}}} g^8 \!\! & g^{r\bar{g}} & g^{r\bar{b}} \\
g^{\bar{r}g} & \!\! {\scriptsize -\!\frac{i}{2}} g^3 \!+\! {\scriptsize \frac{i}{2\sqrt{3}}} g^8 \!\! & g^{g\bar{b}} \\
g^{\bar{r}b} & g^{\bar{g}b} & {\scriptsize -\!\frac{i}{\sqrt{3}}} g^8
}
\rb
\lb \matrix{
q^r \\ q^g \\ q^b
} \rb
$$
corresponding to the strong interactions, such as
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\big[ g^{r\bar{g}}, q^g \big] = q^r
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<td ALIGN=CENTER><img SRC="images/png/quark gluon vertex.png" height=160px></td>
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''Bold''
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2^^3^^=8
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The rank $4$ exceptional group, ''F4'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $52$ dimensional [[Lie algebra]], [[f4]]. An explicit construction can be found in:
Ref:
*[[Cerchiai - Mapping the geometry of the F4 group|papers/Cerchiai - Mapping the geometry of the F4 group.pdf]]
$$
\begin{array}{rcll}
F4 &:& ( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3 ) \;\;\; \p{(B_2, g^3, g^8)} &\left\{
\begin{array}{l}\text{graviweak interactions} \\ \text{three generations}\end{array} \right.
\\
G2 &:& \p{( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3 ) \;\;\;}(B_2, g^3, g^8) &\left\{
\begin{array}{l}\text{strong interactions} \\ \text{anti-particles}\end{array} \right.
\\[-.3em]
\rlap{\hbox{@(hr noshade size="1" style="position:relative; left:-2em;
width:30em; border:0px; border-top:1px solid black")}}\\[-.5em]
E8 &:& ( \ha \om_L^3, \ha \om_R^3, W^3, B_1^3, w, B_2, g^3, g^8) & \, \left\{ \; \text{everything} \right.
\end{array}
$$
Breakdown of E8 to the standard model and gravity:
\begin{eqnarray}
e8 &=& f4 + g2 + 26 \! \times \! 7 \\
&=& so(7,1) + su(3) + (8_{S+}\!+\!8_V+\!8_{S-})\!\times\!(1\!+\!1\!+\!3\!+\!\bar{3}) + 3\!\times\!(3\!+\!\bar{3}) + 2 \\[.4em]
A &=& \big( {\scriptsize \frac{1}{2}} \om + {\scriptsize \frac{1}{4}} e \ph + W + B_1 \big) + g + 3 \! \times \! \Ps + x \Ph + B_2 + w
\end{eqnarray}
Two new quantum numbers and some non-standard particles:
$$
\{ \; w \quad (B_1^3\!+\!B_2) \quad B_1^\pm \quad x_{1/2/3} \Ph^{r/g/b} \quad x_{1/2/3} \Ph^{\bar{r}/\bar{g}/\bar{b}} \; \} \vp{\big(}
$$
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@@
confirmed attendees:
*[[Scott Aaronson|http://www.scottaaronson.com/]], QMI, quantum computing
*[[Fred Adams|http://www.physics.lsa.umich.edu/department/directory/bio.asp?ID=1]]${}^*$, constant change, astrophysics
*[[Anthony Aguirre|http://scipp.ucsc.edu/~aguirre/]], cosmology
*[[Stephon Alexander|http://www.phys.psu.edu/people/display/index.html?person_id=4901]], astrophysics
**Just put out a new paper: Isogravity
**friends with James Bjorken (and everyone else, apparently)
*[[Markus Aspelmeyer|http://homepage.univie.ac.at/Markus.Aspelmeyer/]]${}^*$, QM foundations
*[[Paul A Benioff|http://www.phy.anl.gov/theory/staff/pab.html]], QM foundations, older guy
*[[Caslav Brukner|http://homepage.univie.ac.at/Caslav.Brukner/index.htm]], QM foundations
*[[Dmitry Budker|http://www.fqxi.org/aw-budker2.html]]${}^*$, constant change (experimental)
*[[Gregory Chaitin|http://www.umcs.maine.edu/~chaitin/]], math, complexity, and philosophy of science
*[[Hyung Choi|http://www.zoominfo.com/people/Choi_Hyung_78134925.aspx]], metanexus, QM foundations, science and religion (uh oh)
*[[Louis Crane|http://www.fqxi.org/aw-crane2.html]]${}^*$, QGR, QM histories
*[[Paul Davies|http://cosmos.asu.edu/]], QMI, astrophysics, popular author
*[[John Donoghue|http://www.fqxi.org/aw-donoghue2.html]]${}^*$, emergent symmetry
*[[Richard Easther|http://www.fqxi.org/aw-easther.html]]${}^*$, superstring cosmology
*[[David Ritz Finkelstein|http://www.physics.gatech.edu/people/faculty/dfinkelstein.html#personal]], older particle physicist
**Lie algebra expert. Proponent of stable Lie algebras.
*[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin2.html]]${}^*$, QM GR
*[[Jaume Garriga|http://www.ffn.ub.es/gcg/personal/jaume.html]], cosmology, branes
*[[Steven Gratton|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+ea+gratton%2C+steven]], cosmology, inflation
*[[Alan Guth|http://web.mit.edu/physics/facultyandstaff/faculty/alan_guth.html]], err, invented inflation
*[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]], Causaloids
*[[Adrian Kent|http://www.damtp.cam.ac.uk/user/apak/]], QM foundations
*[[Lawrence Krauss|http://www.phys.cwru.edu/~krauss/]], astrophysics and cosmology, popular author, dislikes KK and strings, lectures a LOT
*[[Matthew Leifer|http://www.fqxi.org/aw-leifer2.html]], QM foundations
*[[Eugene Lim|http://pantheon.yale.edu/~eal48/papers.html]]${}^*$, cosmology
*A. [[Garrett Lisi]]${}^*$
*[[Abraham Loeb|http://www.fqxi.org/aw-loeb2.html]]${}^*$, SETI, astronomy
*[[Fotini Markopoulou|http://www.fqxi.org/aw-markopoulou2.html]]${}^*$, quantum graphity
*[[Laura Mersini|http://en.wikipedia.org/wiki/Laura_Mersini]], cosmology
*[[Farzad Nekoogar|http://www.fqxi.org/aw-nekoogar.html]]${}^*$, popularizer of theoretical physics -- [[multiversal journeys|http://www.multiversaljourneys.com/]]
*[[Ken Olum|http://www.fqxi.org/aw-olum2.html]]${}^*$, GR, wants to rule out wormholes and other GR exotics
*[[Maulik Parikh|http://www.fqxi.org/aw-khoury2.html]]${}^*$, GR boundaries, mach's principle, hep-th and strings
*[[Philip Pearle|http://physerver.hamilton.edu/people/]], QM foundations, older guy
*[[Ekkehard Peik|http://www.fqxi.org/aw-peik2.html]]${}^*$, constant change (experimental)
*[[Simon Saunders|http://www.fqxi.org/aw-saunders2.html]]${}^*$, QM foundations
*Lee Smolin (not going)
*[[Robert Spekkens|http://www.fqxi.org/aw-spekkens2.html]]${}^*$, QM foundations
*[[Max Tegmark|http://web.mit.edu/physics/facultyandstaff/faculty/max_tegmark.html]], astrophysics, cosmology, trouble maker...
*[[Mark Trodden|http://physics.syr.edu/~trodden/]], cosmology, particle physics -- QFT
*[[Roderich Tumulka|http://www.fqxi.org/aw-tumulka2.html]]${}^*$, Bohmian QM
*[[Jos Uffink|http://www.phys.uu.nl/igg/jos/]], QM foundations
*[[Vitaly Vanchurin|http://cosmos.phy.tufts.edu/~vitaly/]], cosmic strings
*[[Xiao-Gang Wen|http://www.fqxi.org/aw-wen2.html]]${}^*$, gravity and light emergent from substrate :P
*[[Serge Winitzki|http://www.theorie.physik.uni-muenchen.de/~serge/]], quantum cosmology
**Likes wiki, and likes ToE.
**Inflation expert -- says $R \ph^2$ term would be great, among others.
***Strong constraints on these coefficients.
*[[Toby Wiseman|http://schwinger.harvard.edu/~wiseman/]], string theory
*[[Wojciech Zurek|http://public.lanl.gov/whz/]], cosmology and astrophysics, chaos, QM foundations
${}^*$ grant winners (19)
Press and Foundation people
*[[Graham P Collins|http://www.sff.net/people/GPC/]], Scientific American Magazine
*[[Valerie Jamieson|http://www.scienceinpublic.com/scienceweek/speakers.htm#Valerie%20Jamieson%20background]], New Scientist Magazine
**particle physics background
*[[Wade Davis|http://en.wikipedia.org/wiki/Wade_Davis]], National Geographic Explorer-in-Residence, ethnobiologist
*[[Charles Harper|http://www.templeton.org/about_us/who_we_are/leadership_team/charles_harper/]], Senior Vice-President, John Templeton Foundation
**He's paying, try not to insult him.
*[[Amanda High|http://www.nptrust.org/about_npt/key_staff.asp#high]], Vice President, National Philanthropic Trust
**What's she doing at the FQXi conference
*[[Howard Burton|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=index.php&e=Founding%20Executive%20Director&cat_id=53&cat_table=2]], Executive Director, Perimeter Institute for Theoretical Physics
**Just got ousted from PI position, even though he founded it. Used to be main PI talent scout.
*[[Christopher Liedel|http://executiveeducation.wharton.upenn.edu/fellows/feb_info/roster_detail.cfm?id=KRSM00000024468]], Executive Vice President & Chief Financial Officer, National Geographic Society
*Robert Kuhn, Kuhn foundation -- makes science documentaries for PBS
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Choosing the anti-Grassmann 3-form to be $\fff{\od{B}} = \nf{e} \od{\Ps} \ve{e} \,$ gives the massive Dirac action in curved spacetime:
\begin{eqnarray}
S_f &=& \int \big< \fff{\od{B}} \udff{F} \big>
= \int \big< \fff{\od{B}} \f{D} \ud{\Ps} \big> \\
&=& \int \big< \nf{e} \od{\Ps} \ve{e} \big( \f{d} \ud{\Ps} + \f{H}{}_1 \ud{\Ps} - \ud{\Ps} \f{H}{}_2 \big) \big> \\
&=& \int \big< \nf{e} \od{\Ps} \ve{e} \big( ( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph + \f{W} + \f{B}{}_1 ) \ud{\Ps}
- \ud{\Ps} ( \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} ) \big) \big> \\
&=& \int \nf{d^4 x} \, |e| \, \big< \od{\Ps} \ga^\mu (e_\mu)^i \big( \pa_i \ud{\Ps} + {\scriptsize \frac{1}{4}} \om_i^{\p{i} \mu \nu} \ga_{\mu \nu} \ud{\Ps} + W_i \ud{\Ps} + B_{1i} \ud{\Ps} \\
&& \hphantom{\int \nf{d^4 x} \, |e| \, \big< \od{\Ps} \ga^\mu (e_\mu)^i \big(} + \ud{\Ps} w_i + \ud{\Ps} B_{2i} + \ud{\Ps} x_i \Ph + \ud{\Ps} g_i \big) + \od{\Ps} \, \ph \, \ud{\Ps} \big>
\end{eqnarray}
The $\od{\Ps} \, \ph \, \ud{\Ps}$ is the standard Higgs mass term.$\vp{\Huge(}$
The $\od{\Ps} \ga^\mu \ud{\Ps} x_\mu \Ph$ term... I don't understand yet -- promising for CKM.
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Fermion states in the Standard Model? Up or down type fermions, either leptons or $3$ colors of quarks, so $8$ fermion types times 24-cell for each, gives $192$ fermion states.
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Only Exceptional Unification accommodates $G_{CPTt}$. E8 Theory with octo-octonionic $\{\om_t, \om_S, U, V, p, x, y, z\}$,
$$
so(8) + so(8) + 64_v + 64_{s+}+ 64_{s-}
$$
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Fermion, $\ud{\ps}$, Higgs, $\ph$, and frame, $\f{e}$, fibers in various representations of the structure group,
$$
Spin(1,3) \times ( SU(2) \times U(1) \times SU(3) ) / Z_6
$$
Connection: $\;\; \f{A} = \f{dx^\mu} A_\mu^{\p{k}B} T_B \s\s\;$ Ehresmann connection: $\;\; \f{\ve{\cal A}}(x,y) = \f{A}^B(x) \ve{\xi}_B(y) + \f{\ve{\cal I}} \;$
Curvature: $\;\; \ff{F} = \f{d} \f{A} + \ha [ \f{A}, \f{A} ] \s\;\;\;\;\;$ Frolicher-Nijenhuis: $\;\; \ff{\ve{\cal F}} = - \ha [ \f{\ve{\cal A}}, \f{\ve{\cal A}} ] = - \f{\ve{\cal A}} ( \f{\pa} \f{\ve{\cal A}} ) + \f{\pa} ( \f{\ve{\cal A}} \f{\ve{\cal A}} ) \;$
Action: $\;\; S(\ff{F}, \f{D} \ud{\ps}, \f{A}, \ud{\ps}, \ph, \f{e}, \dots)^{\p{\big(}} \;$
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A ''Fock space'', $\mathcal{F}$, is an infinite-dimensional [[Hilbert space]] (and therefore also an [[infinite-dimensional unitary representation]] space) corresponding physically to the excitation of quantum particle states. It is the Hilbert space of quantized field amplitudes (sometimes called ''second quantization''), in contrast to the more familiar Hilbert space of quantized particle position (''first quantization''). For each type of elementary particle, such as $\ph$, with a specific spatial [[momentum]], $p$, and mass $m$, there is a basis of [[harmonic energy states|creation and annihilation operators]], $| n^\ph_p \rangle$, with corresponding oscillation energy, $E = \sqrt{p \!\cdot\! p + m^2}$, for each particle. Here $n^\ph_p$ is the ''occupation number'' of particle $\ph$ with momentum $p$. An arbitrary [[ket]] in Fock space can be written as a superposition of basis states,
$$
| n^\ph_{p_1}, n^\ph_{p_2}, n^\ph_{p_3}, ... \rangle \in \mathcal{F}
$$
In this way each type of elementary particle, of each momentum, is a [[quantum harmonic oscillator]]. The ''vacuum'', $| 0 \rangle$, is the ground state of the [[quantum field]] -- this technically corresponds to zero particles, but just as the quantum harmonic oscillator ground state can have non-vanishing vacuum expectation values, so can the quantum field theory vacuum.
A Fock space can be restricted to be symmetric or antisymmetric under the exchange of particles, depending on whether the particles are bosons or fermions. A Fock space,
$$
\mathcal{F} = {\mathbb C} \oplus \mathcal{H} \oplus \mathcal{H} \! \otimes \! \mathcal{H} \, \oplus \, ...
$$
can also be described as the infinite sum of exterior products of single particle Hilbert spaces, with the above ket expressed as
$$
| n_1, n_2, n_3, ... \rangle = | \ph_1 \rangle^{n_1} | \ph_2 \rangle^{n_2} | \ph_3 \rangle^{n_3} ...
$$
and the products symmeterized or antisymmeterized if they're bosons or fermions, such as
$$
| 1_1, 1_2 \rangle = | \ph_1 \rangle | \ph_2 \rangle = \ha \lp | \ph_1 \rangle \otimes | \ph_2 \rangle \pm | \ph_2 \rangle \otimes | \ph_1 \rangle \rp
\in S_\pm \! \lp \mathcal{H} \! \otimes \! \mathcal{H} \rp
$$
This establishes a mysterious identification between the first excited state of a [[quantum field]], $| 1^\ph_{p} \rangle$, and the [[quantum state|quantum mechanics]], $| \ph_p \rangle$, of a single particle.
[[Fourier transform]] pairs satisfy many interesting identities:
Linearity
$$
f(x) = a \, g(x) + b \, h(x) \implies f'(p) = a \, g'(p) + b \, h'(p)
$$
Translation
$$
f(x) = g(x-x_0) \implies f'(p) = e^{-2\pi i x_0 p} g'(p)
$$
Scaling
$$
f(x) = g(a \, x) \implies f'( p) = \fr{1}{|a|} g'(\fr{p}{a})
$$
Conjugation
$$
f(x) = g(x)^* \implies f'(p) = g'(-p)^*
$$
Parseval formula
$$
\int{dx} \, f(x) \, g(x)^* = \int{\! \fr{dp}{2 \pi}} \, f'(p) \, g'(p)^*
$$
Convolution
$$
f'(p) = g'(p)^* \, h'(p) \implies f(x) = \int{dx'} \, g(x')^* \, h(x+x')
$$
The ''Fourier transform'', $f'(p)$, of a function, $f(x$), is
$$
f'(p) = \int{\!dx} \, e^{- i x p} f(x)
$$
which inverts via
$$
f(x) = \int{\! \fr{dp}{2 \pi}} e^{i x p} f'(p)
$$
There are many interesting [[Fourier identities]].
The ''FuN curvature'' (//''Frolicher-Nijenhuis curvature''//), $\ff{\ve{\cal F}}=\f{dx^i} \f{dx^j} \ha {\cal F}_{ij}{}^k \ve{\pa_k}$, of a [[vector valued form]], $\f{\ve{\cal A}}=\f{dx^i} {\cal A}_i{}^j \ve{\pa_j}$, is its [[FuN bracket|FuN derivative]] with itself,
\begin{eqnarray}
\ff{\ve{\cal F}} &=& - \ha \lb \f{\ve{\cal A}}, \f{\ve{\cal A}} \rb_L
= - \lb \f{\ve{\cal A}}, \f{\pa} \rb \f{\ve{\cal A}} \\
&=& - \f{\ve{\cal A}} \lp \f{\pa} \f{\ve{\cal A}} \rp + \f{\pa} \lp \f{\ve{\cal A}} \f{\ve{\cal A}} \rp \\
&=& - \lp \f{\ve{\cal A}} \f{\pa} \rp \f{\ve{\cal A}} + \lp \f{\pa} \f{\ve{\cal A}} \rp \f{\ve{\cal A}}
\end{eqnarray}
In components, this is
$$
{\cal F}_{ij}{}^k = - {\cal A}_i{}^m \pa_m {\cal A}_j{}^k + {\cal A}_j{}^m \pa_m {\cal A}_i{}^k + {\cal A}_m{}^k \pa_i {\cal A}_j{}^m - {\cal A}_m{}^k \pa_j A_i{}^m
$$
If, as often happens, a vector valued form is a [[vector projection]], $\f{\ve{\cal P}} = \f{\ve{\cal P}} \f{\ve{\cal P}}$, its FuN curvature is
\begin{eqnarray}
\ff{\ve{\cal F}} &=& \f{\pa} \f{\ve{\cal P}} - \f{\ve{\cal P}} \lp \f{\pa} \f{\ve{\cal P}} \rp \\
&=& \lp 1 - \f{\ve{\cal P}} \rp \lp \f{\pa} \f{\ve{\cal P}} \rp
\end{eqnarray}
which satisfies $\f{\ve{\cal P}} \ff{\ve{\cal F}}=0$ -- the form part of the FuN curvature is in the kernel (horizontal part) of the projection. If the vector projection is an [[Ehresmann connection]], any two vectors contracted with the FuN curvature give the vertical part of the [[Lie bracket|Lie derivative]] of the horizontal part of the vectors,
$$
\ve{u} \ve{v} \ff{\ve{\cal F}} = - \ha {\lb \ve{u_H} , \ve{v_H} \rb_L}_V = - \ha \lb \ve{u} \lp 1- \f{\ve{\cal P}} \rp , \ve{v} \lp 1- \f{\ve{\cal P}} \rp \rb_L \f{\ve{\cal P}}
$$
//check that//
The ''Frolicher-Nijenhuis Lie derivative'' -- which we refer to as the //''FuN derivative''// -- is a [[natural]] operator generalizing the [[Lie derivative]] to handle [[vector valued form]] fields. The FuN derivative of a vector valued $k$-form field, $\nf{\ve{K}}$, with respect to a vector field, $\ve{v}$, is written terms of [[partial derivative]]s as
\begin{eqnarray}
{\cal L}_{\ve{v}} \nf{\ve{K}} &=& \lim_{t \to 0} \fr{\ph_t^*\nf{\ve{K}} - \nf{\ve{K}}}{t} = \ve{v} \lp \f{\pa} \nf{\ve{K}} \rp + \f{\pa} \lp \ve{v} \nf{\ve{K}} \rp - \lp \nf{\ve{K}} \f{\pa} \rp \ve{v} \\
&=& \lp \ve{v} \f{\pa} \rp \nf{\ve{K}} + \lp \f{\pa} \ve{v} \rp \nf{\ve{K}} - \lp \nf{\ve{K}} \f{\pa} \rp \ve{v}
\end{eqnarray}
This defines the ''Frolicher-Nijenhuis bracket'' (//''FuN bracket''//) for these fields, and enforcing antisymmetry defines the FuN derivative of a vector field with respect to a vector valued form.,
$$
{\cal L}_{\nf{\ve{K}}} {\ve{v}} = \lb \nf{\ve{K}} , \ve{v} \rb_L = - \lb \ve{v} , \nf{\ve{K}} \rb_L = - {\cal L}_{\ve{v}} \nf{\ve{K}}
$$
Similarly, generalizing Cartan's formula for the Lie derivative, the FuN derivative of a differential form is
\begin{eqnarray}
{\cal L}_{\nf{\ve{K}}} \nf{F} &=& \lb \nf{\ve{K}} , \nf{F} \rb_L = \nf{\ve{K}} \lp \f{d} \nf{F} \rp + \lp -1 \rp^k \f{d} \lp \nf{\ve{K}} \nf{F} \rp \\
&=& \lp \nf{\ve{K}} \f{\pa} \rp \nf{F} + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \nf{F}
\end{eqnarray}
which also defines the FuN bracket of these objects. The above expression for the FuN bracket, and Cartan's formula, can also be written using the natural [[exterior derivative]] in a [[commutator]] bracket,
$$
\lb \nf{\ve{K}} , \nf{F} \rb_L = \lb \nf{\ve{K}} , \f{d} \rb \nf{F}
$$
thereby demonstrating the naturalness of the FuN derivative acting on forms. These definitions generalize furthest to give the glorious FuN bracket (and FuN derivative) between vector valued $k$ and $l$ forms
\begin{eqnarray}
\lb \nf{\ve{K}} , \nf{\ve{L}} \rb_L &=& {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \\
&=& \nf{\ve{K}} \lp \f{\pa} \nf{\ve{L}} \rp - \lp -1 \rp^{kl} \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp + \lp -1 \rp^k \f{\pa} \lp \nf{\ve{K}} \nf{\ve{L}} \rp - \lp -1 \rp^{kl+l} \f{\pa} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \\
&=& \lp \nf{\ve{K}} \f{\pa} \rp \nf{\ve{L}} - \lp -1 \rp^{kl} \lp \nf{\ve{L}} \f{\pa} \rp \nf{\ve{K}} + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \nf{\ve{L}} - \lp -1 \rp^{kl+l} \lp \f{\pa}\nf{\ve{L}} \rp \nf{\ve{K}} \\
&=& \lb \nf{\ve{K}} , \f{\pa} \rb \nf{\ve{L}} - \lp -1 \rp^{kl} \lb \nf{\ve{L}} , \f{\pa} \rb \nf{\ve{K}}
\end{eqnarray}
which gives all the FuN brackets and Lie derivatives as special cases. This FuN bracket of vector valued $k$ and $l$ forms is a vector valued $(k+l)$-form, and is defined to satisfy
$$
{\cal L}_{\lb \nf{\ve{K}} , \nf{\ve{L}} \rb_L} = \lb {\cal L}_{\nf{\ve{K}}} , {\cal L}_{\nf{\ve{L}}} \rb
$$
when acting on vectors or forms.
The FuN derivative has a number of other nice [[properties|FuN identities]].
//(Most everything here was learned from talking with [[Michael Edwards]] and reading [[Peter Michor]] et al. (Though the above explicit expression is mine, so if it's wrong, blame [[me|Garrett Lisi]]))//
The [[FuN derivative]] with respect to a [[vector valued k-form|vector valued form]], ${\cal L}_{\nf{\ve{K}}}$, is a grade $k$ [[derivation]] that combines with itself and other operators in a number of ways. Like the [[Lie bracket|Lie derivative identities]], it is linear in both arguments.
The FuN Lie bracket may or may not change sign under the exchange of its vector valued $k$-form and vector valued $l$-form arguments,
$$
\lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L = - \lp -1 \rp^{kl} \lb \nf{\ve{L}}, \nf{\ve{K}} \rb_L
$$
As a derivation, the FuN derivative operates on products of forms via the graded Liebniz rule,
$$
{\cal L}_{\nf{\ve{K}}} \lp \nf{F} \nf{G} \rp
= \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp \nf{G} + \lp -1 \rp^{kf} \nf{F} \lp {\cal L}_{\nf{\ve{K}}} \nf{G} \rp
$$
But it is not a derivation over products of VVFs and forms. Using some [[vector valued form identities]] we get
\begin{eqnarray}
{\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{F} \rp &=& \lp \nf{\ve{K}} \f{\pa} \rp \lp \nf{\ve{L}} \nf{F} \rp + \lp -1 \rp^k \lp \f{\pa} \nf{\ve{K}} \rp \lp \nf{\ve{L}} \nf{F} \rp \\
&=& \lp \lp \nf{\ve{K}} \f{\pa} \rp \nf{\ve{L}} \rp \nf{F}
+ \lp-1\rp^{k\lp l-1\rp} \lb \nf{\ve{L}} \lp \lp \nf{\ve{K}} \f{\pa} \rp \nf{F} \rp
- \lp \nf{\ve{L}} \lp \nf{\ve{K}} \f{\pa} \rp \rp \nf{F} \rb \\
& &
+ \lp-1\rp^k \lp \lp \f{\pa} \nf{\ve{K}} \rp \nf{\ve{L}} \rp \nf{F}
+ \lp-1\rp^{kl} \nf{\ve{L}} \lp \lp \f{\pa} \nf{\ve{K}} \rp \nf{F} \rp
- \lp-1\rp^{kl} \lp \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp \rp \nf{F} \\
&=&
\lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{F}
+ \lp-1\rp^{k\lp l-1\rp} \nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp
- \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{F}
\end{eqnarray}
and
\begin{eqnarray}
{\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{F} &=& \lp \lp \nf{\ve{L}} \nf{\ve{K}} \rp \f{\pa} \rp \nf{F} - \lp-1\rp^{\lp l+k\rp} \lp \f{\pa} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \rp \nf{F} \\
&=& \lp \nf{\ve{L}} \lp \nf{\ve{K}} \f{\pa} \rp \rp \nf{F} - \lp-1\rp^k \lp
\lp-1\rp^l \lp \f{\pa} \nf{\ve{L}} \rp \nf{\ve{K}} - \nf{\ve{L}} \lp \f{\pa} \nf{\ve{K}} \rp + \lp \nf{\ve{L}} \f{\pa} \rp \nf{\ve{K}}
\rp \nf{F} \\
&=&
\nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{F} \rp
+ \lp-1\rp^{k\lp l-1\rp} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{F}
- \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{F} \rp
\end{eqnarray}
which are linked by the last lines of each -- they are the same equation (an equation that may be used to define the FuN bracket of two VVF's in terms of the FuN derivatives of forms). A similar identity exists for three VVF's:
$$
{\cal L}_{\nf{\ve{L}}\nf{\ve{K}}} \nf{\ve{M}}
= \nf{\ve{L}} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{M}} \rp
+ \lp-1\rp^{k\lp l-1\rp} \lp {\cal L}_{\nf{\ve{K}}} \nf{\ve{L}} \rp \nf{\ve{M}}
- \lp-1\rp^{k\lp l-1\rp} {\cal L}_{\nf{\ve{K}}} \lp \nf{\ve{L}} \nf{\ve{M}} \rp
- \lp-1\rp^{m\lp k+ l-1\rp} \lp {\cal L}_{\nf{\ve{M}}} \nf{\ve{L}} \rp \nf{\ve{K}}
$$
When the two VVF's are written as $\nf{\ve{K}}=\nf{K^A} \ve{X_A}$ and $\nf{\ve{L}}=\nf{L^A} \ve{Y_A}$ their FuN bracket is
\begin{eqnarray}
\lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L &=& \nf{K^A} \nf{L^B} \lb \ve{X_A}, \ve{Y_B} \rb_L + \nf{K^A} \lp {\cal L}_{\ve{X_A}} \nf{L^B} \rp \ve{Y_B}
- \lp {\cal L}_{\ve{Y_B}} \nf{K^A} \rp \nf{L^B} \ve{X_A} \\
&+& \lp -1 \rp^k \lp \f{d} \nf{K^A} \rp \ve{X_A} \nf{L^B} \ve{Y_B}
+ \lp -1 \rp^k \lp \ve{Y_B} \nf{K^A} \rp \lp \f{d} \nf{L^B} \rp \ve{X_A}
\end{eqnarray}
and, for the FuN bracket of a vector valued 1-form with itself,
\begin{eqnarray}
\lb \f{\ve{K}}, \f{\ve{K}} \rb_L &=& \f{K^A} \f{K^B} \lb \ve{X_A}, \ve{X_B} \rb_L + 2 \f{K^A} \lp {\cal L}_{\ve{X_A}} \f{K^B} \rp \ve{X_B}
- 2 \lp \f{d} \f{K^A} \rp \ve{X_A} \f{K^B} \ve{X_B}
\end{eqnarray}
When acting on forms, the FuN derivative commutes with the [[exterior derivative]],
$$
0 = \lb {\cal L}_{\nf{\ve{K}}}, \f{d} \rb = {\cal L}_{\nf{\ve{K}}} \f{d} + \lp -1 \rp^k \f{d} {\cal L}_{\nf{\ve{K}}}
$$
In fact, the FuN derivative of a form with respect to the [[identity projection|vector projection]] is the exterior derivative,
$$
{\cal L}_{\nf{\ve{I}}} \nf{F} = \f{d} \nf{F}
$$
and of a VVF is zero, ${\cal L}_{\nf{\ve{I}}} \nf{\ve{K}} = 0$.
Acting on itself twice, the FuN bracket satisfies the ''graded Jacobi identity'',
$$
\lb \nf{\ve{K}}, \lb \nf{\ve{L}} , \nf{\ve{M}} \rb_L \rb_L = \lb \lb \nf{\ve{K}}, \nf{\ve{L}} \rb_L, \nf{\ve{M}} \rb_L - \lp -1 \rp^{kl} \lb \nf{\ve{L}}, \lb \nf{\ve{K}} , \nf{\ve{M}} \rb_L \rb_L
$$
The rank $2$ exceptional group, ''G2'', is a real, [[simple]], compact, connected [[Lie group|Lie group]]. It may be described by [[exponentiating|exponentiation]] its $14$ dimensional [[Lie algebra]], [[g2]].
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
<embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed>
<!-- <embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed> -->
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->
<embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->
<embed src="talks/Perimeter07/anim/g2spin/p20.png" width="462" height="462"></embed>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
<!-- <embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed> -->
<embed src="talks/Perimeter07/anim/g2spin/p46.png" width="462" height="462"></embed>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
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<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>
<th></th>
<th><SPAN class="math">x</SPAN></th>
<th><SPAN class="math">y</SPAN></th>
<th><SPAN class="math">z</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
<th><SPAN class="math">\fr{\sqrt{8}}{\sqrt{3}} B_2</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">1</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">-\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r_I</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b_I</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
<td><SPAN class="math">\fr{1}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\mutr{#0000F7}}{\mtri{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{II}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\mp 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\pm \fr{2}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\stri{#0000F7}}{\sutr{#0000F7}}</SPAN></td>
<td><SPAN class="math">q_{III}</SPAN></td>
<td></td>
<td COLSPAN="3"><SPAN class="math">\pm 1 \;\; \pm \! 1</SPAN></td>
<td></td>
<td>"</td>
<td>"</td>
<td><SPAN class="math">\mp \fr{4}{3}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\rlap{\butr{#999999}}{\btri{#999999}}</SPAN></td>
<td><SPAN class="math">l</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
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<<tiddler HideTags>>
[>img[Garrett at Burning Man, 2004|images/person/Garrett.jpg]]Homepage: http://Li.si
*Email: Gar@Li.si
*Location: Maui, usually
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Lisi_A/0/1/0/all/0/1
Selected work:
*This Wiki.
*[[An Exceptionally Simple Theory of Everything]]
*[[Quantum mechanics from a universal action reservoir|http://arxiv.org/abs/physics/0605068]]
Some talks:
*[[talk for ILQGS 07]]
*[[talk for Perimeter Institute 07]]
*[[talk for FQXi 07]]
*[[talk for Loops 07]]
<<tiddler HideTags>>[>img[talks/CSUF09/images/FiberBundle_200.png]]Base [[manifold]]: $M$
A fiber, $F$, is a representation space of a Lie group, $G$.
Entire space of a [[fiber bundle]]: $E \sim M \times F$
For a [[principal bundle]], $G$ is the fiber: $E \sim M \times G$
[>img[images/png/fiber bundle.png]][[Ehresmann principal bundle connection]] over patches of $E$:
$$
\ve{\f{\cal E}}(x,y) = \f{dx^i} A_i^{\p{a}B}(x) \, \ve{T_B}(y) + \f{dy^p} \ve{\pa_p}
$$
Gauge field [[connection]] over $M$,
$$
\f{A}(x) = \f{dx^i} A_i^{\p{a}B}(x) \, T_B
$$
describes how fibers twist: $\f{D} \ps = (\f{d} + \f{A}) \ps$
[[Curvature|curvature]] is how the connection twists over $M$: $\ff{F} = \f{d} \f{A} + \f{A} \f{A}$
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_\ps \!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ \ha \om_i^{\p{i}\nu\rh} \ha \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ \ha \om_i^{\p{i}\nu\rh} \ha \ga_{\nu\rh}
+ G_i^{\p{i}\al\be} \ha \ga_{\al\be}
+ \fr{1}{4} (e_i)^\nu \ph^\al \ga_\nu \ga_\al
\big) \ud{\ps}
\right\} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
\end{array}
$$
''Unified bosonic connection'':
$$
\f{H} = {\tiny \frac{1}{2}} \f{\om} + {\tiny \frac{1}{4}} \f{e} \ph + \f{G} \;\; \in spin(?)
$$
With SO(10) GUT:
$$
spin(3,1) + 4 \!\times\! 10 + spin(10) = spin(3,11) \mbox{ or } spin(13,1)
$$
or with Pati-Salam GUT:
$$
spin(3,1) + 4 \!\times\! 4 + spin(4) + spin(6) \subset spin(3,11), spin(13,1), spin(7,7) \mbox{ or } spin(9,5)
$$
One generation of fermions:
$$
64_S^{+\mathbb{R}} \mbox{ of } spin(3,11) \mbox{ or } spin(7,7)
$$
The eight [[trace]]less, Hermitian, ''Gell-Mann matrices'', $\la_A$, are
$$
\begin{array}{cccc}
\la_0 = \la_8 = \fr{1}{\sqrt{3}} \left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -2
\end{array}\right]
&
\la_1 = \left[\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0
\end{array}\right]
&
\la_2 = \left[\begin{array}{ccc}
0 & -i & 0\\
i & 0 & 0\\
0 & 0 & 0
\end{array}\right]
&
\la_3 = \left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0
\end{array}\right]
\\
\la_4 = \left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
1 & 0 & 0
\end{array}\right]
&
\la_5 = \left[\begin{array}{ccc}
0 & 0 & -i\\
0 & 0 & 0\\
i & 0 & 0
\end{array}\right]
&
\la_6 = \left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 1 & 0
\end{array}\right]
&
\la_7 = \left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & -i\\
0 & i & 0
\end{array}\right]
\end{array}
$$
and so
$$
v^A \la_A =
\lb\ba{ccc}
v^3 + \fr{1}{\sqrt{3}} v^8 & v^1 - i \, v^2 & v^4 - i \, v^5 \\
v^1 + i \, v^2 & -v^3 + \fr{1}{\sqrt{3}} v^8 & v^6 - i \, v^7 \\
v^4 + i \, v^5 & v^6 + i \, v^7 & -\fr{2}{\sqrt{3}} v^8
\ea\rb
$$
<<tiddler HideTags>>
Three generations of fermions in three copies of $64^\mathbb{R}_{S+}$ of $spin(11,3)$, differing only in mass.
Triality?
$\s\;\;\;$ Maps between three blocks of $64$ in E8.
$\s\;\;\;$ These blocks have different quantum numbers -- doesn't seem viable.
$\s\;\;\;$ Look more closely at symmetry breaking.
Larger Lie group or supergroup?
$\s\;\;\;$ Orthosymplectic, $D(7,3)$ or ?
Larger algebra?
$\s\;\;\;$ E9. Possible relation to QFT.
$\s\;\;\;$ Leech lattice. Three E8's as inner shell.
$\s\;\;\;$ Kac-Moody algebras.
Axions?
$\s\;\;\;$ Use Peccei-Quinn charge, $w$, in E8 (and E6) and scalars in E8. $\;\;\;\; \th \ff{F} \ff{F} \;\;\;\;\; \big< \bar{\ps} \f{e} \th \f{e} \th \f{e} \th \ep \f{D} \ud{\ps} \big>$
$\s\;\;\;$ Fermions of different generations as axion-fermion composites. $\;\;\; \th+\ps$
$\s\;\;\;$ Used successfully in the past for solving the strong CP problem, and dealing with mirror fermions.
$\s\;\;\;$ E8 appears to come with a nice axion model building kit.
Something weirder?
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_D \!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i +\fr{1}{4} \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} + G_i^{\p{i}B} T_B \big) \ud{\ps} + \bar{\ps} \ph \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{d^4x} |e| \big\{ \bar{\ps} \ga^\mu (e_\mu)^i \big( \pa_i + \fr{1}{4} \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} + \ha G_i^{\p{i}\ps\ch} \ga_{\ps\ch} + \fr{1}{4} (e_i)^\mu \ph^\ps \ga_{\mu\ps} \big) \ud{\ps} \big\} \\
\!\!&\!\!=\!\!&\!\! \int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\}
= \int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
= \int \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps}
\end{array}
$$
Gravity: $\;\;\;\; \f{\om} = \ha \f{dx^i} \, \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;\; \in Cl(3,1)^2 = spin(3,1)
\s\;\;\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;\; \in Cl(3,1)^1 = 4 \vp{A_{\big(}}$
GUT: $\;\;\;\; \f{G} \;\; \in \, su(2)_L+u(1)_Y+su(3)$
$\s\s\s \subset su(2)_L+su(2)_R+su(4) = spin(4)+spin(6) \;\; \subset spin(10) \vp{A_{\big(}}$
Fermions: $\;\;\;\; \ud{\ps} \;\; \in 2 \!\times\! (2_L \!+\! 2_R) \!\times\! (1+3) = \mathbb{C}^{32} = \mathbb{R}^{64} \;\;\; (\times 3){}^{\p{\big(}}$
Higgs: $\;\;\;\; \ph=\ph^\ps \ga_\ps \;\; \in Cl(4)^1 = 4 = \mathbb{C}^2 \;$ or $\; Cl(N)^1 = N{}^{\p{\big(}}$
ToE: $\;\;\;\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{G} \;\; \in spin(3,1) + 4 \!\times\! 10 + spin(10) {}^{\p{\Big(}}$
Curvature: $\;\;\;\; \ff{F} = \f{d} \f{H} + \f{H} \f{H}
= \ha (\ff{R} - \fr{1}{8} \f{e}\f{e} \ph^2) + \fr{1}{4} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}^G{}^{\p{\big(}}$
Superconnection: $\;\;\;\; \udf{A} = \f{H} + \ud{\ps} \;\; \in spin(3,11) + 64_S^{+\mathbb{R}} {}^{\p{\big(}}$
$\s\s\s\s\s\s\s\s\;\;\, \subset spin(4,12) + 128_S^{+\mathbb{R}} = E8(-24)$
Supercurvature: $\;\;\;\; \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ff{F} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}{}^{\p{\big(}}$
$$
S = \int \left\{ \ff{\bar{B}} \udff{F} - {\textstyle \fr{\pi G}{4}} \ff{B} \ep \ff{B} + {\small g^2} \ff{B'} \ff{*B'} \right\} \sim \int \left\{ \nf{e} \bar{\ps} D \!\!\!\!/\; \ud{\ps} + {\textstyle \fr{1}{16\pi G}} \nf{e} \big( R - {\textstyle \fr{3}{2}} \ph^2 \big) \ph^2 + {\textstyle \fr{1}{4g^2}} \ff{F'} \ff{*F'} \right\}
$$
<<tiddler HideTags>>Start with a [[Lie group manifold|Lie group geometry]] (//torsor//), $G$, coordinatized by $y^p$.
Two sets of invariant vector fields (//symmetries, [[Killing vector]] fields//):
[>img[images/png/torsor.png]]$$
\ve{\xi^L_A}(y) \, \f{d} g = T_A \, g(y) \s\;\; \ve{\xi^R_A}(y) \, \f{d} g = g(y) \, T_A
$$
[[Lie derivative]]: ~~ ~~ $[ \ve{\xi^R_A}, \ve{\xi^R_B} ] = C_{AB}^{\p{AB}C} \ve{\xi^R_C}$
[[Lie bracket|Lie algebra]]: ~~ ~~ $\lb T_A, T_B \rb = C_{AB}^{\p{AB}C} T_C$
[[Killing form]] (//[[Minkowski metric]]//): ^^ ^^ $g_{AB} = C_{AC}^{\p{AC}D} C_{BD}^{\p{BD}C}$
[[Maurer-Cartan form]] (//[[frame]]//): ^^ ^^ $\f{\cal I} = \f{dy^p} ( \xi^R_p )^A T_A$
Entire space of a [[principal bundle]]: $E \sim M \times G^{\p{\big(}}$
[[Ehresmann principal bundle connection]] over patches of $E$:
[>img[images/png/fiber bundle.png]]$$
\ve{\f{\cal E}}(x,y) = \f{dx^i} A_i^{\p{a}B}(x) \, \ve{\xi^L_B}(y) + \f{dy^p} \ve{\pa_p}
$$
Gauge field [[connection]] over $M$:
$$
\f{A}(x) = \si_0^* \ve{\f{\cal E}} \f{\cal I} = \f{dx^i} A_i^{\p{a}B}(x) \, T_B
$$
<<tiddler HideTags>>
$$
\begin{array}{rcl}
L \!\!&\!\!=\!\!&\!\! \bar{\ps} \ve{e} \lp \f{\pa} + {\small \frac{1}{4}} \f{\om}^{a b} \ga_{a b} + {\small \frac{1}{4}} \f{e}^a \ph^m \ga_{a m} + {\small \frac{1}{2}} \f{W}^{m n} \ga_{m n} + {\small \frac{1}{2}} \f{B}^{m n} \ga_{m n} + {\small \frac{1}{2}} \f{g}^{m n} \ga_{m n} \rp \ps
\end{array}
$$
<html>
<table class="gtable">
<tr>
<td>
<img SRC="talks/CSUF09/images/planes.png">
</td>
<td> </td>
<td>
<SPAN class="math">
\begin{array}{c}
spin(3) \\
\begin{array}{rcl}
\lb \ga_{zy}, \ga_{xz} \rb \!\!&\!\!=\!\!&\!\! \ga_{xy} \\[2em]
\lb \ga_{xy}, \ga_{zt} \rb \!\!&\!\!=\!\!&\!\! 0
\end{array}
\end{array}
</SPAN>
</td>
</tr>
</table>
</html>
<<tiddler HideTags>>
<html>
<table class="gtable">
<tr>
<td>
<img SRC="talks/CSUF09/images/planes.png">
</td>
<td> </td>
<td>
<table class="gtable">
<tr><td>
<SPAN class="math">
\begin{array}{c}
spin(3) \\[1.5em]
\large{ \lb \ga_{zy}, \ga_{xz} \rb = \ga_{xy}} \\[1.5em]
\end{array}
</SPAN>
</tr></td>
</table>
</td>
</tr>
</table>
</html>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">su(5) + \bar{5} + 10 + \bar{"}</SPAN>
</td>
</tr>
<tr>
<td>
Proton decay:
<br><br>
<SPAN class="math">p = u^r + u^b + d^g</SPAN>
<br><br>
<SPAN class="math">d^g \to \bar{e} + X^g_{-\fr{4}{3}}</SPAN>
<br><br>
<SPAN class="math">X^g_{-\fr{4}{3}} + u^b \to \bar{u}{}^{\bar{r}} </SPAN>
<br><br>
<SPAN class="math">
\begin{array}{rcl}
p \!\!&\!\!\to\!\!&\!\! \bar{e} + u^r + \bar{u}{}^{\bar{r}} \\
\!\!&\!\!=\!\!&\!\! \bar{e} + \pi^0
\end{array}</SPAN>
</td>
<td> </td>
<td>
<img SRC="talks/CSUF09/images/SU5Electric.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Georgi-Glashow.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(5) \,\,\oplus\,\, \bar{5} \,\oplus\, 10</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
g = g^A T_A = g^A \fr{i}{2} \la_A
= \fr{i}{2}
\lb
\begin{array}{ccc}
\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \! & g^1 \!-\! ig^2 \!\! & g^4 \!-\! ig^5 \\
g^1 \!+\! i g^2 & \!\!\! -\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \!\! & g^6 \!-\! ig^7 \\
g^4 \!+\! i g^5 & \!\! g^6 \!+\! i g^7 & {\scriptsize -\!\frac{2}{\sqrt{3}}} g^8 \!
\end{array}
\rb
$$
Cartan subalgebra: $\qquad C = g^3 T_3 + g^8 T_8 \quad$ (the diagonal)
Roots and root vectors:
$$
\big[ C , V_{g^{g\bar{b}}} \big] = i \lp \Big( -\!\fr{1}{2} \Big) g^3 + \Big( \fr{\sqrt{3}}{2} \Big) g^8 \rp V_{g^{g\bar{b}}}
\qquad
V_{g^{g\bar{b}}} =
\lb \matrix{
0 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} \rb
$$
for the $g^{g\bar{b}}$ gluon. Weights and weight vectors:
$$
C \, V_{q^r} = i \lp \Big( \fr{1}{2} \Big) g^3 + \Big( \fr{1}{2\sqrt{3}} \Big) g^8 \rp V_{q^r}
\qquad
V_{q^r} = [ 1,0,0 ]
$$
for a red quark, $q^r$, and for their duals acted on by $-C^T$, the anti-quarks.
<<tiddler HideTags>>$$
g = g^A T_A = g^A \fr{i}{2} \la_A
= \fr{i}{2}
\lb
\begin{array}{ccc}
\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \! & g^1 \!-\! ig^2 \!\! & g^4 \!-\! ig^5 \\
g^1 \!+\! i g^2 & \!\!\! -\! g^3 \!+\! {\scriptsize \frac{1}{\sqrt{3}}} g^8 \!\! & g^6 \!-\! ig^7 \\
g^4 \!+\! i g^5 & \!\! g^6 \!+\! i g^7 & {\scriptsize -\!\frac{2}{\sqrt{3}}} g^8 \!
\end{array}
\rb
$$
Cartan subalgebra: $\qquad C = g^3 T_3 + g^8 T_8 \quad$ (the diagonal)
Root and root vector for the $g^{g\bar{b}}$ gluon in $su(3)$:
$$
\big[ C , T_{g^{g\bar{b}}} \big] = i \lp \Big( -\!\fr{1}{2} \Big) g^3 + \Big( \fr{\sqrt{3}}{2} \Big) g^8 \rp T_{g^{g\bar{b}}}
\qquad
T_{g^{g\bar{b}}} = T_7 -i T_6 =
\lb \matrix{
0 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} \rb
$$
Weight and weight vector for the red quark, $q^r$, in the $3$:
$$
C \, \ps_{q^r} = i \lp \Big( \fr{1}{2} \Big) g^3 + \Big( \fr{1}{2\sqrt{3}} \Big) g^8 \rp \ps_{q^r}
\qquad
\ps_{q^r} =
\lb \matrix{
1 \cr
0 \cr
0 \cr
} \rb
$$
The dual anti-quarks in the $\bar{3}$ are acted on by $-C^T$ and have the opposite weights.
<<tiddler HideTags>>Embed the standard model gauge algebra and fermion representation space in the Lie algebra and representation space of a larger group.
[>img[talks/CSUF09/images/coupling.png]]
''Standard Model'' $\p{F^{\big(}}$
\begin{eqnarray}
G_{SM} &=& su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \\[.25em]
\ps_{SM} &=& (2_L\!\,\oplus\,\!2_R) \!\otimes\! (1\!\,\oplus\,\!3) \\[.75em]
\end{eqnarray}
''Georgi-Glashow SU(5)''
\begin{eqnarray}
G_{SM} \subset G_{SU(5)} &=& su(5) \\[.25em]
\ps_{SM} \supset \ps_{SU(5)} &=& \bar{5} \,\oplus\, 10 \\[.75em]
\end{eqnarray}
''Pati-Salam''
\begin{eqnarray}
G_{SM} \subset G_{PS} &=& su(2)_L \,\oplus\, su(2)_R \,\oplus\, su(4) = spin(4) \,\oplus\, spin(6) \\[.25em]
\ps_{SM} = \ps_{PS} &=& (2_L\!\,\oplus\,\!2_R) \!\otimes\! 4 = 4 \!\otimes\! 4 \\[.75em]
\end{eqnarray}
''Spin(10)''
\begin{eqnarray}
G_{SU(5)} \subset G_{Spin(10)} \s\; G_{PS} \subset G_{Spin(10)} \s\; G_{Spin(10)} &=& spin(10) \\[.25em]
G_{SM} = G_{SU(5)} \cap G_{PS} \subset G_{Spin(10)} \s\s\;\; \ps_{SM} &=& 16^\mathbb{C}_{S+}
\end{eqnarray}
<<tiddler HideTags>>Gauge field Lie algebra (not including gravity) embeds in a simpler Lie algebra.
$$\begin{array}{rcl}
\f{A} \,\in\, G_{SM} \!\!&\!\!=\!\!&\!\! su(2)_L \oplus u(1)_Y \oplus su(3) \\[.5em]
\!\!&\!\!\subset\!\!&\!\! su(2)_L \oplus su(2)_R \oplus su(4) \\[.5em]
\!\!&\!\!=\!\!&\!\! spin(4) \oplus spin(6) \\[.5em]
\!\!&\!\!\subset\!\!&\!\! spin(10)
\end{array}
$$
$$\begin{array}{rcl}
\ps_{SM} \!\!&\!\!\in\!\!&\!\! (2_L\!\oplus\!2_R) \!\otimes\! (1\!\oplus\!3) \\[.5em]
\!\!&\!\!=\!\!&\!\! 16^\mathbb{C}_{S+}
\end{array}
$$
Real ''Grassmann numbers'', $\ud{a},\ud{b} \in \mathbb{G}$, are like real numbers but they anti-commute with each other, $\ud{a} \ud{b} = - \ud{b} \ud{a}$, and commute with reals. The square of a Grassmann number is necessarily zero, $\ud{a} \ud{a} = 0$. The product of two real Grassman numbers is a real number (acording to [[Ramond|http://www.amazon.com/Field-Theory-Modern-Frontiers-Physics/dp/0201304503/ref=pd_bbs_sr_1/104-9709999-3726336?ie=UTF8&s=books&qid=1177293245&sr=8-1]]),
$$
\lp \ud{a} \ud{b} \rp^* = - \ud{b}^* \ud{a}^* = - \ud{b} \ud{a} = \ud{a} \ud{b} \in \mathbb{R}
$$
Since Grassmann numbers square to zero, the Taylor expansion of any function of Grassmann variables terminates at the first order,
$$
f(\ud{c}) = a + b \, \ud{c}
$$
Derivatives work as for real numbers (but make sure to change the sign when commuting them past other Grassmann numbers). Using the example above,
$$
\fr{\pa}{\pa \ud{c}} f(\ud{c}) = b
$$
Integrals are effectively the same as derivatives,
$$
\int{\ud{dc} \, f(\ud{c})} = b
$$
Using this rule, for two sets of Grassmann variables, $\ud{a^i},\ud{b^j}$, and a real matrix, $A$, integration gives the [[determinant]],
$$
\int{\ud{da} \, \ud{db} \, \exp(\ud{a^i} A_{ij} \ud{b^j}}) = \det A
$$
For a compex Grassmann number, $\ud{z} = \ud{x} + i \ud{y}$, its square and norm are:
\begin{eqnarray}
\ud{z} \ud{z} &=& i \ud{x} \ud{y} + i \ud{y} \ud{x} = 0 \\
\ud{z}^* \ud{z} &=& i \ud{x} \ud{y} - i \ud{y} \ud{x} = 2 i \ud{x} \ud{y} = - \ud{z} \ud{z}^* \in \mathbb{I}
\end{eqnarray}
An ''anti-Grassmann number'', $\od{a}$, contracts with a Grassmann number to give a real, $\od{a} \ud{b} \in \mathbb{R}$, just like in [[vector-form algebra]]. In fact, a Grassmann number may be thought of as a [[1-form]] in the space of functions. With this interpretation, the product of two Grassmann numbers is not a real, but a ''Grassmann grade two number''.
If [[spin(11,3) GraviGUT fermions]] are to fit in the quaternionic real form of [[e8]] they need to be in the $8^- \times 8^- + 8^+ \times 8^+$ representation space. This requires the introduction of another root coordinate, $w$, with the fermions and anti-fermions having $w = -1$; and so the following e8 root coordinates:
$$
\begin{array}{|l|cccccc|}
\hline
& \om_t & \om_s & u & v & w & x & y & z \\
\hline
\nu_{eL}^{\wedge/\vee} & \mp & \pm & - & + & - & - & - & - \\
\nu_{eR}^{\wedge/\vee} & \pm & \pm & + & + & - & - & - & - \\
e_L^{\wedge/\vee} & \mp & \pm & + & - & - & - & - & - \\
e_R^{\wedge/\vee} & \pm & \pm & - & - & - & - & - & - \\
u^{r\wedge/\vee}_L & \mp & \pm & - & + & - & - & + & + \\
u^{r\wedge/\vee}_R & \pm & \pm & + & + & - & - & + & + \\
d^{r\wedge/\vee}_L & \mp & \pm & + & - & - & - & + & + \\
d^{r\wedge/\vee}_R & \pm & \pm & - & - & - & - & + & + \\
u^{g\wedge/\vee}_L & \mp & \pm & - & + & - & + & - & + \\
u^{g\wedge/\vee}_R & \pm & \pm & + & + & - & + & - & + \\
d^{g\wedge/\vee}_L & \mp & \pm & + & - & - & + & - & + \\
d^{g\wedge/\vee}_R & \pm & \pm & - & - & - & + & - & + \\
u^{b\wedge/\vee}_L & \mp & \pm & - & + & - & + & + & - \\
u^{b\wedge/\vee}_R & \pm & \pm & + & + & - & + & + & - \\
d^{b\wedge/\vee}_L & \mp & \pm & + & - & - & + & + & - \\
d^{b\wedge/\vee}_R & \pm & \pm & - & - & - & + & + & - \\
\hline
\end{array}
\s\s\;
\begin{array}{|l|cccccc|}
\hline
& \om_t & \om_s & u & v & w & x & y & z \\
\hline
\bar{\nu}_{eR}^{\wedge/\vee} & \pm & \pm & + & - & - & + & + & + \\
\bar{\nu}_{eL}^{\wedge/\vee} & \mp & \pm & - & - & - & + & + & + \\
\bar{e}_R^{\wedge/\vee} & \pm & \pm & - & + & - & + & + & + \\
\bar{e}_L^{\wedge/\vee} & \mp & \pm & + & + & - & + & + & + \\
\bar{u}^{r\wedge/\vee}_R & \pm & \pm & + & - & - & + & - & - \\
\bar{u}^{r\wedge/\vee}_L & \mp & \pm & - & - & - & + & - & - \\
\bar{d}^{r\wedge/\vee}_R & \pm & \pm & - & + & - & + & - & - \\
\bar{d}^{r\wedge/\vee}_L & \mp & \pm & + & + & - & + & - & - \\
\bar{u}^{g\wedge/\vee}_R & \pm & \pm & + & - & - & - & + & - \\
\bar{u}^{g\wedge/\vee}_L & \mp & \pm & - & - & - & - & + & - \\
\bar{d}^{g\wedge/\vee}_R & \pm & \pm & - & + & - & - & + & - \\
\bar{d}^{g\wedge/\vee}_L & \mp & \pm & + & + & - & - & + & - \\
\bar{u}^{b\wedge/\vee}_R & \pm & \pm & + & - & - & - & - & + \\
\bar{u}^{b\wedge/\vee}_L & \mp & \pm & - & - & - & - & - & + \\
\bar{d}^{b\wedge/\vee}_R & \pm & \pm & - & + & - & - & - & + \\
\bar{d}^{b\wedge/\vee}_L & \mp & \pm & + & + & - & - & - & + \\
\hline
\end{array}
$$
with fermions in $8^+ \times 8^+$ and anti-fermions in $8^- \times 8^-$.
<<tiddler HideTags>>$$\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + {\textstyle \fr{1}{4}} \f{\om}{}^{ab} \Ga_{ab} + \ha \f{A}{}^{xy} \Ga_{xy} \rp \ps + \ph^x \Ga_x \, \ps \\[.5em]
\!\!&\!\!=\!\!&\!\! \Ga^a \ve{e}{}_a \lp \f{d} + {\textstyle \fr{1}{4}} \f{\om}{}^{bc} \Ga_{bc} + \fr{1}{4} \f{e}{}^b \ph^x \Ga_{bx} + \ha \f{A}{}^{xy} \Ga_{xy} \rp \ps \\[.5em]
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + {\textstyle \fr{1}{2}} \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \rp \ps \\[.5em]
\!\!&\!\!=\!\!&\!\! \ve{e} \lp \f{d} + \f{H} \rp \ps
\end{array}
$$
| $\; \f{H} = \ha \f{\om} + \fr{1}{4} \f{e} \ph + \f{A} \; $ |$\in \; spin(1,3) \,\oplus\, 4 \otimes 10 \,\oplus\, spin(10) = spin(11,3)$ |unified bosonic connection |
| $\; \ps \; $ |$\in \; 64^\mathbb{R}_{S+} \;\;\;\;\; (\otimes 3)$ |all spinor fermion multiplets |
''Curvature''
$$
\ff{F} = \f{d} \f{H} + {\textstyle \fr{1}{2}} [ \f{H}, \f{H} ] = {\textstyle \ha} (\ff{R} - {\textstyle \fr{1}{8}} \f{e}\f{e} \ph^2) + {\textstyle \fr{1}{4}} (\f{T} \ph - \f{e} \f{D} \ph) + \ff{F}{}^A
$$
| $\; \ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om} \; $ |$\in \; spin(1,3) $ |Riemann curvature |
| $\; \ff{T} = \f{d} \f{e} + \ha \f{\om} \f{e} + \ha \f{e} \f{\om} \; $ |$\in \; 4_V $ |Torsion |
| $\; \f{D} \ph = \f{d} \ph + \f{A} \ph - \ph \f{A} \; $ |$\in \; 10_V$ |Covariant Higgs derivative |
| $\; \ff{F}{}^A = \f{d} \f{A} + \f{A} \f{A} \; $ |$\in \; spin(10)$ |Gauge curvature |
<<tiddler HideTags>>$$
\begin{array}{rcl}
S_\ps \!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ \ha \om_\mu^{\p{\mu}bc} \ha \ga_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B^Y_\mu T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \\[.5em]
S^U_\ps \!\!&\!\!=\!\!&\!\!
\int \nf{e} \left\{ \bar{\ps} \Ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ \ha \om_\mu^{\p{\mu}bc} \ha \Ga_{bc}
+ A_\mu^{\p{\mu}xy} \ha \Ga_{xy}
+ \fr{1}{4} (e_\mu)^b \ph^x \Ga_b \Ga_x
\big) \ud{\ps}
\right\} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\int \nf{e} \big\{ \bar{\ps} \ve{e} \big( \f{d} + \f{H} \big) \ud{\ps} \big\} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\int \fr{1}{4!} \bar{\ps} \f{e} \f{e} \f{e} \ep \f{D} \ud{\ps}
\end{array}
$$
''Unified bosonic connection'':
$$
\f{H} = {\tiny \frac{1}{2}} \f{\om} + {\tiny \frac{1}{4}} \f{e} \ph + \f{A} \;\; \in spin(?)
$$
With SO(10) GUT:
$$
spin(1,3) \,\oplus\, 4 \!\otimes\! 10 \,\oplus\, spin(10) = spin(11,3) \mbox{ or } spin(13,1)
$$
or with Pati-Salam GUT:
$$
spin(1,3) \,\oplus\, 4 \!\otimes\! 4 \,\oplus\, spin(4) \,\oplus\, spin(6) \subset spin(11,3), spin(13,1), spin(7,7) \mbox{ or } spin(9,5)
$$
One generation of fermions:
$$
64^\mathbb{R}_{S^+} \mbox{ of } spin(11,3) \mbox{ or } spin(7,7)
$$
<html>
<center>
<table class="gtable">
<tr border=none>
<td><div class="math">
\begin{array}{l}
\om = \ha {\om}{}^{\mu \nu} \ga_{\mu\nu} =
\lb \begin{array}{cc}
{\om}{}_L & \\
& {\om}{}_R
\end{array} \rb \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad
{\om}{}_{L/R} =
\lb \begin{array}{cc}
i {\om}{}_{L/R}^3 & {\om}{}_{L/R}^\wedge \\
{\om}{}_{L/R}^\vee & -i {\om}{}_{L/R}^3
\end{array} \rb \vp{|_{\Big(_{\big(}}}
\\
{e} = {e}{}^\mu \ga_\mu
=
\lb \begin{array}{cc}
& {e}{}_R \\
{e}{}_L &
\end{array} \rb \vp{|_{\Big(}}
\\
\qquad \qquad \qquad \qquad
{e}{}_{L/R} =
\lb \begin{array}{cc}
{e}{}_T^{\wedge/\vee} & \mp {e}{}_S^\wedge \\
\mp {e}{}_S^\vee & {e}{}_T^{\vee/\wedge}
\end{array} \rb \vp{|_{\Big(_{\big(}}}
\\
{f} =
\lb \begin{array}{c}
{f}{}_L \\ {f}{}_R
\end{array} \rb
\qquad \quad \;
{f}{}_{L/R} =
\lb \begin{array}{c}
{f}{}_{L/R}^\wedge \\ {f}{}_{L/R}^\vee
\end{array} \rb
\end{array}
</div></td>
<td> </td>
<td border=none>
<table class="ptable">
<tr>
<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(3,1)</SPAN></th>
<th></th>
<th ALIGN=CENTER><SPAN class="math">\ha \om_L^3</SPAN></th>
<th ALIGN=CENTER><SPAN class="math">\ha \om_R^3</SPAN></th>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_L^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_L^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_R^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\om_R^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_S^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_S^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_T^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr class="butt">
<td ALIGN=CENTER><SPAN class="math">\msqu{#FF5959}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">e_T^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_L^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_L^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_R^\wedge</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td ALIGN=CENTER><SPAN class="math">\btri{#999999}</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">f_R^\vee</SPAN></td>
<td></td>
<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>
<td ALIGN=CENTER><SPAN class="math">-\ha</SPAN></td>
</tr>
</table>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>\begin{eqnarray}
S_s &=& \int \big< \ff{B_s} \ff{F_s} + \Ph_s(\ff{B_s}) \big> \,
= \int \big< \ff{B_s} {\scriptsize \frac{1}{2}} \big( \ff{R} + {\scriptsize \frac{1}{8}}M^2 \f{e}\f{e} \big) - {\scriptsize \frac{1}{4}} \ff{B_s} \ff{B_s} \ga \big> \\
&& \de \ff{B_s} \rightarrow \ff{B_s} = \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \ga^-
\s \text{pseudoscalar:} \;\; \ga = {\ga_0 \ga_1 \ga_2 \ga_3}{\phantom{\Bigg(}} \\
S_s &=& {\scriptsize \frac{1}{4}} \int \big< \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \big( \ff{R} + {\scriptsize \frac{1}{8}} M^2 \f{e}\f{e} \big) \ga^- \big>
= \int \big< \ff{F_s} \ff{F_s} \ga^- \big> \\
&& \big< \ff{R} \ff{R} \ga^- \big> = \f{d} \big< \big( \f{\om} \f{d} \f{\om} + {\scriptsize \frac{1}{3}} \f{\om} \f{\om} \f{\om} \big) \ga^- \big>
\s\s\;\;\;\,
\leftarrow \text{Chern-Simons} \\
&& {\scriptsize \frac{1}{4!}} \big< \f{e}\f{e} \f{e} \f{e} \ga^- \big> = \nf{e}
\s\s\s\s\s\s\;\;\;
\leftarrow \text{volume element} \\
&& \big< \f{e}\f{e} \ff{R} \, \ga^- \big> = \nf{e} R_{\p{\Big(}}
\s\s\s\s\s\s\;\;\;
\leftarrow \text{curvature scalar} \\
S_s &=& {\scriptsize \frac{\La}{12}} \int \nf{e} \lp R + 2 \La \rp
\s\s\;\;\;\;\;
\text{cosmological constant:}_{\p{\big(}} \;\; \La = {\scriptsize \frac{3}{4}} M^2
\end{eqnarray}
<<tiddler HideTags>>\begin{eqnarray}
S_G &=&
\int \big< \ff{B}{}_G \ff{F}{}_G + {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga \big>
\qquad
\ff{F}{}_G = {\scriptsize \frac{1}{2}} \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e} \f{e} \ph^2 \big)
\in \ff{so}(3,1)
\\
&& \de \ff{B}{}_G \rightarrow \ff{B}{}_G = \fr{1}{\pi G} \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \ga
\s\s\s\;\;\;\;
\ga = {\ga_1 \ga_2 \ga_3 \ga_4}{\phantom{\Bigg(}} \\
S_G &=&
{\scriptsize \frac{1}{\pi G}} \int \big< \ff{F}{}_G \ff{F}{}_G \ga \big>
=
{\scriptsize \frac{1}{4 \pi G}} \int \big< \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \big( \ff{R} - {\scriptsize \frac{1}{8}} \f{e}\f{e} \ph^2 \big) \ga \big> \\[.6em]
&& \big< \ff{R} \ff{R} \ga \big> = \f{d} \big< \big( \f{\om} \f{d} \f{\om} + {\scriptsize \frac{1}{3}} \f{\om} \f{\om} \f{\om} \big) \ga \big>
\s\s\s\,
\leftarrow \text{Chern-Simons} \\
&& {\scriptsize \frac{1}{4!}} \big< \f{e}\f{e} \f{e} \f{e} \ga \big> = - \nf{e}
\s\s\s\s\s\s\;\,
\leftarrow \text{volume element} \\
&& \big< \f{e}\f{e} \ff{R} \, \ga \big> = - \nf{e} R_{\p{\Big(}}
\s\s\s\s\s\s\,
\leftarrow \text{curvature scalar} \\
S_G &=& {\scriptsize \frac{1}{16\pi G}} \int \nf{e} \, \ph^2 \lp R - {\scriptsize \frac{3}{2}} \ph^2 \rp
\s
\text{cosmological constant:}_{\p{\big(}} \;\; \La = {\scriptsize \frac{3}{4}} \ph^2
\end{eqnarray}
<<tiddler HideTags>>Using [[chiral]] (//Weyl//) $\mathbb{C}(4 \times 4)$ representation of [[Cl(1,3)]] [[Dirac matrices]]:
$$
\begin{array}{rclrcl}
\ga_0 \!\!&\!\!=\!\!&\!\! \si_1 \otimes 1
=
\lb \begin{array}{cc}
& 1 \\
1 &
\end{array} \rb_{\p{\big(}}
& \;\;\;\;
\ga_\pi \!\!&\!\!=\!\!&\!\! - i \si_2 \otimes \si_\pi
=
\lb \begin{array}{cc}
& -\si_\pi \\
\si_\pi &
\end{array} \rb
\\
\ga_{0\va} \!\!&\!\!=\!\!&\!\! \ga_0 \ga_\va
=
\lb \begin{array}{cc}
\si_\va & \\
& -\si_\va
\end{array} \rb_{\p{(}}
& \;\;\;\;
\ga_{\va\pi} \!\!&\!\!=\!\!&\!\! \ga_\va \ga_\pi
=
\lb \begin{array}{cc}
-i \ep_{\va\pi\ta} \si_\ta & \\
& -i \ep_{\va\pi\ta} \si_\ta
\end{array} \rb
\end{array}
$$
[[Spacetime frame|spacetime frame]] and [[spin connection|spacetime spin connection]]:
$$
\begin{eqnarray}
\f{\om} + \f{e} &=&
\f{dx^a} {\scriptsize \frac{1}{2}} \om_a^{{\p{a}}\mu\nu} \ga_{\mu\nu} + \f{dx^a} ( e_a )^\mu \ga_\mu {}_{\p{(}} \\
&=&
\lb \begin{array}{cc}
( \f{\om^{0 \va}} \si_\va - {\small \frac{i}{2}} \f{\om^{\va \pi}} \ep_{\va \pi \ta} \si_\ta ) & ( \f{e^0} - \f{e^\pi} \si_\pi ) \\
( \f{e^0} + \f{e^\pi} \si_\pi ) & (- \f{\om^{0 \va}} \si_\va - {\small \frac{i}{2}} \f{\om^{\va \pi}} \ep_{\va \pi \ta} \si_\ta )
\end{array} \rb_{\p{(}}
\\
&=&
\lb \begin{array}{cc}
\f{\om_L} & \f{e_L} \\
\f{e_R} & \f{\om_R}
\end{array} \rb_{\p{(}}
\;\; \in \;\; \f{Cl}^{1+2}(1,3)
\end{eqnarray}
$$
Note algebraic equivalence: $Cl^{1+2}(1,3) = Cl^2(1,4) = so(1,4)_{\p{(}}$
<<tiddler HideTags>>
@@display:block;text-align:center;font-size:24pt;Gravity is geometry.@@
@@display:block;text-align:center;"I am convinced, however, that the distinction between geometrical
and other kinds of fields is not logically founded." -- A.E. $\s\s\vp{)^{\big(}}$@@
<<tiddler HideTags>>
<html><center>
<img src="talks/IfA11/images/Gravity.png" height="400">
</center></html>
@@display:block;text-align:center;Gravitational frame bundle over 4D base.@@
<<tiddler HideTags>>The rest frame, $\f{e} \in \f{Cl^1}(1,3)$, is the fiber of spacetime: $\f{e} = \f{dx^i} (e_i)^\mu \ga_\mu$
This determines the metric: $g_{ij} = (e_i)^\mu \eta_{\mu \nu} (e_j)^\nu$
The gravitational spin connection, $\f{\om} \in \f{Cl^2}(1,3) = \f{so}(1,3)$, determines how the frame twists over the spacetime base manifold,
$$
\ff{T} = \f{D} \f{e} = \f{d} \f{e} + {\small \frac{1}{2}} \big[ \f{\om}, \f{e} \big]
$$
But the unified bosonic connection includes the frame:
$$
\f{H} = \ha \f{\om} + \f{G} + {\small \frac{1}{4}} \f{e} \ph
$$
This gives good dynamics for gravity and meshes perfectly with the standard model gauge group, and the Pati-Salam GUT, to act on fermion spinor multiplets:
$$
\big( so(1,3) + su(2)_L + u(1)_R + 4 \!\times\! (2\!+\!2) + u(1)_B + su(3) \big) + 8 \!\times\! 8
$$
<html><center><table class="gtable">
<tr border=none>
<td align="left">A <b>triality</b> rotation, <span class="math">T</span>, of <span class="math">D4</span>:
<div class="math">
{\small
\lb \begin{array}{c}
\ha {\om'}^3_L \\ \ha {\om'}^3_R \\ {W'}^3 \\ {B'}_1^3
\end{array} \rb
=
\lb \begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
1 & 0& 0 & 0
\end{array} \rb
\lb \begin{array}{c}
\ha \om^3_L \\ \ha \om^3_R \\ W^3 \\ B_1^3
\end{array} \rb
=
\lb \begin{array}{c}
\fr{1}{2} \om^3_R \\ B_1^3 \\ W^3 \\ \ha \om^3_L
\end{array} \rb }
</div>
<div class="math">
T \, T \, T \, \om_R^\wedge = T \, T \, \om_L^\wedge = T \, B_1^+ = \om_R^\wedge
</div>
Roots invariant under this <span class="math">T</span>:
<div class="math">
\{
W^+, \,
W^- , \,
e_S^\wedge\ph_+, \,
e_S^\wedge \ph_0, \,
e_S^\vee \ph_-, \,
e_S^\vee \ph_1
\}
</div>
Rotations to triality-equivalent vector and negative chiral spinor representation spaces:
<div class="math">
T \, 8_{S+} = 8_V \quad \; T \, 8_V = 8_{S-} \quad \; T \, 8_{S-} = 8_{S+}
</div>
Three generations, related by triality:
<div class="math">
T \, e_L^\wedge = \mu_L^\wedge
\quad \;
T \, \mu_L^\wedge = \ta_L^\wedge
\quad \;
T \, \ta_L^\wedge = e_L^\wedge
</div>
</td>
<td> </td>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">8_V</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<th COLSPAN="2"> tri </th>
<th></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mtri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{\mu L}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#F2F200}</SPAN></td>
<td><SPAN class="math">\mu_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{\mu R}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mtri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">\mu_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
</table>
<br>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">8_{S-}</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<th COLSPAN="2"> tri </th>
<th></th>
<th><SPAN class="math">B_1^3</SPAN></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\stri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{\ta L}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#F2F200}</SPAN></td>
<td><SPAN class="math">\ta_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{\ta R}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\stri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">\ta_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
</tr>
</table>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr border=none>
<td><div class="math">
\begin{array}{l}
H_1 = (\ha \om + \fr{1}{4} e \ph + W + B_1) \vp{|_{(}} \\[.5em]
\quad \;\; \in so(3,1) + 4 \times 4 + \big( su(2)+su(2) \big) \vp{|_{(}} \\[.5em]
\quad \;\; = Cl^2(7,1) = so(7,1) = d4\\[3em]
8_{S+} \quad \to \qquad \quad H_1 \, (\nu_e + e)\\[.5em]
\qquad \qquad \qquad \qquad \quad =\\[.5em]
{\small
\lb \begin{array}{cccc}
\! \fr{1}{2} \om_L \!+\! \fr{i}{2} W^3 \!\!\! & W^+ & - \! \fr{1}{4} e_R \ph_1 & \fr{1}{4} e_R \ph_+ \\
W^- & \!\!\! \fr{1}{2} \om_L \!-\! \fr{i}{2} W^3 \!\!\! & \p{-} \fr{1}{4} e_R \ph_- & \fr{1}{4} e_R \ph_0 \\
-\fr{1}{4} e_L \ph_0 & \fr{1}{4} e_L \ph_+ & \!\!\! \fr{1}{2} \om_R \!+\! \fr{i}{2} B_1^3 \!\!\! & B_1^+ \\
\p{-}\fr{1}{4} e_L \ph_- & \fr{1}{4} e_L \ph_1 & B_1^- & \!\!\! \fr{1}{2} \om_R \!-\! \fr{i}{2} B_1^3 \!
\end{array} \rb
\lb \begin{array}{c}
\nu_{eL} \\ e_L \\ \nu_{eR} \\ e_R
\end{array} \rb }
\end{array}
</div></td>
<td> </td>
<td>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">D4</SPAN></th>
<th></th>
<th><SPAN class="math">\ha \om_L^3</SPAN></th>
<th><SPAN class="math">\ha \om_R^3</SPAN></th>
<th><SPAN class="math">W^3</SPAN></th>
<th><SPAN class="math">B_1^3</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td><SPAN class="math">\om_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#59FF59} </SPAN></td>
<td><SPAN class="math">\om_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#FFFF00} </SPAN></td>
<td><SPAN class="math">\smash{W^\pm}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#FFFFFF} </SPAN></td>
<td><SPAN class="math">\smash{B_1^\pm}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm 1</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_+</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \ha</SPAN></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#B2B200} </SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_+</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\pm \ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\ha</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_-</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#F2F200} </SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_-</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#F77C00}</SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_0</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\msqu{#F77C00}</SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_0</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mdia{#BF6000}</SPAN></td>
<td><SPAN class="math">e_T^{\wedge/\vee} \ph_1</SPAN></td>
<td></td>
<td><SPAN class="math">\mp \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mdia{#BF6000}</SPAN></td>
<td><SPAN class="math">e_S^{\wedge/\vee} \ph_1</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#B2B200}</SPAN></td>
<td><SPAN class="math">\nu_{eL}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#F2F200}</SPAN></td>
<td><SPAN class="math">e_L^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#999999}</SPAN></td>
<td><SPAN class="math">\nu_{eR}^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D9D9D9}</SPAN></td>
<td><SPAN class="math">e_R^{\wedge/\vee}</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\pm \fr{1}{2}</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
</tr>
</table>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>
|>| !Group Properties |
| $\;$Ordered group product of elements:$\;$|$\; a \, b = c \;\; \in \; G$ |
| Identity element:$\;$|$\; a \,1 = 1 \, a = a$ |
| Inverses:$\;$|$\; a \, a^- = a^- \, a = 1$ |
| Associativity:$\;$|$\; a \, (b \, c)=(a \, b) \, c$ |
The number of elements in a ''finite group'' is the ''order''.
An $n$ dimensional ''Lie group'' is a continuum of elements, $g(x) \in G$, parametrized by $n$ real (or complex) parameters, $x \in \Re^n$. It is also a manifold. Near the identity, $g(0)=1$, Lie group elements may be described by exponentiating $n$ ''Lie algebra'' generators, $T_A \in \mathfrak{g} = {\rm Lie}(G)$,
$$
g(x) = e^{x^A T_A} \simeq 1 + x^A T_A
$$
Finite collections of Lie group elements (maybe or maybe not connected to the identity) can make an embedded finite group.
A ''represenation space'' (or ''G-module''), $V$, is a real or complex vector space upon which a ''group representation'', $\Pi(G) \subset GL(V)$, or Lie algebra representation, $\pi(\mathfrak{g}) \subset GL(V)$, acts linearly. A representation is ''faithful'' iff every $\Pi(g)$ is unique and:
$$
\Pi(g_1 g_2) = \Pi(g_1) \Pi(g_2) \s\;\;\; \pi([X,Y]) = \pi(X) \pi(Y) - \pi(Y) \pi(X)
$$
<<tiddler HideTags>><html><center>
<img SRC="talks/IfA11/images/hst.jpg" height=450px>
</center></html>
<<tiddler HideTags>><html>
<table class="gtable">
<tr>
<td>General Relativity</td>
<td></td>
<td>Quantum Field Theory</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>
<SPAN class="math">
\begin{array}{rcl}
R_{\mu}^{\p{\mu} a} - \ha (e_\mu)^a R \!\!&\!\!=\!\!&\!\! (e_\mu)^a \La - 8 \pi G \, T_{\mu}^{\p{\mu} a} \\[.3em]
R_{\mu}^{\p{\mu} a} \!\!&\!\!=\!\!&\!\! (e_b)^\nu ( \pa_{\lb \nu \rd} \om_{\ld \mu \rb}{}^{ba} + \om_{\lb \nu \rd}{}^{bc} \om_{\ld \mu \rb c}{}^a ) \\[.7em]
\ha T_{\mu \nu}{}^a \!\!&\!\!=\!\!&\!\! \pa_{\lb \mu \rd} (e_{\ld \nu \rb})^a + \om_{\lb \mu \rd}{}^{ab} (e_{\ld \nu \rb})_b \\[.7em]
\end{array}
</SPAN>
</td>
<td> </td>
<td>
<SPAN class="math">
\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \ga^a (e_a)^\mu \big( \pa_\mu +\fr{1}{4} \om_\mu^{\p{\mu}bc} \ga_{bc} + A_\mu^{\p{\mu}B} T_B \big) \ud{\ps} + \ph \ud{\ps} \\[.3em]
0 \!\!&\!\!=\!\!&\!\! D^\mu D_\mu \ph + \mu^2 \ph - \la \ph^3 \\[.7em]
0 \!\!&\!\!=\!\!&\!\! D^\mu F_{\mu \nu}^{\p{\mu \nu} B} + \bar{\ps} \ga^a (e_a)_\nu T^B \ps + (\pa_\nu \ph) T^B \ph \\[.7em]
\end{array}
</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/IfA11/images/hst2.jpg" height=320>
</td>
<td> </td>
<td>
<table class="gtable">
<tr><td>
<img SRC="talks/IfA11/images/lhc2.jpg" height=320>
</tr></td>
</table>
</td>
</tr>
</table>
</html>
In [[quantum mechanics]] a state evolves by a [[unitary]] time evolution operator, $| \ps(t) \rangle = \hat{U}_t | \ps(0) \rangle$, according to the Schrödinger equation. Under this ''Schrödinger picture'', the operators, such as the position operator, $\hat{x}$, are not time dependent. In the ''Heisenberg picture'', we freeze the state vector, $| \ps \rangle = | \ps(0) \rangle$, and introduce time-dependent operators, such as $\hat{x}_t = \hat{U}^-_{t} \hat{x} \, \hat{U_t}$. The expectation values in either picture then are the same, $\langle \ps | \hat{x}_t | \ps \rangle = \langle \ps(t) | \hat{x} | \ps(t) \rangle$. The time dependence of a Heisenberg picture operator is thus governed by
$$
\fr{d}{dt} \hat{A}_t = \fr{i}{\hbar} \lb \hat{H}, \hat{A}_t \rb + \pa_t \hat{A}_t
$$
If we separate the Hamiltonian into a ''free'' part and an ''interaction'' part, $\hat{H}=\hat{H}_0 + \hat{H}_{\mathrm{int}}$, the ''interaction picture'', the state can evolve according to the interaction Hamiltonian and the operators can evolve according to the free Hamiltonian,
$$
\fr{d}{dt} | \ps(t) \rangle = \fr{1}{i \hbar} \hat{H}_{\mathrm{int}}(t) | \ps(t) \rangle \s \hat{A}_t = \hat{U}^{0-}_t \hat{A} \, \hat{U}^{0}_t
$$
in which $\hat{U}^{0}_t = e^{\fr{-i}{\hbar} \hat{H}_0 t}$ and $\hat{H}_{\mathrm{int}}(t) = \hat{U}^{0-}_t \hat{H}_{\mathrm{int}} \hat{U}^{0}_t$. The interaction picture is not valid if the Hamiltonian doesn't separate, such as in curved spacetime, where things can get weird.
The ''Hermitian conjugate'', $\hat{A}^\da$, of an operator, $\hat{A}$, acting on a [[unitary representation]] space is defined by conjugation within the [[Hermitian form]],
$$
\langle v | \hat{A}^\da | u \rangle = \langle \hat{A} v | u \rangle = \langle u | \hat{A} v \rangle^* = \langle u | \hat{A} | v \rangle^*
$$
where we've used bra-[[ket]] notation. If we use basis vectors, this gives the complex conjugate and matrix [[transpose]] for the [[matrix element]]s,
$$
A^\da{}_i{}^j = A^j{}_i{}^*
$$
A matrix or operator is ''Hermitian'' iff $\hat{A}^\da = \hat{A}$, and ''anti-Hermitian'' (or ''skew-Hermitian'') iff $\hat{A}^\da = -\hat{A}$.
A ''Hermitian form'' on a [[vector space]], $V$, is a map (bracket)
$$
\langle \cdot | \cdot \rangle : V \times V \to \mathbb{C}
$$
satisfying
$$
\langle v | a w + b u \rangle = a \langle v | w \rangle + b \langle v | u \rangle
$$
$$
\langle a v + b w | u \rangle = a^* \langle v | u \rangle + b^* \langle w | u \rangle
$$
$$
\langle v | u \rangle^* = \langle u | v \rangle
$$
for all $u,v,w \in V$ and $a,b \in \mathbb{C}$. It is like a [[metric]], but into the complex numbers instead of the reals. A vector space with Hermitian form is a [[Hilbert space]], with this bracket inspiring bra-[[ket]] notation. One also sometimes sees the Hermitian form written more compactly as $\langle u | v \rangle = u^\da v$.
If one has a metric, $g(u,v)$, on a vector space, and a compatible [[complex structure]], $g(u,i v)=-g(i u, v)$, then
$$
\li u | v \ri = g(u,v) - i g(u, i v)
$$
is a Hermitian form on the space.
/%
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A ''Hilbert space'', $\mathcal{H}$, is a usually complex, often infinite-dimensional, [[vector space]] equipped with a [[Hermitian form]]. It is the [[representation space]] (or module) of some [[unitary representation]].
Every [[differential form]] [[field|cotangent bundle]] may be decomposed as
$$
\nf{F} = \nf{\Om} + \f{d} \nf{\Phi} + \ve{\de} \nf{\Psi}
$$
in which $\nf{\Om}$ is [[harmonic]], $\f{d}$ is the [[exterior derivative]], and $\ve{\de}$ is the [[codifferential]].
Every [[closed]] differential form field, $\f{d} \nf{F} = 0$, may be decomposed as
$$
\nf{F} = \nf{\Om} + \f{d} \nf{\Phi}
$$
Therefore, the [[cohomology]] is the same as the space of harmonic forms -- Hodge's theorem. That's kind of strange, since cohomology doesn't require a [[metric]], while the [[Hodge dual]] does.
Ref:
*http://en.wikipedia.org/wiki/De_Rham_cohomology
*[[Vector Calculus and the Topology of Domains in 3-Space|papers/vectorcalc.pdf]]
There are duality transformations in and between the tangent and cotangent spaces similar to [[Clifford dual]]ity. For any [[differential form]] of grade $p$,
$$
\nf{a} = \fr{1}{p!} a_{\al \dots \be} \f{e^\al} \dots \f{e^\be}
$$
its ''vector dual'' (-p)-form is
$$
\bar{a} = \fr{1}{p!} a_{\al \dots \be} \et^{\al \ga} \dots \et^{\be \de} \ve{e_\ga} \dots \ve{e_\de}
$$
By multiplying any p-form by the (-n)-form, $\bar{e} = \ve{e_0} \ve{e_1} \dots \ve{e_{n-1}}$, one gets its ''Hodge vector dual'' (p-n)-form,
$$
\bar{* a} = \bar{e} \nf{a} = \ve{e_0} \ve{e_1} \dots \ve{e_{n-1}} \fr{1}{p!} a_{\al \dots \be} \f{e^\al} \dots \f{e^\be}
= \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \ve{e_\ga} \dots \ve{e_\de}
$$
using [[vector-form algebra]] and the [[permutation symbol]]. And finally, by taking the form dual to this, one gets the ''Hodge dual'' (n-p)-form,
$$
\begin{eqnarray}
* \nf{a} = \nf{*a} &=& \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \f{e_\ga} \dots \f{e_\de} \\
&=& \fr{\ll \et \rl}{\ll e \rl p! \lp n-p \rp!} a_{i \dots j} \va^{i \dots jk \dots l} g_{km} \dots g_{ln} \f{dx^m} \dots \f{dx^n}
\end{eqnarray}
$$
This ''Hodge operator'', $*$, which is only defined in the presence of a [[frame]] or [[metric]], is quite useful and allows the construction of the n-form product of any two p-forms, $\nf{a}$ and $\nf{b}$,
\begin{eqnarray}
\nf{*a}\nf{b} &=& \fr{1}{p! \lp n-p \rp!} a_{\al \dots \be} \ep^{\al \dots \be \ga \dots \de} \f{e_\ga} \dots \f{e_\de} \fr{1}{p!} b_{\ep \dots \up} \f{e^\ep} \dots \f{e^\up}\\
&=& \fr{\ll \et \rl}{p! \lp n-p \rp!} \fr{1}{p!} a_{\al \dots \be} b_{\ep \dots \up} \ep^{\al \dots \be \ga \dots \de} \ep_{\ga \dots \de}{}^{\ep \dots \up} \nf{e}\\
&=& \nf{e} \fr{1}{p!} a_{\al \dots \be} b^{\al \dots \be} = \nf{e} \fr{1}{p!} a_{i \dots j} b^{i \dots j} = \nf{e} \lp \bar{a} \nf{b} \rp = \nf{a} \nf{*b}
\end{eqnarray}
relying on [[permutation identities]]. Just as the Clifford dual squares to $\pm 1$ depending on [[signature|Minkowski metric]], the Hodge dual of a p-form similarly squares to
\[ \nf{**a} = \ll \et \rl \lp -1 \rp^{p \lp n-p \rp} \nf{a} \]
There is an example important enough to address specifically. If $\ff{F} = \ha \f{e^\mu} \f{e^\nu} F_{\mu \nu}$ is a 2-form over a four dimensional space (or spacetime), then its Hodge dual is:
$$
\ff{*F} = \fr{1}{4} F_{\mu \nu} \ep^{\mu \nu \rh \si} \f{e_\rh} \f{e_\si} = \ff{\vv{\ep}} \ff{F}
$$
in which the ''Hodge dual projector'' is a 2-vector valued 2-form defined as
$$
\ff{\vv{\ep}} = - \f{e^\rh} \f{e^\si} \ep_{\rh \si}^{\p{\rh \si} \mu \nu} \ve{e_\mu} \ve{e_\nu} = - \f{e_\rh} \f{e_\si} \ep^{\rh \si \mu \nu} \ve{e_\mu} \ve{e_\nu}
= \left< \f{e} \f{e} \ga \ve{e} \ve{e} \right>
$$
which contracts with $\ff{F}$ via the [[vector-form algebra]]. Other Hodge dual projectors may be built corresponding to other cases.
There is a somewhat awkward but coordinate free expression for the Hodge dual, taking the angle brackets to group the enclosed Clifford elements and the parenthesis to group the form elements in
\begin{eqnarray}
\nf{*a} &=& \fr{1}{p! \lp n-p \rp!} < \lp \ve{e} \rp^p \ga^- ( \lp \f{e} \rp^{n-p} > \nf{a})\\
&=& \fr{1}{p! \lp n-p \rp!} \ve{e_\al} \dots \ve{e_\be} \lp \f{e^\ga} \dots \f{e^\de} \f{e^\ep} \dots \f{e^\up} \rp \li \ga^\al \dots \ga^\be \ga^- \ga_\ga \dots \ga_\de \ri \fr{1}{p!} a_{\ep \dots \up}\\
&=& \fr{\ll \et \rl}{p! \lp n-p \rp!} \ve{e_\al} \dots \ve{e_\be} \nf{e} \ep^{\ga \dots \de \ep \dots \up} \ep_{\ga \dots \de}{}^{\al \dots \be} a_{\ep \dots \up}\\
&=& \fr{1}{p! \lp n-p \rp!} \f{e_\al} \dots \f{e_\be} \ep^{\ga \dots \de \al \dots \be} a_{\ga \dots \de}
\end{eqnarray}
For a [[tangent vector]], $\ve{v}$, and a grade $p$ [[differential form]], $\nf{f}$, the [[vector-form algebra]] contraction can be equated with an expression involving the [[Hodge dual]],
$$\ve{v} \nf{f} = (-1)^{p(n-p)} * \lp \f{v} \nf{*f} \rp$$
in which $\f{v} = v^\al \et_{\al \be} \f{e^\be}$ is the form dual of $\ve{v}$.
(//add more as needed//)
David Ritz Finkelstein
http://arxiv.org/abs/gr-qc/0608086
*proposes a "flexing" of [[Lie algebra]] structure constants to go from one level of physical theory (some struct const = 0) to another (some not 0, or all not 0 ([[simple]])).
*hey, isn't this the same as Lie algebra deformation?
A horizontal dividing line.
----
{{{----}}}
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<<<
A whole block
of text to be quoted.
<<<
or
>>>Multiple levels of indented quotes.
>>Just like [[Bullet Points]].
>yep
>>or like [[Numbered Lists]]
That's what they said.
{{{
<<<
A whole block
of text to be quoted.
<<<
or
>>>Multiple levels of indented quotes.
>>Just like [[Bullet Points]].
>yep
>>or like [[Numbered Lists]]
That's what they said.
}}}
TiddlyWiki lets you write ordinary HTML by enclosing it in {{{<html>}}} and {{{</html>}}}:
<html>
<a href="javascript:;" onclick="onClickTiddlerLink(event);"
tiddlyLink="Welcome"
style="background-color: yellow;">
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</html>
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tiddlyLink="Welcome"
style="background-color: yellow;">
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</html>
}}}
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|Documentation|http://www.TiddlyTools.com/#InlineJavascriptPluginInfo|
|Version|1.9.6|
|Author|Eric Shulman|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|Insert Javascript executable code directly into your tiddler content.|
''Call directly into TW core utility routines, define new functions, calculate values, add dynamically-generated TiddlyWiki-formatted output'' into tiddler content, or perform any other programmatic actions each time the tiddler is rendered.
!!!!!Documentation
>see [[InlineJavascriptPluginInfo]]
!!!!!Revisions
<<<
2010.12.15 1.9.6 allow (but ignore) type="..." syntax
|please see [[InlineJavascriptPluginInfo]] for additional revision details|
2005.11.08 1.0.0 initial release
<<<
!!!!!Code
***/
//{{{
version.extensions.InlineJavascriptPlugin= {major: 1, minor: 9, revision: 6, date: new Date(2010,12,15)};
config.formatters.push( {
name: "inlineJavascript",
match: "\\<script",
lookahead: "\\<script(?: type=\\\"[^\\\"]*\\\")?(?: src=\\\"([^\\\"]*)\\\")?(?: label=\\\"([^\\\"]*)\\\")?(?: title=\\\"([^\\\"]*)\\\")?(?: key=\\\"([^\\\"]*)\\\")?( show)?\\>((?:.|\\n)*?)\\</script\\>",
handler: function(w) {
var lookaheadRegExp = new RegExp(this.lookahead,"mg");
lookaheadRegExp.lastIndex = w.matchStart;
var lookaheadMatch = lookaheadRegExp.exec(w.source)
if(lookaheadMatch && lookaheadMatch.index == w.matchStart) {
var src=lookaheadMatch[1];
var label=lookaheadMatch[2];
var tip=lookaheadMatch[3];
var key=lookaheadMatch[4];
var show=lookaheadMatch[5];
var code=lookaheadMatch[6];
if (src) { // external script library
var script = document.createElement("script"); script.src = src;
document.body.appendChild(script); document.body.removeChild(script);
}
if (code) { // inline code
if (show) // display source in tiddler
wikify("{{{\n"+lookaheadMatch[0]+"\n}}}\n",w.output);
if (label) { // create 'onclick' command link
var link=createTiddlyElement(w.output,"a",null,"tiddlyLinkExisting",wikifyPlainText(label));
var fixup=code.replace(/document.write\s*\(/gi,'place.bufferedHTML+=(');
link.code="function _out(place,tiddler){"+fixup+"\n};_out(this,this.tiddler);"
link.tiddler=w.tiddler;
link.onclick=function(){
this.bufferedHTML="";
try{ var r=eval(this.code);
if(this.bufferedHTML.length || (typeof(r)==="string")&&r.length)
var s=this.parentNode.insertBefore(document.createElement("span"),this.nextSibling);
if(this.bufferedHTML.length)
s.innerHTML=this.bufferedHTML;
if((typeof(r)==="string")&&r.length) {
wikify(r,s,null,this.tiddler);
return false;
} else return r!==undefined?r:false;
} catch(e){alert(e.description||e.toString());return false;}
};
link.setAttribute("title",tip||"");
var URIcode='javascript:void(eval(decodeURIComponent(%22(function(){try{';
URIcode+=encodeURIComponent(encodeURIComponent(code.replace(/\n/g,' ')));
URIcode+='}catch(e){alert(e.description||e.toString())}})()%22)))';
link.setAttribute("href",URIcode);
link.style.cursor="pointer";
if (key) link.accessKey=key.substr(0,1); // single character only
}
else { // run script immediately
var fixup=code.replace(/document.write\s*\(/gi,'place.innerHTML+=(');
var c="function _out(place,tiddler){"+fixup+"\n};_out(w.output,w.tiddler);";
try { var out=eval(c); }
catch(e) { out=e.description?e.description:e.toString(); }
if (out && out.length) wikify(out,w.output,w.highlightRegExp,w.tiddler);
}
}
w.nextMatch = lookaheadMatch.index + lookaheadMatch[0].length;
}
}
} )
//}}}
// // Backward-compatibility for TW2.1.x and earlier
//{{{
if (typeof(wikifyPlainText)=="undefined") window.wikifyPlainText=function(text,limit,tiddler) {
if(limit > 0) text = text.substr(0,limit);
var wikifier = new Wikifier(text,formatter,null,tiddler);
return wikifier.wikifyPlain();
}
//}}}
// // GLOBAL FUNCTION: $(...) -- 'shorthand' convenience syntax for document.getElementById()
//{{{
if (typeof($)=='undefined') { function $(id) { return document.getElementById(id.replace(/^#/,'')); } }
//}}}
[>img[images/person/John Baez.jpg]]Homepage: http://math.ucr.edu/home/baez/
*Location: UCRiverside
A wonderfully prolific mathematical physicist, and all around good guy.
Access keys are shortcuts to common functions accessed by typing a letter with either the 'alt' (PC) or 'control' (Mac) key:
|!PC|!Mac|!Function|
|Alt-F|Ctrl-F|Search|
|Alt-J|Ctrl-J|NewJournal|
|Alt-N|Ctrl-N|NewNote|
|Alt-S|Ctrl-S|SaveChanges|
These access keys are provided by the associated internal [[Macros]] for the functions above. The macro needs to be used in an open note (or the [[MainMenu]] or SideBar) in order for the access keys to work.
While editing a note:
* ~Control-Enter or ~Control-Return accepts your changes and switches out of editing mode (use ~Shift-Control-Enter or ~Shift-Control-Return to stop the date and time being updated for MinorChanges)
* Escape abandons your changes and reverts the note to its previous state
In the search box:
* Escape clears the search term
A [[Lie algebra]], ${\frak g}$, is a [[vector space]] as well as an algebra, spanned by the basis vectors, $T_A$, and one may use the structure constants to build a natural [[metric]] giving a scalar result from two Lie algebra elements,
$$
\lp B, C \rp = {\rm Tr}({\rm Ad}_B {\rm Ad}_C) = T^D \lb B, \lb C, T_D \rb \rb = B^A C^B g_{AB}
$$
using the [[trace]] and Lie algebra adjoint action, ${\rm Ad}_B C = \lb B, C \rb$. The resulting metric coefficients are those of the ''Killing form'',
$$
g_{AB} = \lp T_A, T_B \rp = C_{AC}{}^D C_{BD}{}^C
$$
It is always possible to transform to a new set of generators, $T'_A = L_A{}^B T_B$, producing a new set of structure constants and hence a new metric, $\et_{AB} = L_A{}^C L_B{}^D g_{CD}$. As long as the metric is non-degenerate, which happens iff ${\frak g}$ is semi-[[simple]], it is possible to transform to a set of generators such that this metric is unit diagonal (with $+1$ and $-1$ entries like in the [[Minkowski metric]]) via the methods of [[spectral decomposition|eigen]]. This has already been done for most common generator representations used in physics, up to a constant factor, so that often $g_{AB} = \et_{AB} = \de_{AB}$.
Using the Jacobi identity, the Killing form is symmetric, $g_{AB} = g_{BA}$, and is ''adjoint invariant'',
\begin{eqnarray}
\lp \lb A, B \rb, C \rp &=& \lp A, \lb B, C \rb \rp \\
\lp g A g^-, C \rp &=& \lp A, g^- C g \rp
\end{eqnarray}
which implies the structure constants are antisymmetric in the last two indices,
$$
C_{ABC} = -C_{ACB}
$$
when the Killing forms $g_{AB}$ (and $g^{AB}$) are used to lower (and raise) Lie algebra indices, $C_{ABC}=C_{AB}{}^D g_{DC}$. Since we always have $C_{AB}{}^C = - C_{BA}{}^C$, the structure constants are completely [[antisymmetric|index bracket]],
$$
C_{ABC} = C_{\lb ABC \rb} = C_{BCA} = C_{CAB} = -C_{BAC} = -C_{ACB} = -C_{CBA}
$$
which is useful enough to call the ''Killing form identity''.
The ''inverse Killing form'', $g^{AB}$, is used to define the ''generator duals'', $T^A = g^{AB} T_B$, satisfying $\lp T^A, T_B \rp = \de^A_B$.
For some Lie algebras with [[Clifford algebra]] or matrix generators, the scalar part or trace gives the orthonormality relations
$$
\lp T_A, T_B \rp = g_{AB} \sim \li T_A T_B \ri = \de_{AB}
$$
A ''Killing spinor'' is a [[spinor]] field satisfying
$$
\f{\na} \ps = \la \f{e} \ps
$$
for some constant ''Killing number'', $\la$, in which $\f{\na}$ is the [[spinor covariant derivative]] and $\f{e}$ is the [[coframe|frame]].
The [[tangent vector]] field corresponding to a Killing spinor is a [[Killing vector]],
$$
\ve{\xi} = \left< \bar{\ps} \ga^\al \ps \right> \ve{e_\al}
$$
in which $\bar{\ps}$ is the [[Clifford conjugate]] of the spinor and $\ve{e_\al}$ are [[frame]] vectors.
A ''Killing vector'' field, $\ve{\xi}(x)$, is the generator of a [[flow]], $\ph_t = e^{t {\cal L}_{\ve{\xi}}}$, that leaves the geometry of a [[manifold]] invariant — constituting a symmetry of the geometry. It is a [[tangent vector field|tangent bundle]] satisfying ''Killing's equation'',
$$
L_{\ve{\xi}} \f{e} = B \times \f{e}
$$
The [[Lie derivative]] of the [[frame]] along a Killing vector field gives a [[rotation|Clifford rotation]] of the frame by some corresponding Clifford bivector field, $B (x) \in Cl^2$,
$$
L_{\ve{\xi}} \f{e^\al} = B_\be{}^\al \f{e^\be}
$$
with ''Killing rotation coefficients'', $B_\be{}^\al = - B^\al{}_\be$. This version of Killing's equation, or equivalently,
$$
L_{\ve{\xi}} \ve{e_\al} = - B_\al{}^\be \ve{e_\be}
$$
matches the usual definition that the Lie derivative of the [[metric]] along a Killing vector field vanishes.
Any set of Killing vector fields is related through the [[Lie bracket|Lie derivative]],
$$
\lb \ve{\xi_A}, \ve{\xi_B} \rb_L = C_{AB}{}^C \ve{\xi_C}
$$
with $C_{AB}{}^C$ the set of ''structure constants'' for the symmetries. The manifold [[diffeomorphism]]s ([[isometries|local Lorentz transformation]]) built from the flows generated by a set of Killing vector fields constitute a [[Lie group]].
A Killing vector field has many nice [[properties|Killing vector identities]].
The defining equation of a [[Killing vector]] field, $\ve{\xi}$, is
\begin{eqnarray}
B_\be{}^\al \f{e^\be} &=& {\cal L}_{\ve{\xi}} \f{e^\al} = \ve{\xi} \lp \f{d} \f{e^\al} \rp + \f{d} \xi^\al \\
&=& \ve{\xi} \lp \f{w}{}_\be{}^\al \f{e^\be} + \ff{T^\al} \rp + \f{d} \xi^\al \\
&=& \f{e^\be} \lp \xi^\de w_{\de \be}{}^\al - \xi^\de w_{\be \de}{}^\al + 2 \xi^\de T_{\de \be}{}^\al + \pa_\be \xi^\al \rp
\end{eqnarray}
by virtue of the defining equations for the [[Lie derivative]], the [[cotangent bundle connection]], and the [[torsion]]. This gives a useful expression for the derivative of the Killing vector field coefficients,
$$
\pa_\be \xi_\al = B_{\be \al} - \xi^\de w_{\de \be \al} + \xi^\de w_{\be \de \al} - 2 \xi^\de T_{\de \be \al}
$$
The [[1-form dual|frame]] to the Killing vector field is $\f{\xi} = \xi_\al \f{e^\al} = \xi^\be \et_{\be \al} \f{e^\al}$. The cotangent bundle covariant derivative of this field is
$$
\f{\na} \f{\xi} = \f{e^\be} \na_\be \lp \xi_\al \f{e^\al} \rp
= \f{e^\be} \f{e^\ga} \lp \pa_\be \xi_\ga + \xi_\al w_{\be \ga}{}^\al \rp
= \f{e^\be} \f{e^\ga} \lp B_{\be \ga} - \xi^\de w_{\de \be \ga} - 2 \xi^\de T_{\de \be \ga} \rp
$$
Similarly,
$$
\f{\na} \ve{\xi} = \f{e^\be} \na_\be \lp \xi^\al \ve{e_\al} \rp
= \f{e^\be} \ve{e_\ga} \lp \pa_\be \xi^\ga + \xi^\al w_\be{}^\ga{}_\al \rp
= \f{e^\be} \ve{e_\ga} \lp B_\be{}^\ga - \xi^\de w_{\de \be}{}^\ga - 2 \xi^\de T_{\de \be}{}^\ga \rp
$$
If $\ve{v}$ is the velocity along a [[geodesic]],
$$
0 = \ve{v} \f{\na} \ve{v} = v^\al \lp \pa_\al v^\de + v^\be w_\al{}^\de{}_\be \rp \ve{e_\de}
$$
the component of this velocity along any Killing vector field, $p = \lp \ve{v}, \ve{\xi} \rp = \ve{v} \f{\xi}$, is constant along the geodesic,
$$
\ve{v} \f{d} p = v^\al \pa_\al \lp v^\be \xi_\be \rp
= v^\al \lp \pa_\al v^\be \rp \xi_\be + v^\al v^\be \lp \pa_\al \xi_\be \rp
= v^\al \lp \pa_\al v^\de \rp \xi_\de + v^\al v^\be \lp w_{\al \de \be} \xi^\de - 2 \xi^\de T_{\de \al \be} \rp
= 0
$$
as long as the torsion vanishes, or at least $T_{\de \lp \al \be \rp} = 0$.
If the Killing vector field is of constant length, $\ve{\xi} \f{\xi} = \xi^\al \xi_\al = c$, then
$$
0 = \pa_\be \lp \xi^\al \xi_\al \rp = 2 \xi^\al \lp \pa_\be \xi_\al \rp
= 2 \xi^\al \lp B_{\be \al} - \xi^\de w_{\de \be \al} - 2 \xi^\de T_{\de \be \al} \rp
$$
and the integral curves of the Killing vector field are geodesics,
$$
\ve{\xi} \f{\na} \ve{\xi} = \xi^\be \ve{e_\ga} \lp B_\be{}^\ga - \xi^\de w_{\de \be}{}^\ga \rp = 0
$$
as long as the torsion vanishes or $T_{\de \lp \al \be \rp} = 0$.
The ''Kronecker product'' (//''tensor product''//), $\otimes$, of an $m$ by $n$ [[matrix|linear operator]], $A$, and a $p$ by $q$ matrix, $B$, is an $mp$ by $nq$ matrix, $C = A \otimes B$,
$$
C_{\lp \lp a - 1 \rp p + x \rp}{}^{\lp \lp b - 1 \rp q + y \rp} = A_a{}^b B_x{}^y
$$
It is a ''block'' matrix of $B$'s multiplied by the entries of $A$. For example,
$$
\lb \begin{array}{cc}
1 & 2\\
3 & 1
\end{array} \rb
\otimes
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
=
\lb \begin{array}{cc}
1
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
& 2
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
\\
3
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
& 1
\lb \begin{array}{cc}
0 & 3\\
2 & 1
\end{array} \rb
\end{array} \rb
=
\lb \begin{array}{cccc}
0 & 3 & 0 & 6\\
2 & 1 & 4 & 2\\
0 & 9 & 0 & 3\\
6 & 3 & 2 & 1
\end{array} \rb
$$
This product spawns several identities, including:
$$
\lp A \otimes B \rp \lp C \otimes D \rp = A C \otimes B D
$$
Ref:
http://en.wikipedia.org/wiki/Kronecker_product
<<tiddler HideTags>><html><center>
<img SRC="talks/IfA11/images/lhc2.jpg" height=500px>
</center></html>
<<tiddler HideTags>><html><center><iframe src="talks/IfA11/anim/LHCanim.html" width="560" height="540" frameborder="0"></iframe>
</center></html>
//Use the first method in each example below, unless you have some reason not to.//
Mathematical symbols, such as \(e^{x^2}\), may be inserted inline.
{{{
Mathematical symbols, such as $e^{x^2}$, may be inserted inline.
Mathematical symbols, such as \(e^{x^2}\), may be inserted inline.
}}}
Or as displayed math,$$e^{x^2}$$ on its own line.
{{{
Or as displayed math, \[e^{x^2}\] on its own line.
Or as displayed math, $$e^{x^2}$$ on its own line.
Or as displayed math, \begin{equation}e^{x^2}\end{equation} on its own line.
}}}
Or as an equation array,
\begin{eqnarray}A &=& e^{x^2}\\&=&C\end{eqnarray}
{{{
Or as an equation array,\begin{eqnarray}A &=& e^{x^2}\\&=& C\end{eqnarray}
}}}
Some of the available TeX symbols can be found at [[jsMath|http://www.math.union.edu/~dpvc/jsMath/symbols/welcome.html]], the best method I could find for displaying TeX online. The small button in the lower right corner of this window opens its control planel. I'm not sure how many LaTeX and AMSTeX commands are supported -- play around.
TeX substitution macros such as $\f{A}$, ({{{$\f{A}$}}}), may be inserted into the [[jsMathPlugin]] just before the jsMath.process call. See that plugin for abbreviated commands I've included.
<<tiddler HideTags>><html>
<table class="gtable">
<tr>
<td>
<img SRC="talks/Cate2010/Laser.jpg" width=300>
</td>
<td>
</td>
<td>
<table class="gtable">
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td>
<img SRC="talks/Cate2010/EM field.png" width=300>
</tr></td>
</table>
</td>
</tr>
</table>
</html>
[>img[images/person/Laurent Freidel.jpg]]Homepage: http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=Laurent_Freidel
*Location: $\Pi$
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Freidel_L/0/1/0/all/0/1
Often coauthors with: Artem Starodubtsev.
Selected work:
*[[Quantum gravity in terms of topological observables|papers/0501191.pdf]]
*[[Particles as Wilson lines of gravitational field]]
[>img[images/person/Leonardo Castellani.jpg]]Homepage: http://www.mfn.unipmn.it/~castella/
*Location: Turin
*Papers: http://www-library.desy.de/cgi-bin/spiface/find/hep/www?rawcmd=find++fa+castellani%2Cl+and+not+a+martinelli+or+a+aschieri+and+castellani&FORMAT=WWW&SEQUENCE=
Selected work:
*[[Group Geometric Methods in Supergravity and Superstring Theories|papers/Group Geometric Methods in Supergravity and Superstring Theories.pdf]]
**nice job on [[reductive Lie group geometry]]
**BRST by adding a (Grassmann?) piece to the Lie group geometry
*[[A Geometric Interpretation of BRST symmetry|papers/A Geometric Interpretation of BRST symmetry.pdf]]
**just the BRST part from paper above
**add generator, $Q$, and $\f{d \th} g^A T_A$.
**This reminds me of the geometry of Lagrange multipliers
*[[Gravity on Finite Groups|papers/9909028.pdf]]
**differential geometry over disconnected spaces
*lots of [[Kaluza-Klein]] stuff
A ''Lie algebra'', $\mathfrak{g}$, consists of elements of a $N$ dimensional [[vector space]], $B=B^A T_A \in {\rm Lie}(G) = \mathfrak{g}$, composed of real (or complex, for a complex Lie algebra) coefficients, $B^A$, multiplying $N$ ''Lie algebra generators'', $T_A$. The elements are closed under a ''Lie algebra bracket'', equal to the Lie algebra [[adjoint action|adjoint representation]] of one element on another,
$$
{\rm Ad}_A B = \lb A, B \rb = C
$$
and equivalent to the [[commutator]] relation. Working in some [[representation]], the Lie algebra brackets of the generators,
$$
\lb T_A , T_B \rb = T_A T_B - T_B T_A = C_{AB}{}^C T_C
$$
give the real or complex ''structure constants'' (//''structure coefficients''//), $C_{AB}{}^C = - C_{BA}{}^C$, for the Lie algebra. The generators may, for example, be [[Clifford basis elements]] ($T_A=\ga_A$), square matrices, or abstract operators. The bracket must satisfy the ''Jacobi identity'',
\begin{eqnarray}
0 &=& \lb A, \lb B , C \rb \rb + \lb B, \lb C , A \rb \rb + \lb C, \lb A , B \rb \rb \\
0 &=& C_{AD}{}^E C_{BC}{}^D + C_{BD}{}^E C_{CA}{}^D + C_{CD}{}^E C_{AB}{}^D
\end{eqnarray}
which resembles the product rule for derivatives,
$$
\lb A, \lb B , C \rb \rb = \lb \lb A, B \rb , C \rb + \lb B, \lb A , C \rb \rb
$$
and is identical to the [[Clifford Jacobi identity|Clifford product identities]].
Although the choice of generators is somewhat arbitrary, every Lie algebra has a specific [[Lie algebra structure]].
It has been seen that an $n$ dimensional [[Clifford algebra]] can be interpreted as a $2^n$ dimensional [[Lie algebra]], with the [[Clifford basis elements]] identified as generators, $\ga_{\al \dots \be} \sim T_A$, and the Lie bracket given by the [[Clifford basis identities]]. It is also the case that any $n$ dimensional Lie algebra may be identified as a subalgebra of the bivector algebra of an $n$ dimensional Clifford algebra. This representation may be made as:
$$
T_A = - \fr{1}{4} C_A{}^{BC} \ga_{BC}
$$
employing the Lie algebra structure constants, the Clifford bivector basis elements, and using the unit diagonal [[Killing form]], $g^{BD}=\et^{BD}$, as the Clifford algebra metric and to raise the structure constant indices. This representation faithfully gives
\begin{eqnarray}
\lb T_A, T_D \rb &=& \fr{1}{8} C_A{}^{BC} C_D{}^{EF} \ga_{BC} \times \ga_{EF} \\
&=& \ha C_A{}^{BC} C_{DC}{}^E \ga_{BE} \\
&=& - \fr{1}{4} C_{AD}{}^C C_C{}^{BE} \ga_{BE} \\
&=& C_{AD}{}^C T_C
\end{eqnarray}
via the [[Jacobi identity|Lie algebra]] and Clifford basis identities. These Lie algebra generators are orthogonal,
\begin{eqnarray}
\li T_A T_D \ri &=& \fr{1}{16} C_A{}^{BC} C_D{}^{EF} \li \ga_{BC} \ga_{EF} \ri \\
&=& \fr{1}{16} C_A{}^{BC} C_D{}^{EF} \lp \et_{BF} \et_{CE} - \et_{BE} \et_{CF} \rp \\
&=& \fr{1}{8} C_A{}^{BC} C_{DCB} \\
&=& \fr{1}{8} \et_{AD}
\end{eqnarray}
in the [[scalar part]] (trace) operator. Note that the $2^{[n/2]}$ dimensional [[Clifford matrix representation]] gives a corresponding matrix representation for the Lie algebra.
A ''Lie algebra automorphism'' is a map from a [[Lie algebra]] to itself, $\ph : \mathfrak{g} \mapsto \mathfrak{g}$, that preserves its structure,
$$
\lb \ph A, \ph \, B \rb = \ph \, C \s \forall \;\; A,B,C
$$
This will be the case iff the structure constants are invariant, $C_{AB}{}^C \to C'_{AB}{}^C = C_{AB}{}^C$. The automorphism can be written as a linear matrix operator, $\ph_B{}^C$, acting on either the basis generators or on the components,
$$
A' = \ph A = T_B \, \phi^B{}_C \, A^C = T_B A'^B = T'_C A^C
$$
with $A'^B = \phi^B{}_C \, A^C$ or $T'_C = T_B \, \ph^B{}_C$, but acting on the basis generators makes much more sense. The brackets of the transformed generators are
$$
C'_{AB}{}^C T'_C = \lb T'_A, T'_B \rb = \lb T_C \ph^C{}_A, T_D \ph^D{}_B \rb
= T_E C_{CD}{}^E \ph^C{}_A \ph^D{}_B
$$
with the transformation an automorphism iff
$$
C'_{AB}{}^C = C_{FD}{}^E \ph^F{}_A \ph^D{}_B \ph^-_E{}^C = C_{AB}{}^C
$$
A Lie algebra [[automorphism]] is ''inner'' if it corresponds to the action of a [[Lie group]] element, $A' = \ph A = g_\ph A \, g_\ph^-$ with $g_\ph \in G$.
A ''Lie algebra involution'' is a [[Lie algebra automorphism]] that squares to the identity, $\ph \, \ph = 1$. A Lie algebra involution splits a Lie algebra, $\mathfrak{g}$, into a [[subalgebra]], ${\mathfrak{g}}_+$, and a [[representation space]], ${\mathfrak{h}}_-$, which are [[eigen]]spaces of the involution, $\ph \; {\mathfrak{h}}_\pm = \pm {\mathfrak{h}}_\pm$. Working this through the Lie bracket, we see that elements of these ''involution pair''s must satisfy
\begin{eqnarray}
\left[ {\mathfrak{h}}_+ , {\mathfrak{h}}_+ \right] &=& {\mathfrak{h}}_+ \\
\left[ {\mathfrak{h}}_+ , {\mathfrak{h}}_- \right] &=& {\mathfrak{h}}_- \\
\left[ {\mathfrak{h}}_- , {\mathfrak{h}}_- \right] &=& {\mathfrak{h}}_+
\end{eqnarray}
An involution is a [[Cartan involution|Cartan decomposition]] iff the [[Killing form]] is negative definite for ${\mathfrak{h}}_+$ and positive definite for ${\mathfrak{h}}_-$, and they are orthogonal complements.
The structure of every [[simple]] [[Lie algebra]] may be elucidated by arranging the generators to span different sub-spaces. The ''rank'', $R$, of a $N$ dimensional Lie algebra is the maximal number if inter-commuting orthogonal generators, $T^\mathfrak{C}_I$,
$$
\big[ T^\mathfrak{C}_I, T^\mathfrak{C}_J \big] = 0 \;\;\;\; \forall \;\; 1 \le I,J \le R
$$
These generators span a $R$ dimensional [[vector space]], the ''Cartan subalgebra'', $\mathfrak{C}$, with elements, $C = C^J T^\mathfrak{C}_J \in \mathfrak{C}$. [[Exponentiating|exponentiation]] the Cartan subalgebra gives the ''maximal torus'' -- a $R$ dimensional [[submanifold]] of the [[Lie group manifold|Lie group geometry]]. Although called a torus, this can be a hyperbola, as some basis generators may have positive norm. Generally, a Cartan subalgebra has a [[Cartan decomposition]], $\mathfrak{C} = \mathfrak{k} \oplus \mathfrak{p}$, with compact and noncompact subspaces spanned by ''compact generators'', satisfying $\lp T^\mathfrak{k}_A, T^\mathfrak{k}_A \rp = -1$, and ''hyperbolic generators'', satisfying $\lp T^\mathfrak{p}_M, T^\mathfrak{p}_M \rp = +1$.
The adjoint action of Cartan subalgebra elements, ${\rm Ad}_C$, may be represented by a $(N \times N)$ matrix, $M$, acting on other Lie algebra elements, $B = B^A T_A$, as vectors,
$$
M \, B = [ C , B ] = {\rm Ad}_C \, B = C^I \big[ T^\mathfrak{C}_I , T_A \big] B^A = T_D \big( C^I C_{IA}{}^D \big) B^A = T_D M^D_{\p{D}A} B^A
$$
The components of $M$ may be written in terms of the structure constants as $M^D_{\p{D}A} = C^I C_{IA}{}^D$. Since $M(C)$ is linear in the $C^I$ it can also be written as $M = C^I M_I$, with the mutually-commuting $M_I$ matrices having components $(M_I)^D{}_A = C_{IA}{}^D$. The structure of the Lie algebra may be understood in terms of the complex [[eigen]]values and eigenvectors of $M$, or equivalently the mutual eigenvectors of the $M_I$ and corresponding eigenvalues. Geometrically, these eigenvalues describe how the eigenvector directions twist over the maximal torus.
Since $M \, C = 0$ for any $C$ in the Cartan subalgebra, $R$ of the eigenvalues of $M$ will be $0$, with this null eigenspace equal to $\mathfrak{C}$. The remaining $(N-R)$ eigenvectors of $M$ come in pairs, $V_{\pm j}$, with each corresponding to a unique complex eigenvalue,
$$
M(C) \, V_{\pm j} = \pm \al_j(C) V_{\pm j} \s \s
M_I \, V_{\pm j} = \lb T^\mathfrak{C}_I , V_{\pm j} \rb = \pm \al_{jI} V_{\pm j} = \pm \lp \al_{jI}^\mathbb{R} + i \, \al_{jI}^\mathbb{I} \rp V_{\pm j}
$$
(no sum over $j$). The $\pm \al_j$ are called positive or negative ''roots'', and the eigenvalues, $\pm \al_{jI}$, the ''root components'', with corresponding eigenvectors, $V_{\pm j}$, called the positive or negative ''root vector''s. The root components corresponding to compact generators, $T^\mathfrak{k}_A$, are imaginary, $\al_{jA} = i \, \al_{jA}^\mathbb{I} \in \mathbb{I}$, while the root components corresponding to hyperbolic generators, $T^\mathfrak{p}_M$, are real, $\al_{jM} = \al_{jM}^\mathbb{R} \in \mathbb{R}$. (For a generalized, infinite-dimensional Lie algebra this is not necessarily the case!)
Getting a bit fancier, the roots may be identified as elements, $\al_j \in \mathfrak{C}^*$, of the complex [[dual space]] to the Cartan subalgebra, $\al_j : \mathfrak{C} \to \mathbb{C}$. Written in terms of dual basis elements, $T_\mathfrak{C}^K = g^{KJ} T^\mathfrak{C}_J$, they are $\al_j = \al_{jK} T_\mathfrak{C}^K$, so
$$
\al_j(C) = \al_j \, C = \al_{jK} T_\mathfrak{C}^K \, C^I T^\mathfrak{C}_I = \al_{jK} C^K \in \mathbb{C}
$$
Via the dual to the [[Killing form]], the inner product on this space is naturally Euclidean (for a finite dimensional Lie group) and gives an ''inner product of roots'',
$$
\al_i \cdot \al_j = \, <\! \al_i, \al_j \!> \; = \lp \al_i, \al_j \rp = \al_{iJ} \al_{jK} g^{JK}
= \al_{iJ}^\mathbb{R} \al_{jK}^\mathbb{R} \de^{JK} + \al_{iJ}^\mathbb{I} \al_{jK}^\mathbb{I} \de^{JK}
$$
The collection of roots form a [[root system]] in $R$ dimensional ''root space'' and completely determine the Lie algebra. Because compact Lie groups are favored in physics, imaginary root components correspond to real elementary particle charges. Also, this inner product is positive definite for Lie algebras, but singular for an [[affine Lie algebra]] and indefinite for a [[Generalized Lie algebra|Borcherds algebra]].
At its ''core'', every real Lie algebra consists of interlinked [[su(2)]] and/or [[sl(2)]] subalgebras. To elucidate this structure, we can define complex Cartan subalgebra elements dual to the roots,
$$
h_i = \al_{iJ} g^{JK} T^\mathfrak{C}_K = \al^\mathbb{R}_{iM} \de^{MN} T^\mathfrak{p}_N + i \, \al^\mathbb{I}_{iA} \de^{AB} T^\mathfrak{k}_B
$$
satisfying
$$
\al_j (h_i) = \;<\! \al_j, \al_i \!>\; = \lp h_j, h_i \rp
$$
If we normalize our root vectors, $\lp e_{+i}, e_{-i} \rp = 1$, a complete set of nonvanishing Lie brackets in the ''Cartan-Weyl basis'', $\{ T^\mathfrak{k}_A, T^\mathfrak{p
}_M, e_{\pm i} \}$, is
\begin{eqnarray}
\lb T^\mathfrak{k}_A , e_{\pm j} \rb &=& \pm \, i \, \al_{jA}^\mathbb{I} \, e_{\pm j} \\
\lb T^\mathfrak{p}_M , e_{\pm j} \rb &=& \pm \, \al_{jM}^\mathbb{R} \, e_{\pm j} \\
\lb e_{+i} , e_{-i} \rb &=& h_i = \al^\mathbb{R}_{iM} \de^{MN} T^\mathfrak{p}_N + i \, \al^\mathbb{I}_{iA} \de^{AB} T^\mathfrak{k}_B \\
\lb e_{\pm i} , e_{\pm j} \rb &=& n_{\pm i,\pm j} \, e_{\pm k} \\
\end{eqnarray}
In the last bracket above the only non-vanishing brackets have roots that add to give another root, $\pm \al_i \pm \al_j = \pm \al_k$, and the normalization coefficients, $n_{\pm i, \pm j}$, must be determined consistently. For a non-split Lie algebra, the Cartan-Weyl basis results in complex structure constants which can be brought into real form by transforming to a [[real Cartan-Weyl basis]].
The $h_i$ are a complex, non-orthogonal set of Cartan subalgebra generators. The $h_a$, corresponding to simple roots, span the Cartan subalgebra, and can be used in the ''Chevalley-Cartan basis'', $\{ h_a, e_{\pm i} \}$, with non-vanishing brackets
$$
\lb h_a , e_{\pm j} \rb = \pm <\! \al_a, \al_j \!> e_{\pm j} \s \s
\lb e_{+i} , e_{-i} \rb = \al_i{}^a h_a
$$
as well as the brackets between the non-conjugate $e_{\pm i}$. They $h_a$ are similar in function to the gravitational [[frame]] that determines a [[metric]].
If we define ''co-roots'', $\al^\vee = \fr{2}{< \al, \al >} \al$, we can use these to define corresponding complex Cartan elements, $H_i = \fr{2}{< \al_i, \al_i>} h_i$, and re-scale our root vectors, $E_{\pm i} = \sqrt{\fr{2}{< \al_i, \al_i >}} e_{\pm i}$, with the resulting ''Chevalley basis'', $\{ H_a, E_{\pm i} \}$, having brackets
\begin{eqnarray}
\big[ H_a , H_b \big] &=& 0 \\
\big[ H_a , E_{\pm i} \big] &=& \pm \al_i{}^b A_{ab} E_{\pm i} \\
\big[ E_{+i} , E_{-i} \big] &=& H_i = \al_i{}^a \fr{<\! \al_a, \al_a \!>}{<\! \al_i, \al_i \!>} H_a \\
\big[ E_{\pm i} , E_{\pm j} \big] &=& N_{\pm i, \pm j} \, E_{\pm k} \\
Ad_{(E_{\pm a})^{1-A_{ab}}} \lp E_{\pm b} \rp &=& 0
\end{eqnarray}
in which $A_{ab} = 2 \fr{< \al_a , \al_b >}{< \al_a , \al_a >}$ is the [[Cartan matrix|root system]] and $\al_i = \al_i{}^a \al_a$ in terms of simple roots. The ''normalization coefficients'', $N_{\pm i, \pm j}$, are zero if $\pm \al_i \pm \al_j = \pm \al_k$ is not a root and otherwise are $\pm (p+1)$, with $p$ equal to the largest integer such that $\pm \al_j \mp p \al_i$ is a root. The normalization coefficient signs are determined via an [[asymmetry function]], and we have $N_{+i, +j} = N_{-i, -j} = -N_{+j, +i}$ and $N_{+i, -j} = N_{-i, +j} = -N_{-j, +i}$. If the root components in the Chevalley brackets are all real, the Lie algebra is a split real form, containing $R$ distinct [[sl(2)]] [[subalgebra]]s spanned by the $\{ H_a, E_{\pm a} \}$.
A ''real Chevalley basis'' for a compact real form, containing $R$ distinct [[su(2)]] subalgebras, is obtained by defining the real and imaginary parts of the root vectors, $E^\mathbb{R}_j =
\ha (E_{+j} + E_{-j})$ and $E^\mathbb{I}_j = \fr{i}{2} (E_{+j} - E_{-j})$, as well as $H'_a = i H_a$, with brackets
\begin{eqnarray}
\big[ H'_a , E^{\mathbb{R}/\mathbb{I}}_j \big] &=& \pm \al^{\mathbb{I}}_j{}^b A_{ab} E^{\mathbb{I}/\mathbb{R}}_j \\
\big[ E^{\mathbb{R}}_j , E^{\mathbb{I}}_j \big] &=& - \ha \, H'_j \\
\big[ E^{\mathbb{R}}_i , E^{\mathbb{R}}_j \big] &=& \ha N_{+i, +j} \, E^{\mathbb{R}}_k + \ha N_{+i, -j} \, E^{\mathbb{R}}_{k'} \\
\big[ E^{\mathbb{R}}_i , E^{\mathbb{I}}_j \big] &=& \ha N_{+i, +j} \, E^{\mathbb{I}}_k - \ha N_{+i, -j} \, E^{\mathbb{I}}_{k'} \\
\big[ E^{\mathbb{I}}_i , E^{\mathbb{I}}_j \big] &=& -\ha N_{+i, +j} \, E^{\mathbb{R}}_k + \ha N_{+i, -j} \, E^{\mathbb{R}}_{k'} \\
\end{eqnarray}
This has an evident [[complex structure|structures on a real representation space]], $J^C E^{\mathbb{R}/\mathbb{I}}_i = \pm E^{\mathbb{I}/\mathbb{R}}_i$, compatible with the Cartan subalgebra, $[ H'_a, J^C E^{\mathbb{R}/\mathbb{I}}_b ] = J^C [ H'_a, E^{\mathbb{R}/\mathbb{I}}_b ]$, so the root vectors are also $J_C$ eigenvectors, $J^C E_{\pm j} = \pm i E_{\pm j}$. All real Lie algebras have, at their core, a set of $R$ stitched together [[sl(2)]]'s and [[su(2)]]'s, with the corresponding [[structures on a real representation space]]. A good example of a real Lie algebra with mixed compact and non-compact structure is [[spin(1,3)]].
link from [[Maurer-Cartan form]]
ref:
http://en.wikipedia.org/wiki/Lie_algebroid
[[Differential Operators and Actions of Lie Algebroids|papers/0209337.pdf]]
The ''Lie derivative'' is the rate of change of any field perceived by an observer as she moves along a path with some [[velocity|tangent vector]], $\ve{v}$. Basically, the field where she is going is pulled back and compared with the field where she's at. This description is extended to give the Lie derivative, ${\cal L}_{\ve{v}}$, with respect to a [[flow]], $\ph_t$, giving the rate of change of any field perceived by observers at every manifold point as they move according to the velocity field, $\ve{v}(x)$. The parameterized flow is given to first order in $t$ by
$$
\ph_t^i(x) \simeq x^i + t v^i(x)
$$
The Lie derivative of any field, $X$, is a [[natural]] operator defined as
$$
{\cal L}_{\ve{v}} X = \fr{d}{d t} \ph_t^* X = \fr{d}{d t} X(t) = \lim_{t \to 0} \fr{\ph_t^*X - X}{t} = \lim_{t \to 0} \fr{ \lp \ph_{-t} \rp_* X - X}{t}
$$
in which
$$
\ph_t^*X = \ph_t^*\lb X \rl_{\ph_t} \simeq \ph_t^* \lp \lb X \rl_x + t v^i \pa_i \lb X \rl_x \rp
$$
is the [[pullback]] of the field from where the flow is going back to the initial points, expanded to first order in $t$.
If the field is a [[function]] over the manifold, the Lie derivative of this field is the same as the [[directional derivative|tangent vector]] of this function with respect to the velocity field at every point,
$$
{\cal L}_{\ve{v}} f = \fr{d}{d t} f(x) = \lim_{t \to 0} \fr{\ph_t^*f - f}{t} = v^i \pa_i f = \ve{v} \f{d} f
$$
If the field is a [[1-form|cotangent bundle]] field, $\f{f} = \f{dx^i} f_i(x)$, the pullback along the flow is
$$
\ph_t^*\f{f} = \f{dx^i} \lb \fr{\pa \ph_t^j}{\pa x^i} \rb f_j(\ph_t)
\simeq \f{dx^i} \lb \de^j_i + t \pa_i v^j \rb \lb f_j(x) + t v^k \pa_k f_j(x) \rb
\simeq \f{dx^i} \lp f_i + t v^k \pa_k f_i + t f_j \pa_i v^j \rp
$$
and the Lie derivative is thus
\begin{eqnarray}
{\cal L}_{\ve{v}} \f{f} &=& \lim_{t \to 0} \fr{\ph_t^*\f{f} - \f{f}}{t}
= \f{dx^i} \lp v^k \pa_k f_i + f_j \pa_i v^j \rp \\
&=& \lp \ve{v} \f{\pa} \rp \f{f} + \lp \f{\pa} \ve{v} \rp \f{f} \\
&=& \ve{v} \lp \f{d} \f{f} \rp + \f{d} \lp \ve{v} \f{f} \rp
\end{eqnarray}
using the [[exterior derivative]], [[partial derivative]], and [[vector-form algebra]]. For any differential form or [[Clifform]] field with form grade greater than zero this generalizes to give ''Cartan's formula'' for the Lie derivative,
\begin{eqnarray}
{\cal L}_{\ve{v}} \, \nf{F}
&=& \lp \ve{v} \f{\pa} \rp \nf{F} + \lp \f{\pa} \ve{v} \rp \nf{F} \\
&=& \ve{v} \lp \f{d} \nf{F} \rp + \f{d} \lp \ve{v} \nf{F} \rp
\end{eqnarray}
and another nice formula (easier for computations) obtained via the power of vector-form algebra and the partial derivative operator.
If the field is a [[vector|tangent bundle]] field, $\ve{u}(x)$, the pushforward along the negative flow of the velocity at where the flow goes is
$$
\lp \ph_{-t} \rp_* \ve{u} = u^j(\ph_t) \lb \fr{\pa \ph_{-t}^i}{\pa x^j} \rb \ve{\pa_i}
\simeq \lb u^j + t v^k \pa_k u^j \rb \lb \de^i_j - t \pa_j v^i \rb \ve{\pa_i}
\simeq \lp u^j + t v^k \pa_k u^i - t u^j \pa_j v^i \rp \ve{\pa_i}
$$
and the the Lie derivative of a vector field is thus
$$
{\cal L}_{\ve{v}} \ve{u} = \lim_{t \to 0} \fr{ \lp \ph_{-t} \rp_* \ve{u} - \ve{u}}{t}
= \lp v^k \pa_k u^i - u^j \pa_j v^i \rp \ve{\pa_i}
= \ve{v} \f{\pa} \ve{u} - \ve{u} \f{\pa} \ve{v}
$$
The Lie derivative of one velocity field with respect to another defines the ''Lie bracket'',
$$
\lb \ve{v}, \ve{u} \rb_L = {\cal L}_{\ve{v}} \ve{u} = - \lb \ve{u}, \ve{v} \rb_L
$$
The Lie derivative has a number of nice [[properties|Lie derivative identities]].
The [[Lie derivative]] is a natural, fundamental derivative operator on any geometric field on a manifold.
By virtue of Cartan's formula, it commutes with the [[exterior derivative]] operator when acting on [[function]]s, [[differential form]]s or [[Clifform]]s,
$$
{\cal L}_{\ve{v}} \f{d} \nf{F} = \f{d} \lp \ve{v} \lp \f{d} \nf{F} \rp \rp = \f{d} {\cal L}_{\ve{v}} \nf{F}
$$
but not always when acting on vector fields.
It is linear in both the velocity field,
$$
{\cal L}_{\ve{v} + \ve{u}} X = {\cal L}_{\ve{v}} X + {\cal L}_{\ve{u}} X
$$
and argument,
$$
{\cal L}_{\ve{v}} \lp X + Y \rp = {\cal L}_{\ve{v}} X + {\cal L}_{\ve{v}} Y
$$
The Lie derivative is a grade $0$ [[derivation]], acting on various products of fields via the Liebniz rule,
\begin{eqnarray}
{\cal L}_{\ve{v}} \lp a \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} a \rp \nf{F} + a \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
{\cal L}_{\ve{v}} \lp \f{a} \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} \f{a} \rp \nf{F} + \f{a} \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
{\cal L}_{\ve{v}} \lp \ve{a} \nf{F} \rp &=& \lp {\cal L}_{\ve{v}} \ve{a} \rp \nf{F} + \ve{a} \lp {\cal L}_{\ve{v}} \nf{F} \rp \\
\end{eqnarray}
Scaling the velocity field by a function results in
\begin{eqnarray}
{\cal L}_{f \ve{v}} \nf{F} &=& f {\cal L}_{\ve{v}} \nf{F} + \lp \f{d} f \rp \lp \ve{v} \nf{F} \rp \\
{\cal L}_{f \ve{v}} \ve{u} &=& f {\cal L}_{\ve{v}} \ve{u} - \lp \ve{u} \f{d} f \rp \ve{v}
\end{eqnarray}
The [[commutator]] of Lie derivatives with respect to two velocity fields acting on anything is equal to the Lie derivative with respect to the Lie bracket of the two velocity fields,
$$
\lb {\cal L}_{\ve{v}}, {\cal L}_{\ve{u}} \rb = {\cal L}_{\ve{v}} {\cal L}_{\ve{u}} - {\cal L}_{\ve{u}} {\cal L}_{\ve{v}} = {\cal L}_{\lb \ve{v}, \ve{u} \rb_L}
$$
This gives the [[Jacobi identity|Lie algebra]] for the Lie bracket when acting on a vector field,
$$
\lb \lb \ve{v}, \ve{u} \rb_L, \ve{w} \rb_L = - \lb \lb \ve{u}, \ve{w} \rb_L, \ve{v} \rb_L + \lb \lb \ve{v}, \ve{w} \rb_L, \ve{u} \rb_L
$$
The Lie derivative of [[fiber bundle]] basis elements (other than tangent or cotangent bundle basis), such as [[Clifford basis elements]], is zero,
$$
{\cal L}_{\ve{v}} \ga_\al = 0
$$
The definition of the Lie derivative in terms of the flow allows the flow to be written as the [[exponentiation]] of the Lie derivative,
$$
X(t) = \ph_t^* X = e^{t {\cal L}_{\ve{v}}} X
$$
For a surface, $\Si_t$, carried along with the flow, the time derivative of an integral over that surface is
$$
\fr{d}{d t} \int_{\Si_t} \nf{F} = \int_{\Si_t} {\cal L}_{\ve{v}} \nf{F}
$$
which combines consistently with [[Stoke's theorem|integration]],
$$
\fr{d}{d t} \int_{\pa \Si_t} \nf{F}
= \fr{d}{d t} \int_{\Si_t} \f{d} \nf{F}
= \int_{\Si_t} {\cal L}_{\ve{v}} \f{d} \nf{F}
= \int_{\pa \Si_t} {\cal L}_{\ve{v}} \nf{F}
$$
An $n$ dimensional ''Lie group'' is a [[group]] of infinitely many elements, $g(x) \in G$, parametrized by $n$ real (or complex, for a complex Lie group) parameters, $x \in \Re^n$. A Lie group is also a [[manifold]], with points, $x$, corresponding to the parameters (in various patches).
Near the identity, $1=g(0)$, group elements may be described in terms of coordinates multiplying the [[Lie algebra]] generators, $T_A \in \mathfrak{g} = {\rm Lie}(G)$, associated with the group,
$$
g(x) \simeq 1 + x^A T_A
$$
The Lie algebra completely describes the local geometry of the group. And Lie algebra elements may be acted on by Lie group elements, in the [[adjoint representation]]. For a ''connected'' manifold and Lie group, all manifold points (and corresponding group elements) may be connected to the identity by smooth paths. [[Exponentiation|exponentiation]] of Lie algebra generators gives the ''universal cover'' of the corresponding connected Lie group,
$$
g(x) = e^{x^A T_A} \in G
$$
Points of a connected Lie group manifold may be described by the $x^A$ coordinates, with global group structure (manifold topology) determined by the ranges and matchings of $x^A$. A connected manifold, and Lie group, is ''simply connected'' if all paths are contractible. For example, a sphere is simply connected while a torus is not. The universal cover of a connected Lie group is simply connected.
The [[Lie group geometry]] is the Lie group manifold with geometry and symmetries corresponding to the action of the Lie algebra generators as vector fields. (Lie group geometry may alternatively be described as a [[Lie group bundle]], with the base manifold taken to be the Lie group manifold, the Lie group as typical fiber, and a special "identity" section, $g_I(x)$, defined.)
A [[Lie group]] is an $n$ dimensional group, $G$, with elements, $g$, that can be identified with points, $x$, on an $n$ dimensional manifold, $M$. Typically the group elements, $g(x) \in G$, are understood to be written as a function of parameters; however, it is possible to consider this identification as a bijective section, the ''identity section'', $g_I(x)$, of a [[fiber bundle]]. This bundle clearly has $M$ as base and $G$ as typical fiber. But what is the structure group and action? It is desirable to preserve the group structure of the Lie group fiber: if any three fiber/group elements satisfy $g_1 g_2 = g_3$ the corresponding elements, after transformation by an element of the structure group, should satisfy $g'_1 g'_2 = g'_3$. The structure group and action is thus identified as the [[automorphism]] group and action for $G$, and the ''Lie group bundle'' is defined as the [[automorphism bundle]] for an $n$ dimensional $G$ over an $n$ dimensional base, along with an identity section.
Since the Lie group bundle comes with a special identity section, $g_I(x)$, it has a particular automorphism bundle connection, the ''Lie group connection'', $\f{A}$, similar to the [[Maurer-Cartan connection]], such that the identity section is horizontal,
$$
0 = \f{\na} g_I = \f{d} g_I + \ha \f{A} g_I - \ha g_I \f{A}
$$
This Lie group connection can be calculated in a few steps. Presuming the matrix of connection coefficients has an [[inverse|matrix inverse]], $A_B{}^i$, this set of vectors is identified as the adjoint action vectors of the [[Lie group geometry]],
$$
\ve{A_B} = \ve{\xi^A_B} = \ha \ve{\xi^L_B} - \ha \ve{\xi^R_B}
$$
in which the left and right action vector field matrices may be calculated via the defining equations for their inverses, $\f{\xi_L^B} T_B = \lp \f{d} g_I \rp g_I^-$ and $\f{\xi_R^B} T_B = g_I^- \lp \f{d} g_I \rp$. The curvature of the Lie group connection vanishes. //(check that)// The identity section transforms under a gauge transformation to $g'_I = h g_I h^-$.
//Will outer automorphisms make things more complicated?//
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/torifibers.png" height="400">
</td></tr>
<tr><td>
Maximal tori of <SPAN class="math">SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes Spin(1,3)</SPAN> over spacetime.
</td></tr>
</table>
</center></html>
[<img[images/png/torsor.png]]A [[Lie group]] has elements $g(x) \in G$ specified by the points, $x$ (corresponding to coordinates, $x^i$), of an $n$ dimensional [[manifold]]. This manifold naturally acquires a geometry corresponding to the structure of the Lie group.
The [[group action|group]] -- left, right, or adjoint -- of any group element, $h$, on all other elements of the group provides a set of [[autodiffeomorphism|diffeomorphism]]s, $ \left\{ \ph^L_h,\ph^R_h,\ph^A_h \right\} $, on the group manifold. However, only the adjoint action is a group [[automorphism]] as well as a manifold autodiffeomorphism. A group element, $h$, near the identity,
$$h \simeq 1 + t v^B T_B$$
is specified by a [[Lie algebra]] vector, $v=v^B T_B \in Lie \, G$, and a small parameter, $t$. These parameterized actions on the group correspond to [[flow]]s,
$$
e^{t \ve{\xi} \f{d}}
$$
on the group manifold, with each flow corresponding to a vector field generator, $\ve{\xi}$, over the manifold. For example, for the right action,
\begin{eqnarray}
R_h g &=& g h = g(\ph^R_h(x)) \\
&\simeq& g + t v^B g T_B = g + t v^B \lp \xi^R_B \rp^i \pa_i g(x)
\end{eqnarray}
So, for each group action, $\left\{ L, R, A \right\}$, and each Lie algebra generator, $T_B$, there is a corresponding vector field,
\begin{eqnarray}
T_B g &=& {\cal L}_{\ve{\xi^L}} g = \ve{\xi^L_B} \f{d} g = \lp \xi^L_B \rp^i \pa_i g(x) \\
g T_B &=& {\cal L}_{\ve{\xi^R}} g = \ve{\xi^R_B} \f{d} g = \lp \xi^R_B \rp^i \pa_i g(x) \\
\ha T_B g - \ha g T_B &=& {\cal L}_{\ve{\xi^A}} g = \ve{\xi^A_B} \f{d} g = \lp \xi^A_B \rp^i \pa_i g(x)
\end{eqnarray}
which acts via the [[Lie derivative]] and corresponds to a flow on the group manifold. The action of the Lie algebra generators and the flow by the corresponding right action vector field generators may be conceptually identified as the same, $T_B \sim \ve{\xi^R_B}$. When working with a specific group representation and coordinatization, the vector field component matrices, $\lp \xi_B \rp^i$, may be found explicitly by solving the above defining equations. Using the defining equations, the [[Lie bracket|Lie derivative]]s between vector field generators are:
\begin{eqnarray}
\lb \ve{\xi^R_B}, \ve{\xi^R_C} \rb_L &=& C_{BC}{}^D \ve{\xi^R_D} \\
\lb \ve{\xi^L_B}, \ve{\xi^L_C} \rb_L &=& - C_{BC}{}^D \ve{\xi^L_D} \\
\lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L &=& 0
\end{eqnarray}
with the same structure constants as from the Lie algebra brackets. Also tiddler that
$$\ve{\xi^A_B} = \ha \ve{\xi^L_B} - \ha \ve{\xi^R_B}$$
allows the Lie brackets of adjoint action vector fields with the others to be easily determined. To make things more confusing, the ''left action vector fields'', $\ve{\xi^L_B}$, are ''[[right invariant]] vector fields'', while the ''right action vector fields'', $\ve{\xi^R_B}$, are ''[[left invariant]] vector fields'',
\begin{eqnarray}
R_{h*} \ve{\xi^L_B}(x) &=& \ve{\xi^L_B}(\Phi^R_h(x)) \\
L_{h*} \ve{\xi^R_B}(x) &=& \ve{\xi^R_B}(\Phi^L_h(x))
\end{eqnarray}
These vector fields have [[1-form]] duals, the ''right invariant 1-forms'', $\f{\xi_L^B}=\f{dx^i} \lp \xi^L_i \rp^B$, satisfying $\ve{\xi^L_A} \f{\xi_L^B} = \de_A^B$ and
$$
\f{\xi_L^B} T_B = \lp \f{d} g \rp g^-
$$
and the ''left invariant 1-forms'', $\f{\xi_R^B} = \f{dx^i} \lp \xi^R_i \rp^B$, satisfying $\ve{\xi^R_A} \f{\xi_R^B} = \de_A^B$ and
$$
\f{\xi_R^B} T_B = g^- \f{d} g = \f{\cal I}
$$
in which $\f{\cal I}$ is the [[Maurer-Cartan form]] over the Lie group manifold. (Note that the $L$ and $R$ are just labels that can move around, and not [[indices]] to be summed over.)
A [[geodesic]] in a Lie group is found by [[exponentiation]] of the flow from a point, resulting in a parameterized [[path]] with [[tangent vector]] $\vec{\xi}$ satisfying the geodesic equation, $0 = (\vec{\xi} \f{\na}) \vec{\xi}$, which for a Lie group is
$$
\pa_t \vec{\xi} = ad^*_{\vec{\xi}} \vec{\xi}
$$
in which the ''coadjoint operator'' is defined as
$$
<ad^*_{\vec{\xi}} \vec{u}, \vec{w}> = <\vec{u}, ad_{\vec{\xi}} \vec{w}> = <\vec{u}, [\vec{\xi},\vec{w}]>
$$
producing the local coordinate expression
$$
(ad^*_{\vec{\xi}} \vec{u})^A = u^E \xi^B C_{BC}{}^D g_{ED} g^{CA}
$$
The left invariant vector fields, along with their natural [[Lie algebra metric|Lie algebra]], provide a natural [[Lie group tangent bundle geometry]], including a frame, connection, and curvature. Note that on the ''Lie group geometry'', a manifold with a collection of special vector fields on it, all points are identical since the Lie brackets are the same between the vector fields at every point. In particular, there is no special ''identity point'' on the Lie group geometry -- a Lie group geometry is also called a //''torsor''//, //''G-torsor''//, or //''principal homogeneous space''//.
<<tiddler HideTags>>[>img[talks/CSUF09/images/group.png]]The $N$ generators, $\ve{T_A}$, are orthogonal vector fields on the $N$ dimensional Lie group manifold, $G$.
$$
\lb \ve{T_A} , \ve{T_B} \rb =
{\cal L}_{\ve{T_A}} \ve{T_B} = C_{AB}^{\p{AB}C} \ve{T_C}
$$
Following a generator around $G$ from any point, the resulting path is a circle.
[>img[talks/CSUF09/images/spiral.png]]Eigen-generators (root vectors) twist integrally around this circle.
$$
{\cal L}_{\ve{T_3}} \ve{T_+} = i \al_{3+} \ve{T_+}
$$
[>img[talks/CSUF09/images/torus.png]]Following all the commuting generators in the Cartan subalgebra produces a $R$ dimensional ''maximal torus'' in the Lie group. The $R$ twist numbers of weight vectors around each orthogonal circle in the maximal torus are the coordinates of that weight.
Weight diagrams specify how generators are twisting around each other.
A [[metric]] for the [[tangent bundle]] over the [[Lie group]] manifold may be defined such that the [[left invariant]] vector fields of a [[Lie group geometry]] have the same scalar product as the [[Lie algebra]] generators using the [[Killing form]]:
$$
\lp \ve{\xi^R_B}, \ve{\xi^R_C} \rp = \lp T_B, T_C \rp = g_{BC} = C_{BD}{}^E C_{CE}{}^D
$$
The Lie algebra metric, $g_{BC}$, may be made diagonal (like the [[Minkowski metric]] and Kronecker delta) by transforming the generators by a constant matrix (as long as $G$ is semi-[[simple]]), found via the methods of [[spectral decomposition|eigen]]. In this way, the left invariant vector fields are identified as the set of [[orthonormal basis vector fields|frame]] on the Lie group manifold, $\ve{e_B} = \ve{\xi^R_B}$, and the left invariant 1-form coefficients,
\begin{eqnarray}
\lp e_i \rp^B &=& \lp \xi^R_i \rp^B = {\cal I}_i{}^B \\
\f{e^B} &=& \f{\xi_R^B} = \f{ {\cal I}^B }
\end{eqnarray}
are the coefficients of the frame 1-form and the [[Maurer-Cartan form]], $\f{\cal I} = g^- \f{d} g$. The resulting metric on the manifold is
$$
g_{ij} = \lp e_i \rp^B \lp e_j \rp^C g_{BC}
$$
The right invariant vector fields, $\ve{\xi_B} = \ve{\xi^L_B}$, are [[Killing vector]]s for this Lie group geometry, since
$$
{\cal L}_{\ve{\xi_B}} \ve{e_C} = \lb \ve{\xi^L_B}, \ve{\xi^R_C} \rb_L = 0
$$
And, since
$$
{\cal L}_{\ve{e_B}} \ve{e_C} = \lb \ve{\xi^R_B}, \ve{\xi^R_C} \rb_L = C_{BC}{}^A \ve{e_A}
$$
the left invariant vector fields (the orthonormal basis vectors) are also Killing, by the [[Killing form]] identity, $\lp B_B \rp_C{}^A = C_{BC}{}^A = -C_B{}^A{}_C$.
The [[torsion]]less [[tangent bundle connection]], $\f{w}{}^A{}_B$, for the Lie group manifold may be found by solving [[Cartan's equation]],
$$
0 = \f{d} \f{e^C} + \f{w}^C{}_B \f{e^B}
$$
We can cheat a little by seeing that, since $\f{e^B}=\f{{\cal I}^B}$, this is the same as the Maurer-Cartan equation,
$$
0 = \f{d} \f{{\cal I}^C} + \ha \f{{\cal I}^A} \f{{\cal I}^B} C_{AB}{}^C
$$
giving the ''Lie group tangent bundle connection'',
$$
\f{w}^C{}_B = \ha \f{e^A} C_{AB}{}^C = - \ha \f{e^A} C_A{}^C{}_B
$$
with the indices of the structure constants raised and lowered by the diagonal matrices, $g^{AB}$ and $g_{AB}$, and using the Killing form identity. The connection coefficients directly relate to the structure constants, $w_{AC}{}^B = - \ha C_{AC}{}^B$. Alternatively, a connection with torsion could be defined if desired.
Using one of the [[Killing vector identities]], the covariant derivative of any of the right invariant vector fields or some combination, $\ve{\xi} = \xi^B \ve{\xi^L_B}$, along itself vanishes
$$
\ve{\xi} \f{\na} \ve{\xi} = \ve{\xi} \f{e^A} \ve{e_C} \lp \xi^B \lp B_B \rp_A{}^C - \xi^B w_{B A}{}^C \rp
= \fr{3}{2} \xi^A \xi^B \ve{e_C} C_{BA}{}^C
= 0
$$
This implies the integral curves of the [[flow]]s along any right invariant field are [[geodesic]]s. By another Killing vector identity, all the right invariant vector fields are constant length.
The [[Riemann curvature]] for the Lie group tangent bundle connection is
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& - \ha \lp \f{d} \f{e^C} \rp C^A{}_B{}_C + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \ha \lp \f{w}^C{}_D \f{e^D} \rp C^A{}_B{}_C + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \fr{1}{4} \f{e^F} \f{e^D} \lp - C^C{}_{DF} C^A{}_{BC} + C^A{}_{CF} C^C{}_{BD} \rp \\
&=& - \fr{1}{4} \f{e^F} \f{e^D} C_{BCF} C_D{}^{AC}
\end{eqnarray}
using the Jacobi identity. The [[Ricci curvature]] is
$$
\f{R}{}_B = \ve{e_A} \ff{R}^A{}_B = - \fr{1}{4} \ve{e_A} \f{e^F} \f{e^D} C_{BCF} C_D{}^{AC}
= - \fr{1}{4} \f{e^D} C_{BCA} C_D{}^{AC} = - \fr{1}{4} \f{e^D} g_{BD} = - \fr{1}{4} \f{e}{}_B
$$
showing that a Lie group geometry is an [[Einstein space|Einstein's equation]]. The [[curvature scalar]] is $R = \ve{e^B} \f{R}{}_B = - \fr{1}{4} n$.
The [[volume form]] over the Lie group manifold is the ''Haar measure'',
$$
\f{e^1} \dots \f{e^n} = \nf{d^n x} \left| e \right|
$$
The complete list of real, [[simple]], compact, connected [[Lie group]]s was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups. Sorted by rank, $r$, they are:
| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | even [[special orthogonal group]] |
| $2$ | $G_2$ | | $14$ | [[G2]] |
| $4$ | $F_4$ | | $52$ | [[F4]] |
| $6$ | $E_6$ | | $78$ | [[E6]] |
| $7$ | $E_7$ | | $133$ | E7 |
| $8$ | $E_8$ | | $248$ | [[E8]] |
<<tiddler HideTags>>The complete list of real, [[simple]], compact, connected [[Lie group]]s was completed around 1890. They fall into four infinite families (the classical Lie groups) and five exceptional groups. Sorted by rank, $r$, they are:
| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $Spin(2r+1)$ | $r(2r+1)$ | odd spin group |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $Spin(2r)$ | $r(2r-1)$ | even spin group |
| $2$ | $G_2$ | | $14$ | [[G2]] |
| $4$ | $F_4$ | | $52$ | [[F4]] |
| $6$ | $E_6$ | | $78$ | [[E6]] |
| $7$ | $E_7$ | | $133$ | E7 |
| $8$ | $E_8$ | | $248$ | [[E8]] |
A ''Lieform'' is a ''[[Lie algebra]] valued [[differential form]]'', having a single form grade, $p$. In terms of [[coordinate basis forms]] and Lie algebra generators, an arbitrary Lieform may be written as
$$
\nf{A} = \f{dx^i} \dots \f{dx^k} \fr{1}{p!} A_{i \dots k}{}^B T_B \in \nf{\mathfrak{g}}
$$
The basis forms and Lie algebra generators act in different algebras. By convention, the form basis elements will be collected on the left and the Lie algebra generators on the right. The most common type of Lie form is a ''Lie algebra valued 1-form'', $\f{A} = \f{dx^i} A_i{}^B T_B \in \f{\mathfrak{g}}$.
The most common operation between Lieforms is the graded [[commutator]], equivalent to the graded [[Lie algebra bracket|Lie algebra]],
$$
\lb \nf{A}, \nf{B} \rb = \nf{A^C} \nf{B^D} \lb T_C, T_D \rb = \nf{A^C} \nf{B^D} C_{CD}{}^E T_E = \nf{A} \nf{B} - \lp -1 \rp^{pq} \nf{B} \nf{A}
$$
which produces a grade $(pq)$ Lieform from the bracket of grade $p$ and $q$ Lieforms.
Link to notes, such as [[Horizontal Rule]].
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}}}
Link to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,
without the messiness of the full URL appearing.
{{{
Link to [[external sites|http://www.osmosoft.com]] or [[ordinary notes|Horizontal Rule]] with ordinary words,
without the messiness of the full URL appearing.
}}}
Or just type out http://www.osmosoft.com and it will be automatically linkified.
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This is a revision of the listTags plugin — [[I|Garrett Lisi]] mashed this up with Udo's big plugin.
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[[Loops '07|http://www.matmor.unam.mx/eventos/loops07/]] was held in Morelia, Mexico. My [[talk for Loops 07]] has links to my slides and the accompanying audio.
Here are some talks I went to, and some personal impressions (these are just brief tiddlers to myself, please don't take them too seriously)
*Monday
**[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]]${}^*$, The causaloid formalism: a tentative framework for quantum gravity
***Compression of measurement data
***Obtain probabilistic distribution over temporal orderings of measurements maybe?
**[[Rafael Sorkin|http://www.phy.syr.edu/~sorkin/]], Quantum reality and anhomomorphic logic
***Wants to use a "quantum" version of probability by discarding preclusion or inference rules
****discarding logical "and" (multiplication) and/or discarding logical "or" (addition)
***Talked to him about complex probability distributions over paths.
**[[John F. Donoghue|http://www.fqxi.org/aw-donoghue.html]]${}^*$, Effective field theory and quantum general relativity
***He argued that the Effective Field Theory of gravity could be used to perturb around a Newtonian potential to get the classical (GR) corrections and the quantum GR correction proportional to $h$ -- in two different ways.
***Quantum GR will have to reproduce this.
**[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin.html]]${}^*$, Relational physics with real rods and clocks, (A)
***Works with Jorge Pullin
***Err, it was hard to understand what he was saying, and his talk was kind of scattered.
**[[Johannes Tambornino|http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=4735]], Taming observables in GR: A perturbative approach, (B)
***Tries to quantize symmetry reduced theory the right way.
**[[Frederic P. Schuller|http://www.nucleares.unam.mx/~f.p.schuller/]], Area metric gravity, (B)
***Reformulate everything in terms of $G_{[ab][cd]}$ "area metric."
**[[Merced Montesinos|http://www.fis.cinvestav.mx/~geogravi/gr_mphysics/Nuevos/faculty.html]], Cartan's equations define a topological field theory, (B)
***Main idea: include Riemann curvature and Torsion as independent degrees of freedom. Them put extra "topological" terms, like Euler characteristic and Pontryagin characteristic, in the action to ensure $\ff{R}$ and $\ff{T}$ satisfy Cartan equations.
***Possibly related paper: http://arxiv.org/abs/gr-qc/0603076
**Marat Reyes, Generalized path dependent representations for gauge theories, (B)
****Rats, I missed this one.
**Yuya Sasai, Braided quantum field theories and their symmetries, (A)
***Err, very hard to tell what the heck he was doing since his English wasn't very good.
**[[Garrett Lisi|http://sifter.org/~aglisi/]]${}^*$, Deferential Geometry, (A)
*** :) My talk went very well.
***"This is very interesting." -- L.S. "It's not bullshit." -- S.H.
*Tuesday
**Thomas Thiemann, Elements of Loop Quantum Gravity
***Has a new book coming out in September -- should be good.
***Talk was packed full of good stuff.
***Start wit Palatini formulation, combine constraints into master $M$, convert variables to $su(2)$ holonomies and fluxes, build solutions using coherent states, minimize expectation value of master constraint, $M$, rather than making it strictly $0$.
**Abhay Ashtekar, LQG: Lessons from models
***Symmetry reduced models. Spherically symmetric black holes, 1+1.
**[[Carlo Rovelli]]${}^*$, Vertex amplitude and propagator in loop quantum gravity
***Work in $so(4)$. Barrett/Crane model has problem with off diagonal (vertex) terms -- B.C. doesn't give intertwiners in perturbation calc. so it's wrong. Need to use GFT to get better model.
**Jan Ambjorn, 4d quantum gravity as a sum over histories
***His talk went long and he got cut off.
***I'm kind of unimpressed with CDT -- it's an odd approximation that forces plausible numerical results.
**Dan Christensen, Computations involving spin networks, spin foams, quantum gravity and lattice gauge theory (B)
***Kind of odd: gets positive real values for QM amplitude of spinfoam areas.
**[[Wade Cherrington|http://arxiv.org/abs/0705.2629]], Numerical Spin Foam Computation of Pure Yang-Mills Theory (B)
***YM on a lattice, in an odd way. Holonomies. Expand amplitudes into group characters.
**Seth Major
***Kodama state, approximating flat spacetime, from exponentiating Chern-Simons action.
**John Swain, Spin Networks and Simplicial Quantum Gravity (B)
***Cool guy. 3+1 D Regge -> simplicial, areas and angles.
*Wednesday
**Moshe Rozali, Background Independence in String Theory
***No action for $g$ in string theory. Have to choose a metric and perturb around that.
**Klaus Fredenhagen, General covariance in quantum field theory and the background problem in perturbative quantum gravity
***Sited [[new Stefan Hollands paper|http://arxiv.org/abs/0705.3340]] on BRST and Yang-Mills in curved spacetime.
***Path integral is not covariant when you define it in detail.
***Admissible embeddings are ones that preserve causal curves.
**Alejandro Perez, Regulator dependence in quantum gravity and non perturvative renormalizability: possible new perspectives
***missed it to talk to S.H.
**Martin Reuter, Asymptotically safe quantum gravity and cosmology
***Missed it. :( I came down with flu. But watched it online.
***Using truncated action, with running coupling constants $G(k)=g(k)/k^2$ and $\La(k)=k^2 \la(k)$, there is a Gaussian fixed point at $k \to 0$ and a non-Gaussian fixed point, $\big( g(\infty), \la(\infty) \big) \to {\rm const}$.
***Using more action terms, with many running parameters, the EH action appears to emerge as the unique fixed point at high energy. Just comes from fields and symmetries.
***Large cosmo constant at high energy can drive inflation, without inflaton.
***His recent [[paper|http://arxiv.org/abs/0708.1317]] on this just came out.
**No parallel sessions
**Quarantined myself because of flu.
*Thursday
**Feeling better -- think I'll venture out of hotel room again.
**Daniele Oriti, Group field theory: spacetime from quantum discreteness to an emergent continuum
***G's live at vertices. Graphically beautiful slides -- equations and figures.
***My guess on what GFT is: start with many copies of a Lie gp, G. State is a collection of reps for each gp. When these states "match" these gps are linked, giving a spin network approximating a base manifold for a principal G-bundle.
****No, I talked with Thomas Thiemann and he said this isn't how it works. There is only a G for each dimension of the spacetime (e.g. four, or three for just space), $(D-1)=4?$
**Artem Starodubtsev, Some physical results from spinfoam models
***According to a discussion at Physics Forums, this talk will actually be about BF gravity.
***He didn't show up to the conference. Had some travel holdup in the US.
**Martin Bojowald, Loop quantum cosmology and effective theory
***Skipped it to talk with L and S.
**Jorge Pullin${}^*$, Uniform discretizations and spherically symmetric loop quantum gravity
**Sundance Bilson-Thompson, Braids, loops, and the emergence of the standard model (B)
***Leader of the Braidy Bunch.
***Poor guy had a plague of audio problems.
***gluons are two stacked braids, with a plus and minus charge.
**Jonathan Hackett, ribbon networks, (B)
***reduced link invariants
**Yidun Wan, ribbons, (B)
***nice circle diagrams for tetra, and links could describe topology, but why are they braided?
**Jonathan Engle, cosmology, (A)
**Ileana Naish-Guzman, On the regularizability of the Ponzano-Regge model, (A)
***Works with John Barrett. Soft Brittish accent. Twisted cohomology groups over a cell complex.
**James Ryan, Aspects of Group Field Theory (A)
***Excellent intro to GFT. with group su(1,1) or su(2)
**Winston J. Fairbairn, Quantization of string-like sources coupled to BF theory: transition amplitudes and topological invariance, (A)
***(d-3) branes.
*Friday
**Fotini Markopoulou${}^*$, Quantum gravity and emergent locality
***Quantum Graphity
**Lee Smolin${}^*$, Chiral excitations of quantum geometries as a possible route to unification
***"New theories should include surprises."
***Ribbons as framed graphs, consistent with a cosmological constant.
***We don't really know the relationship between spin network Hamiltonian constraint and spin foam with evolution moves.
***Working with Sundance's braids
**Sabine Hossenfelder, Phenomenological Quantum Gravity
***"Top down inspired bottom up approaches"
***Pessimistic Freeman Dyson quote on QFT/GR independence.
***Colider constraints on KK models. (Black hole production)
***Zero black holes in standard setup (I'm not sure that's true).
***Minimal length scale as UV cutoff. (Doesn't this relate to discrete deSitter modes?)
**William Donnelly, Entanglement Entropy in Loop Quantum Gravity (B)
***Black hole entropy. Spin networks as the boundary/horizon in two part spinfoam/spacetime.
**Olaf Dreyer${}^*$, Internal Relativity: A progress report (A)
***Start with something like Ising model on lattice with Lorentz group as internal structure.
**Florian Girelli, 2-Groups and Topological Action (changed talk title!) (A)
***parallel transp of strings. defined 2-group, 2-Lie algebra, 2-principal bundle. Need 2-Peter-Weil theorem.
***one of not many DSR talks...
**Roberto Pereira, The loop-quantum-gravity vertex-amplitude (A)
***Works with Rovelli.
**Emanuele Alesci, Graviton propagator: the non diagonal terms (A)
***Works with Rovelli. (I get the impression this guy does the grind work of the calculations.) on {10J} propogator.
***Had to make up a term so that intertwiners would be involved when calculating the off diagonal part of the propogator.
**[[Isabeau Premont-Schwarz|http://arxiv.org/abs/hep-th/0508168]], Quantum Evolution in an Expanding Hilbert Space (Talk title at last minute) (A)
***Is this just using a non-square $U$ for evolution?
*Saturday
**John Stachel, Projective and Conformal Structures in General Relativity
***Older guy. Einstein biographer.
***Cecile deWitt has new book out, "Functional Integration"
***Affine space, affine connection and curvature. Need to break geometric variables into smaller pieces before quantizing.
***(His slides were messed up, missing most of the math symbols, which kinda wrecked it)
**Michael Reisenberger, Canonical gravity with free null initial data
***Free (unconstrained) gravitational initial data variables are known for initial hypersurfaces consisting of two intersecting null hypersurfaces. Recently the Poisson bracket on functions of such data has been obtained. This opens the prospect of a constraint free canonical formulation of general relativity.
**David Rideout, Can the supercomputer provide new insights into quantum gravity?
***Cactus. Causal sets and spin networks.
${}^*$ FQXi member -- might see the same talk in Iceland
(A,B) detiddlers which parallel session
During last two hours, questions were asked of the Plenary speakers, based on a book that had been passed around the audience. Carlo Rovelli moderated.
#You're at Loops '17, presenting your talk. Presuming your research program has been fully successful, describe your talk. What likelihood do you estimate for this happening? (This question was saved until the end, so speakers could prepare.)
#Topology change in QG?
##Abhay: not possible in canonical framework, but possible in LQG spin network.
#Does QG say anything about QM? Do we need a deeper framework, or a new interpretation, or different GR?
##Lucien: Need new math.
##John S: Need process QMI. (use paths)
##Thomas Thiemann: No. (just use conservative approach)
##Bianca: Relational framework
##John D: Path integrals OK, need to change GR.
#QG has fluctuating causal structure -- would this have any measurable effect?
##Sabine: photon spreading
##Michael: Matter sees only one spacetime. (yep)
#What is finite in spin foam models?
##Alejandro Perez: There are ambiguities in the theory.
#What would be a graph theory of nature? Spinfoam fundamental, or embedded in a manifold?
##Thomas: Topology change, so not embedded. He thinks spinfoam.
##Sundance: Braids.
#The first question was answered last:
##Lucien: Causaloids.
##John D: Background dependent, GR + SM emergent from spin substrate, with perturbative breaking of general covariance. 60%
##Thomas T: Complete description of QGR, as well as experiments done by himself. (ha) 5%
##Ashtekar: Resolution of all ambiguities in LQG, and establishment of all relationships between branches.
##A Perez: Thiemann was wrong. (ha!) "We have to make the road by walking."
##M Reuter: Same underlying theory, with non-Gaussian fixed point. 20%
##Daniele:
###100% - statistical GFT, with low temperature equilibrium phase
###80% - derive effective dynamics from GFT
###50% - one model singled out as successful
##Sabine: her talk would have the same title (ha) Measurements to support one model or another.
###New, unexpected data. 50%
###She'd find a permanent position. (ha)
##John S: Quantized Conformal Structures. But odds were low he'd be with it in 10 years. :(
##David Rideout: Causel sets give QM and GR. Probability: epsilon. (ha)
##Carlo Rovelli: closing words. "Let each of a hundred flowers bloom" -- quote from Michael's talk.
Random tiddlers:
*Many people used [[Beamer|http://latex-beamer.sourceforge.net/]] for their slides.
Loops '08 will be in Nottingham, England, in July (unless I misheard?)
Loops '09 probably in Beijing (or maybe P.I.)
I met a LOT of people
**PI grad students
***Chanda
***Jean Christian Boileau
***Jonathan Hackett (social guy at the end of table)
***Joel Brownstein (inflation guy)
***Bruno Hartmann (skinny sharp german guy with glasses, works with Thiemann)
***Isabeau Premont-Schwarz (funny german guy with glasses)
***Sean (ultimate frisbee guy)
***Cecilia
***Alejandro Satz (http://realityconditions.blogspot.com/)
***Joel Brownstein (inflation)
***William Donnely (http://williamdonnelly.blogspot.com/)
**other PI people
***Lucien Hardy
***Fotini
***Olaf?
***Lee Smolin
***Sabine
***Sundance
***Hans Westman (big baldish german with glasses)
***Rafael Sorkin
**misc
***John Stachel
***Daniele Oriti
****Alejandro Perez (kind of wild looking)
****Florian Gireli
***John Swain
***Michael Reisenberger
***Jonathon Engle
***Wayne Bomstad (grad student, works with John Klauder in Florida, said he admires for being well rounded)
The ''Lorentz algebra'', $spin(1,n\!-\!1)$, is the $\ha n (n\!-\!1)$-dimensional [[Lie algebra]] of the [[Lorentz group]], with $1$ time-like and $(n\!-\!1)$ space-like dimensions. It is a special case of the [[spin Lie algebra]] of mixed signature. It is also isomorphic to the Lie algebra, $so(1,n\!-\!1)$, of the generalized [[special orthogonal group]]. Lie algebra basis elements, $T_A$, may be represented by $\ha n (n-1)$ antisymmetric $n \times n$ matrices, $T_A \sim J_{ij}$, antisymmetric in the $ij$ labels, or alternatively by [[bivectors|Clifford basis elements]], $T_A \sim \ga_{\al \be}$, of the $Cl(1,n\!-\!1)$ [[Clifford algebra]]. The bivectors satisfy the [[Clifford basis identities]]:
$$
\lb \ga_{\al \be}, \ga_{\ga \de} \rb = 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} = C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb} \ga_{\ep \up}
$$
giving the structure constants,
$$
C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb}
= 2 \left\{ - \et_{\al \ga} \de^{\lb\ep \up\rb}_{\be \de} + \et_{\al \de} \de^{\lb\ep \up\rb}_{\be \ga} + \et_{\be \ga} \de^{\lb\ep \up\rb}_{\al \de} - \et_{\be \de} \de^{\lb\ep \up\rb}_{\al \ga} \right\}
$$
with $\et_{\al \ga}$ the generalized [[Minkowski metric]]. The [[Killing form]] (diagonal in this basis) is
$$
g_{\lb \al \be \rb \lb \ga \de \rb} = C_{\lb \al \be \rb \lb \ep \up \rb}{}^{\lb \ze \et \rb} C_{\lb \ga \de \rb \lb \ze \et \rb}{}^{\lb \ep \up \rb}
= 8 \lp n-2 \rp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp
$$
"Lorentz algebra" often refers more specifically to the [[spacetime]] case, [[spin(1,3)]].
A ''Lorentz boost'' is a [[Lorentz rotation]] of a [[spacetime]] vector or [[Dirac spinor]] along a spatial ''boost vector'', $v = \ga_\pi v^\pi$, and corresponds to a [[Clifford rotation]] by the [[Cl(1,3) bivector]], $B$, corresponding (but not equal to) to the time-$v$ plane, $\ga_0 v$. Specifically, $B = \ga_0 n \ze$, in which the ''rapidity'', $\ze$, is defined via
$$
\be = \fr{\left| v \right|}{c} = \tanh{\ze}
$$
in which $c$ is the speed of light, and the ''boost direction'' is defined as $n = \fr{v}{\left| v \right|}$, so $v = n \, c \tanh{\ze}$. The ''boost rotor'' is then
$$
U_\nu = e^{\ha B} = e^{- \ha \ga_0 n \ze} = \cosh{\fr{\ze}{2}} - \ga_0 n \sinh{\fr{\ze}{2}}
$$
(It is also sometimes useful to define the ''rapidity vector'', $\nu = n \, \ze = \fr{v}{|v|} \tanh^- \fr{|v|}{c}$, so $B = \ga_0 \nu$.)
Any [[spacetime]] Clifford vector, $u$, decomposes into its temporal and spatial parts, and its spatial part splits into parts parallel and perpendicular to the boost direction,
$$
u = u^\mu \ga_\mu = u^0 \ga_0 + u^\pi \ga_\pi
= u_t + u_s = u_t + u_\parallel + u_\perp
$$
with $u_\parallel = n u^n$ and $u_\perp = u_s - u_\parallel$, in which $u^n = n \cdot u$ is the magnitude of $u$ along the boost direction. Working through some hyperbolic trig and Clifford algebra, we see that the ''Lorentz boosted vector'' is
$$
\begin{array}{rcl}
u' \!\!&\!\!=\!\!&\!\! u^\mu \ga'_\mu = u^\mu L^\nu{}_\mu \ga_\nu = U_v \, u \, U_v^- \\
\!\!&\!\!=\!\!&\!\! \lp \cosh{\fr{\ze}{2}} - \ga_0 n \sinh{\fr{\ze}{2}} \rp \lp u_t + u_\parallel + u_\perp \rp \lp \cosh{\fr{\ze}{2}} + \ga_0 n \sinh{\fr{\ze}{2}} \rp \\
\!\!&\!\!=\!\!&\!\! u_\perp + \ga_0 \lp u^0 \cosh{\ze} + u^n \sinh{\ze} \rp + n \lp u^n \cosh{\ze} + u^0 \sinh{\ze} \rp
\end{array}
$$
which is more familiar after identifying the [[Lorentz factor|Lorentz rotated coordinates]], $\cosh{\ze} = \ga = 1 / \sqrt{1-\be^2}$, and $\sinh{\ze} = \ga \be$.
A ''Lorentz boosted Dirac spinor'', using the Weyl rep of the [[Dirac matrices]] and the corresponding [[Cl(1,3) bivector]]s, is
$$
\ps' = U_v \, \ps =
\lb \begin{array}{cc}
\cosh{\fr{\ze}{2}} - n^\pi \si_\pi \sinh{\fr{\ze}{2}} & 0 \\
0 & \cosh{\fr{\ze}{2}} + n^\pi \si_\pi \sinh{\fr{\ze}{2}}
\end{array} \rb
\lb \begin{array}{c}
\ps_L \\
\ps_R
\end {array} \rb
$$
in which the ''left and right chiral boost rotor''s are $U^{L/R}_\nu = \cosh{\fr{\ze}{2}} \mp n^\pi \si_\pi \sinh{\fr{\ze}{2}}$. These are not [[unitary]], but are [[Hermitian]] and inverses, $U^{L/R}_\nu U^{R/L}_\nu = 1$.
The ''special orthochronous Lorentz group'', $\mbox{SO}{}^+(1,n-1)$, is a [[Lie group]] composed of [[Lorentz rotation]]s in $n$ dimensions, including one of time. The ''Lorentz group'', $\mbox{O}(1,n-1)$, has four disconnected components — two are special ($\mbox{S}$), linked by [[parity conjugation|parity conjugate]], and two are orthochronous (${}^+$), linked by [[time conjugation|time conjugate]]. The special orthochronous Lorentz group is the connected subgroup component containing the identity. The Lorentz group thus has ''components'',
$$
\mbox{O}(1,n-1) = \mbox{SO}(1,n-1) \rtimes \{1,P \} = \mbox{O}^+(1,n-1) \rtimes \{1, PT \} = \mbox{SO}{}^+(1,n-1) \rtimes \{1,P,T,PT \}
$$
which has double covers,
$$
\mbox{Pin}(1,n-1) = \mbox{Spin}(1,n-1) \rtimes \{1,P \} = \mbox{Pin}{}^+(1,n-1) \rtimes \{1, PT \} = \mbox{Spin}{}^+(1,n-1) \rtimes \{1,P,T,PT \}
$$
related to [[Clifford rotation]]s. The double cover of the special orthochronous Lorentz group, $\mbox{SO}{}^+(1,n-1)$, is the orthochronous [[spin group]], $\mbox{Spin}{}^+(1,n-1)$ (with $\mbox{Spin}{}^+(1,3) = SL(2,\mathbb{C})$), the double cover of the ''special Lorentz group'', $\mbox{SO}(1,n-1) = \mbox{SO}{}^+(1,n-1) \rtimes \{1,PT \}$ (or ''orientation preserving Lorentz group''), a generalized [[special orthogonal group]], is the generalized [[spin group]], $\mbox{Spin}(1,n-1)$, and the double cover of the Lorentz group, $\mbox{O}(1,n-1)$, is the [[pin group]], $\mbox{Pin}(1,n-1)$. Note that, despite the isomorphism of $\mbox{Spin}(1,3)$ to $\mbox{Spin}(3,1)$, and that they are subgroups of $\mbox{Pin}(1,3)$ and $\mbox{Pin}(3,1)$, the group $\mbox{Pin}(1,3)$ is not isomorphic to $\mbox{Pin}(3,1)$. For $\mbox{Pin}(1,3)$ we have $P^2=1$ and $T^2=1$, while for $\mbox{Pin}(3,1)$ we have $P^2=-1$ and $T^2=-1$. This description of the Lorentz group is general as to dimension, but "Lorentz group" is often used to refer specifically to the ''spacetime Lorentz group'', or [[spacetime spin group]], with $n=4$. The [[Lie algebra]] of the Lorentz group is the [[Lorentz algebra]], and the Lie algebra of the spacetime Lorentz group is [[spin(1,3)]].
The Wikipedia article is quite thorough:
http://en.wikipedia.org/wiki/Lorentz_group
The measure in Quantum Field Theory expressions comes from integrating the "on shell" delta function of [[momentum]] over positive energies,
$$
\ba{rcl}
\int d^4 p \, \de((p^0)^2 - p_s^2 - m^2) \, \th(p^0) &=& \int d^4 p \, \de((p^0)^2 - E_p^2) \, \th(p^0) \\
&=& \int d^4 p \, \fr{\lp \de(p^0 + E_p) + \de(p^0 - E_p) \rp}{2 E_p} \, \th(p^0) \\
&=& \int \fr{d^3 p}{2 E_p} \\
\ea
$$
A ''Lorentz coordinate transformation'' is a [[coordinate change]], $x^i \to x'^i(x)$, in a [[rest frame]], to a different set of coordinates corresponding to a [[reference frame|rest frame]] that is rotated and moving at velocity $\ve{v}$ with respect to the first. This transformation corresponds to a [[Lorentz rotation]] of the coordinate frame, $x'^i = L_j{}^i x^j$, and preserves ''spacetime interval''s,
$$
{x'^0}^2 - {x'^1}^2 - {x'^2}^2 - {x'^3}^2 = {x^0}^2 - {x^1}^2 - {x^2}^2 - {x^3}^2
$$
If we assume no spatial rotation, and a ''boost'' to velocity $\ve{v} = v \ve{\pa_3}$, the corresponding Lorentz coordinate transformation is
\begin{eqnarray}
x'^0 &=& \ga \, (x^0 - \be x^3) \\
x'^1 &=& x^1 \\
x'^2 &=& x^2 \\
x'^3 &=& \ga \, (x^3 - \be x^0)
\end{eqnarray}
in which the ''Lorentz factor'' is $\ga = 1 / \sqrt{1-\be^2} > 1$ and $\be = \fr{v}{c}$ is the ratio of the boost magnitude to the speed of light. In a rest frame, these transformed coordinates are [[Lorentz boost]]ed vectors from the origin. Physically, if an observer in a [[rest frame]] with coordinates $x^i$ is observing a ticking clock in the moving frame with coordinates $x'^i$, the observer will see the clock ticks happening more slowly by a factor of $\ga$ (''time dilation'') and the length of the clock along the direction of motion contracted by a factor of $\ga$ (''length contraction''). From the perspective of an observer in the moving frame, the rest frame is the moving frame and they will similarly observe dilation and contraction of events in that frame.
A rotation is a smoothly operating linear transformation acting on vectors that leaves the scalar product between vectors invariant. ("Vectors" in this case may stand for [[tangent vector]]s, [[1-form]]s, [[Clifford vectors|Clifford element]], or any other appropriately [[indexed|indices]] object.) For example, two vectors with components $u^\al$ and $v^\al$ may have the scalar product, $u^\al \et_{\al \be} v^\be$. A linear transformation of vector components by a ''Lorentz matrix'' maps the vector components to
\begin{eqnarray}
{u'}^\al &=& L^\al {}_\be u^\be\\
{v'}^\al &=& L^\al {}_\be v^\be
\end{eqnarray}
which must preserve the scalar product,
\[ u^\al \et_{\al \be} v^\be = {u'}^\al \et_{\al \be} {v'}^\be = L^\al{}_\be u^\be \et_{\al \ga} L^\ga{}_\de v^\de \]
So the Lorentz matrix must be ''orthogonal'',
\[ L^\al{}_\be \et_{\al \ga} L^\ga{}_\de = \et_{\be \de} \]
or, with the [[Minkowski metric]] raising and lowering indices, $L_{\ga \be} L^{\ga \de} = \de_\be^\de$ or $L^T L = I$. A transformation satisfying this restriction is a [[Lorentz transformation]]. But such a transformation could also include reflections, and would then not be "smoothly operating" (not connected to the identity). To exclude this possibility, $L$ is restricted to have positive determinant, $|L|=1$, in which case it is called "special" or "proper", and $L$ is also restricted to preserver the direction of time (no reflection of the $0$ components), in which case it is called "orthochronous". A special orthochronous Lorentz transformation is called a ''Lorentz rotation''. It is "smoothly operating" or "connected to the identity" in that it may be built up by many small rotations,
\[ L = \lim_{N \to \infty} \lp I + \fr{1}{N} l \rp^N \]
in which $l$ is an antisymmetric matrix, $l_{\al \be} = l_{\lb \al \be \rb}$. A Lorentz transformation built this way is special and orthochronous.
The group of Lorentz rotations forms the [[special orthochronous Lorentz group|Lorentz group]].
Although this matrix representation is more standard, Lorentz rotations are usually better described and carried out as [[Clifford rotation]]s, by which [[Lorentz boost]]s and [[spatial rotation]]s can be described more easily.
If we start with a field, $\ph(x)$, in a [[rest frame]], we can imagine actively rotating that field -- so the new field is the same field-shape as the old field, but rotated in space,
$$
\ph'(x) = \ph(L^- x)
$$
in which $L^-$ is the inverse of a [[Lorentz rotation]] of the rest frame coordinates, $x^\mu$.
Rotating a [[1-form]] field, such as the [[electromagnetic field]], the ''Lorentz transformation'' of the field is
$$
\f{A}'(x) = \f{dx}^\mu L_\mu{}^\nu A_\nu (L^- x)
$$
This is an auto[[diffeomorphism]] in flat spacetime, and not a [[coordinate change]]. Rather, the new field is in a rotated direction, and is the same shape as the old field but a function of [[Lorentz rotated coordinates]], $x'^\mu = L_\nu{}^\mu x^\nu$. If this transformation is in curved spacetime, we imagine this is happening locally near an origin point on a [[manifold]] coordinate patch. If the field corresponds to a quantum field operator in an [[infinite-dimensional unitary representation]], that quantum field transforms as
$$
\hat{A}_\mu(x) \mapsto \hat{A'_\mu}(x) = \hat{U}(L) \hat{A_\mu}(x) \hat{U}(L)^- = L_\mu{}^\nu \hat{A_\nu} (L^- x)
$$
For an infinitesimal [[Lorentz rotation]], $L_\mu{}^\nu = \de_\mu{}^\nu + \ep l_\mu{}^\nu$, with $l_{\mu\nu} = l_{[\mu\nu]}$ antisymmetric, after using the [[Minkowski metric]] to lower indices, $l_{\mu\nu} = l_\mu{}^\rh \et_{\rh\nu}$. The infinitesimal quantum field transformation is thus
$$
\lb \hat{J}\!(l) , \hat{A_\mu} \rb = l_\mu{}^\nu \hat{A_\nu}(x) - x^\nu l_\nu{}^\rh \pa_\rh \hat{A_\mu}(x)
$$
in which $\hat{J}$ is an element of the six-dimensional [[spin(1,3)]] [[Lorentz algebra]].
If we consider a Cliffordized [[electromagnetic field]], $A = \ve{e} \f{A}$, the Lorentz transformation of the [[quantum electromagnetic field]] and Cliffordized electromagnetic field is
$$
\hat{A}(x) \mapsto \hat{A}'(x) = \hat{U}(L) \hat{A}(x) \hat{U}(L)^- = U(L) \hat{A}(L^- x) U^-(L)
$$
with $U(L) = e^{\ha B}$ the [[rotor|Clifford rotation]] corresponding to the Lorentz transformation. For an infinitesimal transformation this becomes
$$
\lb \hat{J}(l) , \hat{A} \rb = B(l) \times \hat{A} - x^\nu l_\nu{}^\rh \pa_\rh \hat{A}(x)
$$
The Lorentz Lie algebra elements break into two parts, $\hat{J}\!(l) = \hat{S}(l) + \hat{L}(l)$, corresponding to the ''spin angular momentum operator'' and ''orbital angular momentum operator''. The basis elements of these operators satisfy
$$
\lb \hat{S}^{\mu \nu} , \hat{A} \rb = \ga^{\mu \nu} \times \hat{A} \s\;\;\; \lb \hat{L}^{\nu \rh} , \hat{A} \rb = - x^{[\nu} \pa^{\rh]} \hat{A}
$$
If we rotate a [[Dirac spinor]] field, the Lorentz transformation of the [[quantum Dirac spinor]] and Dirac spinor is
$$
\ud{\hat{\Ps}}(x) \mapsto \ud{\hat{\Ps}}'(x) = \hat{U}(L) \ud{\hat{\Ps}}(x) \hat{U}(L)^- = U(L) \ud{\hat{\Ps}} (L^- x)
$$
For an infinitesimal transformation this becomes
$$
\lb \hat{J}(l) , \ud{\hat{\Ps}} \rb = \ha B(l) \ud{\hat{\Ps}} - x^\nu l_\nu{}^\rh \pa_\rh \ud{\hat{\Ps}}(x)
$$
and the spin angular momentum operator and orbital angular momentum operator satisfy
$$
\lb \hat{S}^{\mu \nu} , \ud{\hat{\Ps}} \rb = \ha \ga^{\mu \nu} \ud{\hat{\Ps}} \s\;\;\; \lb \hat{L}^{\nu \rh} , \ud{\hat{\Ps}} \rb = - x^{[\nu} \pa^{\rh]} \ud{\hat{\Ps}}
$$
leading us to define the [[spin operator]] on a Dirac spinor or [[Weyl spinor]].
Lorentz transformations, described by $L_\mu{}^\nu$, are elements of a [[representation]] of the [[Lorentz group]], $O(1,3)$ -- a generalized [[orthogonal group|special orthogonal group]], with $L_\mu{}^\nu L_\mu{}^\nu = \et_f$. These are equivalent to [[Clifford rotation]]s, $\ga'_\al = \ga_\be L^\be{}_\al = U \ga_\al U^-$. But the allowed [[Clifford group]] transformations also include elements of the [[pin group|spacetime spin group]], $U \in Pin(1,3)$, the double cover of the [[Lorentz group]], which includes elements of the proper orthochronous spin group as well as [[CPT symmetry]].
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The left-chiral [[Majorana equation|Majorana spinor]] is
$$
0 = \lp i \pa_0 - i \si_\va \pa_\va \rp \ps_L + m \ep \ps^{*}_L
$$
We make an ansatz that solutions have the form
$$
\ps_L = \ch_1 e^{- i p_\mu x^\mu} + \ch_2 e^{+ i p_\mu x^\mu}
$$
with $p$ the [[momentum]]. This reduces the left-chiral Majorana equation to
\begin{eqnarray}
0 &=& + \lp E + p^\va \si_\va \rp \ch_1 + m \ep \ch_2^* \\
0 &=& - \lp E + p^\va \si_\va \rp \ch_2 + m \ep \ch_1^*
\end{eqnarray}
These are both solved if
$$
\ch_2 = \fr{1}{m} \lp E - p^\va \si_\va \rp \ep \ch_1^*
$$
with two basis solutions allowed,
$$
\ch_1 = \ch^\wedge = \lb \begin{array}{c}
1 \\
0
\end {array} \rb
\s \text{or} \s
\ch_1 = \ch^\vee = \lb \begin{array}{c}
0 \\
1
\end {array} \rb
$$
giving two basis solutions to the left-Chiral Majorana equation,
$$
\ps_L^{\wedge/\vee} = w_L^{\wedge/\vee} = \ch^{\wedge/\vee} e^{- i p_\mu x^\mu} + \fr{1}{m} \lp E - p^\va \si_\va \rp \xi^{\wedge/\vee} e^{+ i p_\mu x^\mu}
$$
in which $\xi^{\wedge/\vee} = \ep \ch^{\wedge/\vee}$ are the basis [[flipped spin]]ors. Similarly, the right-chiral Majorana equation is
$$
0 = \lp i \pa_0 + i \si_\va \pa_\va \rp \ps_R - m \ep \ps^{*}_R
$$
and has two basis solutions,
$$
\ps_R^{\wedge/\vee} = w_R^{\wedge/\vee} = \ch^{\wedge/\vee} e^{- i p_\mu x^\mu} - \fr{1}{m} \lp E + p^\va \si_\va \rp \xi^{\wedge/\vee} e^{+ i p_\mu x^\mu}
$$
These ''Majorana equation solutions'', $w_L^{\wedge/\vee}$ and $w_R^{\wedge/\vee}$, match [[Dirac solutions]] via the definition of a [[Majorana spinor]]. Specifically,
\begin{eqnarray}
\Ps_M^{\wedge/\vee} &=&
\lb \begin{array}{c}
w_L^{\wedge/\vee} \\
-\ep w_L^{\wedge/\vee*}
\end {array} \rb
=
\lb \begin{array}{c}
\ch^{\wedge/\vee} \\
\fr{1}{m} (E+p^\ep \si_\ep) \ch^{\wedge/\vee}
\end {array} \rb
e^{-i p_\mu x^\mu}
+
\lb \begin{array}{c}
\fr{1}{m} (E-p^\ep \si_\ep) \xi^{\wedge/\vee} \\
-\xi^{\wedge/\vee} \\
\end {array} \rb
e^{+i p_\mu x^\mu} \\
& = &
\fr{1}{m^2} u'{}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \fr{1}{m^2} v'{}_p^{\wedge/\vee} e^{+i p_\mu x^\mu}
\end{eqnarray}
and similarly if we use the $w_R^{\wedge/\vee}$ solution.
A [[Dirac spinor]] has a $U(1)$ symmetry manifesting as multiplication by a complex phase, $e^{i \ph}$, usually corresponding to the electromagnetic interaction. If we wish to consider an uncharged fermion, this must be described by half of a Dirac spinor. Such an uncharged spinor is called a ''Majorana spinor'', $\Ps^M$, and is equal to its [[charge conjugate]], $(\Ps^M)^C = \Ps^M$ -- the ''Majorana condition''. If we work with the Weyl representation of the [[Dirac matrices]] and allow the left-chiral part of the Dirac spinor to vary freely, then the right-chiral part is restricted by the Majorana condition, $ i \ga_2 (\Ps^M)^* = \Ps^M$, or vice versa, so that a Majorana spinor can be written as
$$
\Ps^M =
\lb \begin{array}{c}
\ps^M_L \\
( \ps^{M}_L )^C
\end {array} \rb
=
\lb \begin{array}{c}
\ps^M_L \\
- \ep \ps^{M*}_L
\end {array} \rb
=
\lb \begin{array}{c}
( \ps^{M}_R )^C \\
\ps^M_R
\end {array} \rb
=
\lb \begin{array}{c}
\ep \ps^{M*}_R \\
\ps^M_R
\end {array} \rb
$$
in which $\ep$ is the [[skew]]. The kinetic part of the flat spacetime [[Dirac Lagrangian]], inputing this Majorana spinor, is
\begin{eqnarray}
{\cal L}_K &=& \bar{\Ps}^M i \ga^\mu \pa_\mu \Ps^M \\
&=& i \ps^{M\da}_L \si^\mu \pa_\mu \ps^M_L + i (\ep \ps^{M*}_L)^\da \bar{\si}^\mu \pa_\mu (\ep \ps^{M*}_L) \\
&=& 2 i \ps^{M\da}_L \si^\mu \pa_\mu \ps^M_L = 2 i \ps^{M\da}_R \bar{\si}^\mu \pa_\mu \ps^M_R
\end{eqnarray}
after using [[Pauli matrix|Pauli matrices]] magic and taking only the real part. A lone Majorana spinor cannot have a charge but it can have a mass. The mass term in the Dirac Lagrangian, with this Majorana spinor determined by its left or right chiral part, is
$$
{\cal L}_m = - m \bar{\Psi}^M \Psi^M = - m \lp \ps^{MT}_L \ep \ps^M_L - \ps^{M \da}_L \ep \ps^{M*}_L \rp =
m \lp \ps^{MT}_R \ep \ps^M_R - \ps^{M \da}_R \ep \ps^{M*}_R \rp
$$
using the [[transpose]], and in which the second terms are the Hermitian conjugates of the first, so the Lagrangian is real. Since a Lagrangian, including mass, can be written in terms of exclusively the left or the right chiral parts of a Majorana spinor, we can refer disingenuously to these Majorana components as left-chiral or right-chiral Majorana fields and treat them as [[Weyl spinor]]s satisfying a ''Majorana equation'',
$$
0 = i \si^\mu \pa_\mu \ps_L + m_L \ep \ps^{*}_L
\s \s
0 = i \bar{\si}^\mu \pa_\mu \ps_R - m_R \ep \ps^{*}_R
$$
If we have two Weyl spinors, $\ps_L$ and $\ps_R$, we can build a Dirac spinor and have the usual kinetic and Dirac mass terms,
$$
{\cal L}_m = - m \lp \ps_R^\da \ps_L + \ps_L^\da \ps_R \rp
$$
If we treat $\ps_L$ and $\ps_R$ as components of independent Majorana spinors, then they can have independent Majorana masses, $m_L$ and $m_R$, as well as a shared Dirac mass, $m$.
$$
{\cal L}_{m_L} = m_L \lp \ps_L^\da \ep \ps_L^* - \ps_L^{\da*} \ep \ps_L \rp
\s \s
{\cal L}_{m_R} = - m_R \lp \ps_R^\da \ep \ps_R^* - \ps_R^{\da*} \ep \ps_R \rp
$$
(Note that Majorana mass terms violate $U(1)$ gauge symmetry, since $\ps_L \to e^{i \ph} \ps_L$ implies $\ps_L^\da \ep \ps_L^* \to e^{-2 i \ph} \ps_L^\da \ep \ps_L^*$.) These Lagrangian mass terms can be combined as
$$
{\cal L}_M = -
\lb
\begin{array}{c}
\ps_L \\
(\ps_R)^C
\end{array}
\rb ^ \da
\lb
\begin{array}{cc}
m_L & m \\
m & m_R
\end{array}
\rb
\lb
\begin{array}{c}
(\ps_L)^C \\
\ps_R \\
\end{array}
\rb
+ \textrm{h.c.}
$$
in which "h.c." is the harmonic conjugate. The eigenvalues of this mass matrix are
$$
M_\pm = \ha \lp (m_L + m_R) \pm \sqrt{4 m^2 + (m_L - m_R)^2} \rp \simeq m_{L/R} \pm \fr{m^2}{\left| m_L-m_R \right|}
$$
in which the approximation is for small $m$. If $m_L$ is zero and $m_R$ is large and $m$ is small, then this gives the celebrated ''seesaw mechanism'', with the two resulting fermions (mostly $\ps_R$ and mostly $\ps_L$) having masses $m_R$ and $\fr{m^2}{m^R}$.
<<tiddler HideTags>>\begin{eqnarray}
\lp D \!\!\!\! / + \ph \rp \ud{\ps} &=& \ga^\mu \lp e_\mu\rp^a \lp \pa_a + \fr{1}{4} \om_a^{\p{a}\nu\rh} \ga_{\nu\rh} + B,W,G_a^{\p{a}A} T_A \rp \ud{\ps} + \ph \, \ud{\ps} \\
&=& \ga^\mu \lp e_\mu\rp^a \lp \pa_a + \fr{1}{4} \om_a^{\p{a}\nu\rh} \ga_{\nu\rh}
+ \fr{1}{4} \lp e_a \rp^\nu \ga_\nu \ph
+ B,W,G_a^{\p{a}A} T_A \rp \ud{\ps}
\end{eqnarray}
| $\; \ga_\mu \; $ |[[Clifford basis vectors]] for [[Cl(1,3)]] |
| $\; \ga_{\mu\nu} = \ga_\mu \ga_\nu \; $ |[[Clifford basis bivectors|Clifford basis elements]] |
| $\; T_A \; $ |[[Lie algebra]] basis elements (//generators//) |
| $\; ( e_\mu )^a \; $ |[[orthonormal basis vector|frame]] components (//frame, vierbein//) |
| $\; \om_a^{\p{a}\nu\rh} \; $ |[[spin connection]] components |
| $\; B_a^{\p{a}A}, W_a^{\p{a}A}, G_a^{\p{a}A} \; $ |Yang-Mills [[gauge field|principal bundle]] components (//connections//) |
| $\; \ph \; $ |Higgs scalar field multiplet |
| $\; \ud{\ps} \; $ |[[Grassmann|Grassmann number]] valued [[spinor]] field multiplet |
$$
\begin{array}{rcl}
{\rm Clifford \; algebra} \!\!&\!\! \longleftrightarrow \!\!&\!\! {\rm Lie \; algebra}^{\phantom{(}} \\
\searrow \!\!\!\!\!\! \nwarrow \!\!&\!\! \!\!&\!\! \swarrow \!\!\!\!\!\! \nearrow \\
& {\rm Matrices} &
\end{array}
$$
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<<tiddler HideTags>>First $\mathbb{C}(8\times8)$ quadrant of a $\mathbb{C}(16\times16)$ [[chiral]] rep of $Cl(1,7)$ bivectors:
\begin{eqnarray}
\f{H^+} &=& \big( \ha \f{w} + \fr{1}{4} \f{e} \ph + \f{W} + \f{B} \big)^+ \\
&=& \fr{1}{4} \f{w^{\mu\nu}} \ga^+_{\mu\nu} + \fr{1}{4} \f{e^\mu} \ph^\ph \ga^+_{\mu\ph} \\
&& - \f{W^\pi} \fr{1}{4} \big( \ep_{\pi (\ph-4)(\ps-4)} \ga^+_{\ph \ps} + \ga^+_{(\pi+4)8} \big)
+ \f{B} \ha \big( \ga^+_{78} - \ga^+_{56} \big)_{\phantom{\Big(}} \\
&=&
\lb \begin{array}{cccc}
\ha \f{\om_L} \!+\! i \f{W^3} & i \f{W^1} \!+\! \f{W^2} & - \fr{1}{4} \f{e_R} \ph_0^* & \fr{1}{4} \f{e_R} \ph_+ \\
i \f{W^1} \!-\! \f{W^2} & \ha \f{\om_L} \!-\! i \f{W^3} & \fr{1}{4} \f{e_R} \ph_+^* & \fr{1}{4} \f{e_R} \ph_0 \\
-\fr{1}{4} \f{e_L} \ph_0 & \fr{1}{4} \f{e_L} \ph_+ & \ha \f{\om_R} \!+\! i \f{B} & \\
\fr{1}{4} \f{e_L} \ph_+^* & \fr{1}{4} \f{e_L} \ph_0^* & & \ha \f{\om_R} \!-\! i \f{B}
\end{array} \rb^{\phantom{\big(}}
\end{eqnarray}
with $\ph_0 = (\ph^7 + i \ph^8)$ abd $\ph_+ = (-\ph^5 + i \ph^6)$.
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ {\tiny \frac{1}{2}} \om_i^{\p{i}\nu\rh} {\tiny \frac{1}{2}} \ga_{\nu\rh}
+ W_i^{\p{i}\pi} T^W_\pi
+ B_i T^Y
+ g_i^{\p{i}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
\begin{array}{lcrcc}
\ga_1 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_1 \\
\ga_2 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_2 \\
\ga_3 \!\!&\!\! = \!\!&\!\! \si_2 \!\!&\!\! \otimes \!\!&\!\! \si_3 \\
\ga_4 \!\!&\!\! = \!\!&\!\! i \si_1 \!\!&\!\! \otimes \!\!&\!\! 1
\end{array}
\s
\begin{array}{l}
Cl(3,1) \subset GL(4,\mathbb{C}) \\[.25em]
\ga_{\mu \nu} = \ga_\mu \ga_\nu \in spin(3,1) \\[.25em]
\ep = \ga_1 \ga_2 \ga_3 \ga_4 = i \si_3 \otimes 1 \\[.25em]
P_{R/L} = \ha (1 \pm i \ep)
\end{array}
\s
\begin{array}{rcl}
\La_A \!\!&\!\! = \!\!&\!\!
\lb
\matrix{
0 & 0 \\
0 & \la_A
}
\rb \\
\!\!&\!\! \in \!\!&\!\! GL(4,\mathbb{C})\vp{A^{\big(}}
\end{array} \vp{A_{\Big(}}
$$
$$
\begin{array}{lcl}
T^\om_{\mu \nu} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{\mu \nu} \s\; \in GL(32,\mathbb{C}) \\[.5em]
T^W_\pi \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_\pi \otimes P_L \\[.5em]
T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes i \si_3 \otimes P_R \\[.25em]
\!\!&\!\! \!\!&\!\! - i \, \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3}) \otimes 1 \otimes 1
\end{array}
\s\;
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$
$$
\mbox{complex structure:} \;\; i \to
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 \;\; \Rightarrow \; T \in GL(64,\mathbb{R}), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl^1(11,3) \mbox{ basis elements in } \mathbb{R}(128) \\[.25em]
\begin{array}{rcrccccccccccccc}
\Ga_1 \!\!&\!\! = \!\!&\!\! i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_2 \!\!&\!\! = \!\!&\!\! -i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_3 \!\!&\!\! = \!\!&\!\! i \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_4 \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_5 \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_6 \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_7 \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\Ga_8 \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\Ga_9 \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{10} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\Ga_{11} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{12} \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\Ga_{13} \!\!&\!\! = \!\!&\!\! \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\Ga_{14} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
\\[.75em]
\Ga = \Ga_1 \Ga_2 \dots \Ga_{14} = \si_3 \otimes 1 \\[.75em]
P_\pm = \ha (1 \pm \Ga) \\[.75em]
\Ga^+_{ij} = P_+ \Ga_i \Ga_j \, \mbox{ in } \, \mathbb{R}(64)
\end{array}
\s\s\;\;\;\;
\begin{array}{l}
\!\!\!\!\!\! \mbox{18 Standard Model generators in } spin(11,3) \\[.25em]
\begin{array}{lcl}
T^\om_{ab} \!\!&\!\! = \!\!&\!\! \Ga^+_{a \, b} \\[.4em]
T^W_1 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{5 \, 8} - \fr{1}{4} \Ga^+_{6 \, 7} \\
T^W_2 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{5 \, 7} - \fr{1}{4} \Ga^+_{6 \, 8} \\
T^W_3 \!\!&\!\! = \!\!&\!\! - \fr{1}{4} \Ga^+_{5 \, 6} + \fr{1}{4} \Ga^+_{7 \, 8} \\[.4em]
T^g_1 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{9 \, 12} - \fr{1}{4} \Ga^+_{10 \, 11} \\
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 11} - \fr{1}{4} \Ga^+_{10 \, 12} \\
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 10} + \fr{1}{4} \Ga^+_{11 \, 12} \\
T^g_4 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{9 \, 14} + \fr{1}{4} \Ga^+_{10 \, 13} \\
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{9 \, 13} + \fr{1}{4} \Ga^+_{10 \, 14} \\
T^g_6 \!\!&\!\! = \!\!&\!\! \fr{1}{4} \Ga^+_{11 \, 14} - \fr{1}{4} \Ga^+_{12 \, 13} \\
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{1}{4} \Ga^+_{11 \, 13} - \fr{1}{4} \Ga^+_{12 \, 14} \\
T^g_8 \!\!&\!\! = \!\!&\!\! \fr{1}{4\sqrt{3}} ( - \Ga^+_{9 \, 10} - \Ga^+_{11 \, 12} + 2 \, \Ga^+_{13 \, 14} ) \\[.4em]
T^Y \!\!&\!\! = \!\!&\!\! \fr{1}{4} ( \Ga^+_{5 \, 6} + \Ga^+_{7 \, 8} ) \\
& &\!\! + \fr{1}{6} ( \Ga^+_{9 \, 10} + \Ga^+_{11 \, 12} + \Ga^+_{13 \, 14} )
\end{array} \\[.5em]
\ps = \ps^{\, \io} Q_\io \in 64^\mathbb{R}_{S+}
\end{array}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl(3,11) \mbox{ basis elements in } GL(128,\mathbb{R}) \\[.25em]
\begin{array}{lcrccccccccccccc}
\Ga_1 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_2 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_3 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_4 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_5 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \\
\Ga_6 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \\
\Ga_7 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_8 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_9 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{10} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{11} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{12} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{13} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{14} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\end{array}
\\[.75em]
\begin{array}{rcl}
\Ga \!\!&\!\!\! = \!\!\!&\!\! \Ga_1 \Ga_2 \dots \Ga_{14} \\
\!\!&\!\!\! = \!\!\!&\!\! \si_3 \otimes 1 \otimes 1 \otimes 1 \otimes 1 \otimes 1 \otimes 1
\end{array} \\[.75em]
P_\pm = \ha (1 \pm \Ga) \\[.75em]
\Ga^+_{\al \be} = P_+ \Ga_\al \Ga_\be \mbox{ in } GL(64,\mathbb{R})
\end{array}
\s\s\;\;\;\;
\begin{array}{l}
\!\!\!\!\!\! \mbox{18 Standard Model generators in } spin(3,11) \\[.25em]
\begin{array}{lcl}
T^\om_{\mu \nu} \!\!&\!\! = \!\!&\!\! \sqrt{2} \, \Ga^+_{\mu \nu} \\[.25em]
T^W_1 \!\!&\!\! = \!\!&\!\! \Ga^+_{5 , 8} - \Ga^+_{6 , 7} \\
T^W_2 \!\!&\!\! = \!\!&\!\! \Ga^+_{5 , 7} + \Ga^+_{6 , 8} \\
T^W_3 \!\!&\!\! = \!\!&\!\! - \Ga^+_{5 , 6} + \Ga^+_{7 , 8} \\[.25em]
T^g_1 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 12} - \Ga^+_{10 , 11} \\
T^g_2 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 11} + \Ga^+_{10 , 12} \\
T^g_3 \!\!&\!\! = \!\!&\!\! -\Ga^+_{9 , 10} + \Ga^+_{11 , 12} \\
T^g_4 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 14} - \Ga^+_{10 , 13} \\
T^g_5 \!\!&\!\! = \!\!&\!\! \Ga^+_{9 , 13} + \Ga^+_{10 , 14} \\
T^g_6 \!\!&\!\! = \!\!&\!\! \Ga^+_{11 , 14} - \Ga^+_{12 , 13} \\
T^g_7 \!\!&\!\! = \!\!&\!\! \Ga^+_{11 , 13} + \Ga^+_{12 , 14} \\
T^g_8 \!\!&\!\! = \!\!&\!\! \fr{1}{\sqrt{3}} ( - \Ga^+_{9 , 10} - \Ga^+_{11 , 12} + 2 \, \Ga^+_{13 , 14} ) \\[.25em]
T^Y \!\!&\!\! = \!\!&\!\! \fr{\sqrt{3}}{\sqrt{5}} ( \Ga^+_{5 , 6} + \Ga^+_{7 , 8} ) \\
& &\!\! + \fr{2}{\sqrt{15}} ( \Ga^+_{9 , 10} + \Ga^+_{11 , 12} + \Ga^+_{13 , 14} )
\end{array} \\[.5em]
\ps \in 64^{+\mathbb{R}}_S
\end{array}
$$
<<tiddler HideTags>>$$
\begin{array}{c}
\mbox{Real } Cl(4,12) \mbox{ basis elements in } GL(256,\mathbb{R}) \\[.25em]
\begin{array}{lcrccccccccccccccc}
\Ga_1 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \\
\Ga_2 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_3 \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \\
\Ga_4 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_5 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_6 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_7 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_8 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \\
\Ga_9 \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{10} \!\!&\!\!\! = \!\!\!&\!\! \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{11} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{12} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{13} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{14} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{15} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_3 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\Ga_{16} \!\!&\!\!\! = \!\!\!&\!\! i \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_2 \!\!&\!\!\! \otimes \!\!\!&\!\! \si_1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \!\!&\!\!\! \otimes \!\!\!&\!\! 1 \\
\end{array}
\end{array}
\s
\begin{array}{l}
\mbox{120 generators in } spin(4,12) \\[.25em]
\s \mbox{91 in } spin(3,11) \\[.25em]
\s \s \mbox{6 for } \om \mbox{ in } spin(3,1) \\[.25em]
\s \s \mbox{45 in } spin(10) \\[.25em]
\s \s \s \mbox{12 for } W, B, g \\[.25em]
\s \s \s \mbox{3 for } W', Z' \\[.25em]
\s \s \s \mbox{30 for colored } X \mbox{ bosons} \\[.25em]
\s \s \mbox{40 for } e\ph \mbox{ frame (4)} \times \mbox{Higgs (10) } \\[.25em]
\s \mbox{1 for Peccei-Quinn } w \mbox{ in } spin(1,1) \\[.25em]
\s \mbox{8 for } e\th \mbox{ ''axion'' frame (4)} \times \mbox{Higgs (2)}\\[.25em]
\s \mbox{20 for more } X \mbox{ bosons} \\[.25em]
\mbox{128 generators in } 128_S^{+\mathbb{R}} \mbox{ of } spin(4,12) \\[.25em]
\s \mbox{64 for SM fermions in } 64_S^{+\mathbb{R}} \mbox{ of } spin(3,11) \\[.25em]
\s \mbox{64 for ''mirror'' fermions, with opposite } w \\[.25em]
\end{array}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
\begin{array}{rcrccc}
\ga_1 \!\!&\!\!=\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_2 \!\!&\!\!=\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_3 \!\!&\!\!=\!\!&\!\! i \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_4 \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\end{array}
\s\;\;\;
\begin{array}{l}
Cl(1,3) \subset \mathbb{C}(4) \\[.25em]
\ga_{ab} = \ha ( \ga_a \ga_b - \ga_b \ga_a) \in spin(1,3) \\[.25em]
\ep = \ga_1 \ga_2 \ga_3 \ga_4 = -i \, \si_3 \otimes 1 \\[.25em]
P_{L/R} = \ha (1 \pm i \, \ep)
\end{array}
\s\;\;\;
\begin{array}{rcl}
\La_A \!\!&\!\! = \!\!&\!\!
\lb
\matrix{
0 & 0 \\
0 & \la_A
}
\rb \\
\!\!&\!\! \in \!\!&\!\! \mathbb{C}(4)\vp{A^{\big(}} \\
\end{array}
$$
$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \s\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R - \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c}
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$
$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
{\small
\begin{array}{rcrccc}
\ga_{23} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{13} \!\!&\!\!=\!\!&\!\! i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{12} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_{14} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{24} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{34} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
}
\;\;\;
\begin{array}{l}
{\small
\si_1 \!=\! \lb \begin{matrix} 0 & 1 \cr 1 & 0 \cr \end{matrix} \rb
\;
\si_2 \!=\! \lb \begin{matrix} 0 & -i \cr i & 0 \cr \end{matrix} \rb
\;
\si_3 \!=\! \lb \begin{matrix} 1 & 0 \cr 0 & -1 \cr \end{matrix} \rb
}
\\[1em]
P_{L/R} = \ha (1 \pm \si_3) \otimes 1 \s P_L= \lb \begin{matrix} 1 & 0 \cr 0 & 0 \cr \end{matrix} \rb
\end{array}
\;\;\;
g^A \La_A =
{\small
\lb
\matrix{
0 & 0 & 0 & 0 \\
0 & g^3 \!+\! \fr{1}{\sqrt{3}} g^8 \! & g^1\!-\!ig^2 & g^4\!-\!ig^5 \\
0 & g^1\!+\!ig^2 & \! -g^3 \!+\! \fr{1}{\sqrt{3}} g^8 & g^6\!-\!ig^7 \\
0 & g^4\!+\!ig^5 & g^6\!+\!ig^7 & \fr{-2}{\sqrt{3}} g^8
}
\rb
}
$$
$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \s\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R - \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c}
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$
$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ {\textstyle \frac{1}{2}} \om_\mu^{\p{\mu}bc} {\textstyle \frac{1}{2}} T^\om_{bc}
+ W_\mu^{\p{\mu}I} T^W_I
+ B_\mu^Y T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\} \vp{A_{\Big(}}
$$ $$
{\small
\begin{array}{rcrccc}
\ga_{23} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{13} \!\!&\!\!=\!\!&\!\! i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{12} \!\!&\!\!=\!\!&\!\! -i \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\ga_{14} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \cr
\ga_{24} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\ga_{34} \!\!&\!\!=\!\!&\!\! \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \cr
\end{array}
}
\;\;\;
\begin{array}{l}
{\small
\si_1 \!=\! \lb \begin{matrix} 0 & 1 \cr 1 & 0 \cr \end{matrix} \rb
\;
\si_2 \!=\! \lb \begin{matrix} 0 & -i \cr i & 0 \cr \end{matrix} \rb
\;
\si_3 \!=\! \lb \begin{matrix} 1 & 0 \cr 0 & -1 \cr \end{matrix} \rb
}
\\[1em]
P_{L/R} = \ha (1 \pm \si_3) \otimes 1 \s P_L= \lb \begin{matrix} 1 & 0 \cr 0 & 0 \cr \end{matrix} \rb
\end{array}
\;\;\;
g^A \La_A =
{\small
\lb
\matrix{
0 & 0 & 0 & 0 \\
0 & g^3 \!+\! \fr{1}{\sqrt{3}} g^8 \! & g^1\!-\!ig^2 & g^4\!-\!ig^5 \\
0 & g^1\!+\!ig^2 & \! -g^3 \!+\! \fr{1}{\sqrt{3}} g^8 & g^6\!-\!ig^7 \\
0 & g^4\!+\!ig^5 & g^6\!+\!ig^7 & \fr{-2}{\sqrt{3}} g^8
}
\rb
}
$$
$$
\begin{array}{rlcl}
spin(1,3) \;\;\;\; \!\!&\!\! T^\om_{ab} \!\!&\!\! = \!\!&\!\! 1 \otimes 1 \otimes \ga_{ab} \s\; \in \mathbb{C}(32) \\[.5em]
su(2)_L \;\;\;\; \!\!&\!\! T^W_I \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_I \otimes P_L \\[.5em]
su(3) \;\;\;\; \!\!&\!\! T^g_A \!\!&\!\! = \!\!&\!\! \fr{i}{2} \La_A \otimes 1 \otimes 1 \\[.5em]
u(1)_Y \;\;\;\; \!\!&\!\! T^Y \!\!&\!\! = \!\!&\!\! 1 \otimes \fr{i}{2} \si_3 \otimes P_R - \fr{i}{2} \,{\small \mbox{diag}(1,-\fr{1}{3},-\fr{1}{3},-\fr{1}{3})} \otimes 1 \otimes 1 \\[.25em]
\!\!&\!\! \!\!&\!\! = \!\!&\!\! \fr{i}{2} \,{\small \mbox{diag}(-1,0,-1,-2,\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3},\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}.\fr{1}{3},\fr{4}{3},\fr{1}{3},-\fr{2}{3}) } \otimes 1 \\
\end{array}
\,
\ud{\ps} =
\lb
\begin{array}{c}
\nu \\ e \\ u^r \\ d^r \\ u^g \\ d^g \\ u^b \\ d^b
\end{array}
\rb
\begin{array}{c}
\! \left\{
\,
\lb
\begin{array}{c}
u_{L}^{r\wedge} \\
u_{L}^{r\vee} \\
u_{R}^{r\wedge} \\
u_{R}^{r\vee} \\
\end{array}
\rb
\right.
\\[4em]
\end{array}
\, \in 32^\mathbb{C} \vp{A_{\Big(}}
$$
$$
\mbox{complex structure:} \;\; i \mapsto
\lb
\matrix{
0 & -1 \\
1 & 0 }
\rb
= -i \si_2 , \;\; u^{r\wedge}_L \mapsto
\lb
\matrix{
u^{r\wedge}_{Lr} \\
u^{r\wedge}_{Li} }
\rb
\;\;
\Longrightarrow \; T \in \mathbb{R}(64), \; \ud{\ps} \in 64^\mathbb{R}
$$
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<embed src="talks/TED08/images/G_tahiti_620.jpg" width="827" height="620"></embed>
</center></html>@@
The Maurer-Cartan connection is like the [[Lie group bundle]] connection, but for a [[principal bundle]]. The relevant ''principal Lie group bundle'' is a fiber bundle with $n$ dimensional base manifold, $M$, and $n$ dimensional Lie group, $G$, as typical fiber and structure group acting on the fiber from the right -- it is a principal bundle. The structure group action (and maybe the structure group) differs from that of a Lie group bundle. Like for a Lie group bundle, a bijective identity section, $g_I(x)$, maps base manifold points to group/fiber elements. The ''Maurer-Cartan connection'', $\f{M}$, is the connection such that the covariant derivative of the identity section is horizontal,
$$
0 = \f{\na} g_I = \f{d} g_I - g_I \f{M}
$$
which gives
$$
\f{M} = g_I^- \f{d} g_I
$$
The [[exterior derivative]] of an inverse element comes from
$$
0 = \f{d} \lp g g^- \rp = g \f{d} g^- + \lp \f{d} g \rp g^-
$$
and is $\f{d} g^- = - g^- \lp \f{d} g \rp g^-$. So the Maurer-Cartan connection satisfies the ''Maurer-Cartan equation'',
$$
\f{d} \f{M} = \f{d} \lp g_I^- \lp \f{d} g_I \rp \rp = \lp \f{d} g_I^- \rp \lp \f{d} g_I \rp = - \f{M} \f{M} = -\f{M} \times \f{M}
$$
and the related principal bundle curvature vanishes,
$$
\ff{F} = \f{d} \f{M} + \f{M} \f{M} = 0
$$
Interestingly, the Maurer-Cartan connection may also be related to the vielbein arising from [[Lie group geometry]],
$$
\f{M} = \f{\xi_R^B} T_B = g_I^- \f{d} g_I = \f{e^B} T_B
$$
In this way, the Maurer-Cartan connection is a frame as well as a connection for a Lie group.
To reiterate: The Maurer-Cartan connection is a connection for a $2n$ dimensional principal bundle. This is just something new to try playing with. It's not the [[Maurer-Cartan form]], which is a 1-form over the $n$ dimensional Lie group manifold.
The ''Maurer-Cartan form'' (//''M-C form''//) is a [[Lieform]] field, $\f{\cal I}(x) = \f{dx^i} {\cal I}_i{}^A T_A$, defined over a [[Lie group]] manifold, corresponding to the local geometry of the Lie group action. Every two Lie group elements, $g',g \in G$, can be related by the right action of another Lie group element,
$$
g' = g \, \th(g,g')
$$
If the two elements are near each other they are related by a [[tangent vector]], $\ve{v}$, on the Lie group manifold,
$$
g' \simeq g + t \ve{v} \f{d} g
$$
and by the corresponding
$$
\th(g,g') \simeq 1 + t \, \Th(g,v) = 1 + t \ve{v} \, \f{\cal I}(g)
$$
in which, after plugging these two into the first equation, the ''Maurer-Cartan form'' is
$$
\f{\cal I} = g^- \f{d} g
$$
a [[Lie algebra]] valued [[1-form]] over the Lie group manifold.
More precisely, the Maurer-Cartan form arises in [[Lie group geometry]] from the equation for the right action vector field,
$$
\ve{\xi_A^R} \f{d} g = g T_A
$$
The inverse matrix of this vector field's components gives the components of the Maurer-Cartan form, ${\cal I}_i{}^A = \lp \xi^R_i \rp^A$, which are the same as the components of the natural Lie group geometry vielbein. However, unlike the vielbein, which is a set of 1-forms, the Maurer-Cartan form is Lie algebra valued. The components of the Maurer-Cartan form may be found from its defining equation, or equivalently by solving for the right action vector fields and inverting. With a nondegenerate [[Killing form]], the calculation of the components is usually done via
$$
\lp \xi^R_i \rp^A = \lp \et^{AB} T_B, g^- \pa_i g(x) \rp
$$
The M-C form is a sort of identity map from vectors on the Lie group manifold to Lie algebra elements, $\f{\cal I} = \f{{\cal I}^A} T_A = \f{\xi_R^A} T_A$ -- specifically, it maps right acting vector fields at a point to their corresponding Lie algebra element, $\ve{\xi^R_A} \f{\cal I} = T_A$. The M-C form provides an explicit isomorphism from vector valued fields to Lie algebra valued fields, $\ve{v} \f{\cal I} = v \in {\mathfrak{g}}$, and thus acts as the anchor of a [[Lie algebroid]]. If the equivalence between Lie algebra elements and their corresponding right acting vector fields is taken seriously, then the resulting ''Ehresmann-Maurer-Cartan [[vector valued form]]'' (//''E-M-C VVF''//), $\f{\ve{\cal I}}$, is nothing but the [[identity projection|vector projection]] on the Lie group manifold:
$$
\f{\ve{\cal I}} = \f{\xi_R^A} \ve{\xi^R_A} = \f{dx^i} \ve{\pa_i}
$$
Looking at it in a weird way, the E-M-C VVF is the [[Ehresmann connection]] for a fiber bundle with the Lie group manifold as fiber and a single point as the base.
There is a manifold [[diffeomorphism]], $x \mapsto y = y_h(x)$, corresponding to any choice of right acting group element, $h \in G$, according to $g(y_h(x)) = R_h g(x) = g(x) h$. The [[pullback]] of the M-C form under this diffeomorphism is
$$
R_h^* \f{\cal I} = \f{dx^i} \fr{\pa y_h^j}{\pa x^i} {\cal I}_j{}^A(y_h(x)) T_A = \f{dx^i} h^- g^-(x) \pa_i g(x) h = h^- \f{\cal I}(x) h
$$
which is often taken as a defining property of the M-C form. Similarly, the pullbacks of the M-C form under the left and adjoint actions are $L_h^* \f{\cal I} = h \f{\cal I} h^-$ and $A_h^* \f{\cal I} = h \f{\cal I} h^-$. Note that this relates to the fact that, as the identity projection, the E-M-C VVF is invariant under any diffeomorphism, $\phi^* \f{\ve{\cal I}} = \f{\ve{\cal I}}$.
A formula for the exterior derivative of an inverse element, $\f{d} g^- = - g^- \lp \f{d} g \rp g^-$, comes from
$$
0 = \f{d} \lp g g^- \rp = g \f{d} g^- + \lp \f{d} g \rp g^-
$$
So the exterior derivative of the Maurer-Cartan form is
$$
\f{d} \f{\cal I} = \f{d} \lp g_I^- \lp \f{d} g_I \rp \rp = \lp \f{d} g_I^- \rp \lp \f{d} g_I \rp = - \f{\cal I} \f{\cal I} = -\f{\cal I} \times \f{\cal I} = - \ha \lb \f{\cal I} , \f{\cal I} \rb
$$
This gives the ''Maurer-Cartan equation'',
$$
\begin{eqnarray}
0 &=& \f{d} \f{\cal I} + \ha \lb \f{\cal I}, \f{\cal I} \rb = \ff{\cal F} \\
0 &=& \f{d} \f{{\cal I}^C} + \ha \f{{\cal I}^A} \f{{\cal I}^B} C_{AB}{}^C
\end{eqnarray}
$$
which gives vanishing [[curvature]] for the M-C form. Of course, the [[FuN curvature]] of the E-M-C VVF (which is the identity projection) also vanishes, $\ff{\ve{\cal F}} = - \ha \lb \f{\ve{\cal I}}, \f{\ve{\cal I}} \rb_L = 0$.
[>img[images/person/Max Tegmark.jpg]]Homepage: http://space.mit.edu/home/tegmark/index.html
*Location: MIT
Selected work:
*[[The Mathematical Universe|papers/0704.0646v1.pdf]]
**Computable Universe Hypothesis
**Complexity based measure on space of possible mathematics
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/tori.png" height="400">
</td></tr>
<tr><td>
Maximal tori inside <SPAN class="math">SU(3) \otimes SU(2)_L \otimes U(1)_Y \otimes Spin(1,3)</SPAN>
</td></tr>
</table>
</center></html>
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<img src="talks/CSUF09/images/tortoise.png">
</center></html>
@@
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<img src="talks/Zuck09/images/Maximilian Tortoise.png" width="659" height="480">
</center></html>
[[Maxwell's equations]], $\f{J} = \ve{\de} \f{d} \f{A}$, for the $U(1)$ [[connection]], $\f{A}$, (the [[electromagnetic field]]) in a [[rest frame]] with no current reduce to
$$
0 = d \times (d \times A) = (d \cdot d) \, A - d \, (d \cdot A)
$$
in which the connection 1-form was replaced by its associated [[Cl(1,3)]] [[Clifford vector]] field, $A = \ve{e}\f{A} = \ga^\mu A_\mu$, and the [[codifferential]] and [[exterior derivative]] were replaced by $d=\ga^\mu \pa_\mu$ and [[Clifford algebra]] multiplication. This equation has positive and negative energy ''plane wave solutions'',
$$
A = \ep' e^{\mp i p_\mu x^\mu} = \ep' e^{\mp i (E t - p^\va x^\va)}
$$
(with it understood that we take the real part of this complex expression), in which $p$ is the [[momentum]] and $\ep' = \ep'{}^\mu \ga_\mu $ is the constant ''[[polarization]] vector''. (Note that we are assuming we are in a flat spacetime [[rest frame]].) With this ansatz, Maxwell's equations reduce to
$$
0 = - (p \cdot p) \, \ep' + p \, (p \cdot \ep')
$$
in which the momentum vector is $p = \ga^\mu p_\mu = \ga^0 E - \ga^\va p^\va$. This is solved if
$$
0 = p \cdot p = E^2 - p_s \cdot p_s \s \text{and} \s 0 = (p \cdot \ep') = E \, \ep'{}^0 - p_s \cdot \ep_s
$$
in which $p_s = p^\va \si_\va$ and $\ep_s = \ep'{}^\va \si_\va = \ep{}^\va \si_\va$ are vectors in the spatial Clifford algebra, [[Cl(3)]], and we use the [[Pauli matrices]] for our Clifford representation. If we restrict to [[Weyl gauge|Maxwell solutions]], $A_0 = \ph = 0$, this implies $\ep'{}^0=0$, and the polarization must be spatial and orthogonal to the wave momentum. We can span this space of solutions by choosing two spatial orthonormal [[polarization]] basis vectors, $\ep_1$ and $\ep_2$, perpendicular to $p_s$. These are then combined into right and left handed complex circular polarization basis vectors, $\ep_\pm = \ep_1 \pm i \ep_2$. Using these, a basis of positive-energy electromagnetic plane wave solutions is $A_\pm = \ep'_\pm e^{- i p_\mu x^\mu}$, with $E = \sqrt{p_s p_s}$. The corresponding electric and magnetic fields, as spatial Clifford vector and bivector fields, from
$$
E \ga^0 + B = F = d \times A = \ga^0 \pa_t a_s + d_s \times a_s
$$
are
\begin{eqnarray}
E_\pm &=& - \pa_t a = i E \, \ep_\pm e^{- i p_\mu x^\mu} \sim \fr{1}{\sqrt{2}} E \, ( \ep_1 \sin{px} \mp \ep_2 \cos{px}) \\
B_\pm &=& d_s \times a = i p_s \ep_\pm e^{- i p_\mu x^\mu} = \pm E \si \ep_\pm e^{- i p_\mu x^\mu} \sim \pm \fr{1}{\sqrt{2}} E \, \si (\ep_1 \cos{px} \pm \ep_2 \sin{px} ) \\
\end{eqnarray}
(with their real parts) in which we trust the reader not to confuse the energy with the electric field.
The resulting ''Poynting vector'' is
$$
P_\pm = E_\pm \times B_\pm = \pm \ha E^2 (\pm \si \ep_1 \ep_2) = \ha E \, p_s
$$
The ''spin angular momentum density of an electromagnetic wave'' is the spatial vector
\begin{eqnarray}
s_\pm &=& \si ( E_\pm \times A_\pm ) = \ha \si E \, ( \ep_1 \sin{px} \mp \ep_2 \cos{px}) \times ( \ep_1 \cos{px} \pm \ep_2 \sin{px}) \\
&=& \ha \si E \, ( \pm \ep_1 \ep_2 ) = \pm \ha p_s \\
\end{eqnarray}
From the [[Yang-Mills Lagrangian]], the [[Euler-Lagrange equation]] for a $U(1)$ [[connection]] (the [[electromagnetic field]]) in curved [[spacetime]] is $\f{d} * \ff{F}=0$. With a ''current'' 1-form source, $\f{J} = \rh \f{e}^0 + \f{j}$, with $\rh$ the ''charge density'' and $\f{j}$ the ''spatial current density'', we have ''Maxwell's equations in curved spacetime'',
$$
\f{d} * \ff{F} = * \f{J} \s \f{d} \ff{F} = 0
$$
This gives the equation of motion for the electromagnetic field, $\ve{\de} \ff{F} = \f{J}$, using the [[codifferential]] operator -- either for $\f{A}$ or for the electric and magnetic fields via the [[curvature]], $\ff{F} = \f{d} \f{A} = \f{E} \f{e}^0 + \ff{B}$.
In a [[rest frame]] with negative spatial [[Minkowski metric]], we have $\ve{\de} = \ve{d} = \ve{e}{}^\mu \pa_\mu$, and so we have
\begin{eqnarray}
\f{J} = \rh \f{e}^0 + \f{j} &=& \lp \ve{e}^0 \pa_t + \ve{d_s} \rp \lp \f{E} \f{e}^0 + \ff{B} \rp = \ve{d} \ff{F} \\
0 &=& \lp \f{e}^0 \pa_t + \f{d}{}_s \rp \lp \f{E} \f{e}^0 + \ff{B} \rp = \f{d} \ff{F}
\end{eqnarray}
using [[vector-form algebra]], and thus we get ''Maxwell's equations in flat spacetime'' (or just ''Maxwell's equations''),
$$
\begin{array}{rclcrcl}
\rh \!\!&\!\!=\!\!&\!\! \ve{d_s} \f{E}
& \s &
0 \!\!&\!\!=\!\!&\!\! \f{d}{}_s \ff{B} \\
\f{j} \!\!&\!\!=\!\!&\!\! - \pa_t \f{E} + \ve{d_s} \ff{B}
& \s &
0 \!\!&\!\!=\!\!&\!\! \pa_t \ff{B} + \f{d}{}_s \f{E}
\end{array}
$$
It is also possible to write Maxwell's equations as a second order PDE, $\f{J} = \ve{\de} \f{d} \f{A} = \ve{\de} \f{d} \f{A}$, for the [[electromagnetic field]], $\f{A} = \ph \f{e}^0 + \f{a}$. In flat spacetime,
\begin{eqnarray}
\rh \f{e}^0 + \f{j} &=& \lp \ve{e}{}^0 \pa_t + \ve{d_s} \rp \lp \f{e}^0 \pa_t + \f{d}{}_s \rp \lp \ph \f{e}^0 + \f{a} \rp \\
&=& \lp \ve{e}{}^0 \pa_t + \ve{d_s} \rp \lp \f{e}^0 \pa_t \f{a} + \f{d}{}_s \ph \f{e}^0 + \f{d}{}_s \f{a} \rp \\
&=& \pa^2_t \f{a} - \pa_t \f{d}{}_s \ph - \f{e}^0 \ve{d_s} \pa_t \f{a} + \f{e}^0 \ve{d_s} \f{d}{}_s \ph + \ve{d_s} \f{d}{}_s \f{a}
\end{eqnarray}
which separates into
\begin{eqnarray}
\rh &=& - \pa_t \ve{d_s} \f{a} + \ve{d_s} \f{d}{}_s \ph \\
\f{j} &=& \pa^2_t \f{a} - \pa_t \f{d}{}_s \ph + \ve{d_s} \f{d}{}_s \f{a}
\end{eqnarray}
This can also be obtained using the [[Clifford covariant derivative]] and converting to [[Cl(1,3)]] [[Clifford algebra]] valued operators and fields, in which Maxwell's equations are $J = \na \times (\na \times A)$. In flat spacetime this becomes
$$
J = d \times (d \times A) = (d \cdot d) A - d (d \cdot A)
$$
using [[Clifford product identities]], with $A = \ga^\mu A_\mu$ and $d=\ga^\mu \pa_\mu$.
Maxwell's equations are invariant under a [[gauge transformation]], $\f{A} \to \f{A}' = \f{A} - \f{d} \Ph$. This gauge freedom can be ''fixed'', either by hand or by the [[BRST technique]], resulting in a restriction, such as $0=(\f{e}^1\f{e}^2\f{e}^3)\f{A}$ -- which implies $A_0 = \ph =0$, ''Weyl gauge''.
In a vacuum with no current, these equations have plane wave [[Maxwell solutions]].
//check signs in the above equations//
[>img[images/person/Michael Edwards.jpg]]
*Location: Santa Cruz
Persuaded me that the [[FuN derivative]] was worth thinking about in the context of [[Ehresmann connection]]s, and wrote a nice [[synopsis|papers/BRST2-6.pdf]].
The ''Minkowski metric'', $\et_{\mu \nu}$, is a $4 \times 4$ diagonal matrix with unit magnitude real entries. The choice of ''signature'' is somewhat arbitrary, and mostly a matter of taste. For positive time and negative space signature $(p=1, q=3)$, generally preferred by field theorists,
\[ \et_{\mu \nu}
=
\lb \begin{matrix}
1& 0& 0& 0\\
0& -1& 0& 0\\
0& 0& -1& 0\\
0& 0& 0& -1
\end{matrix} \rb
= \cases {
1&\text{if $\mu=\nu=0$}\cr
-1&\text{if $\mu=\nu>0$}}
= \et^{\mu \nu}
\]
while for negative time and positive space signature $(p=3,q=1)$, generally preferred by relativists,
\[ \eta_{\mu \nu}
=
\lb \begin{matrix}
-1& 0& 0& 0\\
0& 1& 0& 0\\
0& 0& 1& 0\\
0& 0& 0& 1
\end{matrix} \rb
= \cases {
-1&\text{if $\mu=\nu=0$}\cr
1&\text{if $\mu=\nu>0$}}
= \et^{\mu \nu}
\]
(This "matrix" notation is a bit sloppy, since $\et_{\mu \nu}$ is not really a matrix but just a collection of indexed coefficients.) All computations should be signature ambivalent. When they're not, the signature may be accommodated by including $\et_{00}= \pm 1$ in expressions.
The ''generalized Minkowski metric'' is $n \times n$ — accommodating extra spatial dimensions (of signature $-\et_{00}$).
The Minkowski metric may be used to raise or lower label [[indices]], such as in "$\ga^\al = \et^{\al \be} \ga_\be$" and "$v_\mu = \et_{\mu \nu} v^\nu$". The Minkowski metric with one index raised or lowered is just the Kronecker delta, $\et^\al_\be = \et^{\al \ga} \et_{\ga \be} = \de^\al_\be$.
Inline {{{monospaced text}}} with no editing commands executed inside the brackets.
Inline <html>{{{</html>monospaced text<html>}}}</html> with no editing commands executed inside the brackets.
{{{
monospaced
blocks
(useful for source code)
}}}
By putting "<html>{{{</html>" and "<html>}}}</html>" on their own lines.
For a [[division algebra]], such as the [[octonion]]s, the ''Moufang identities'' are
1. $z(x(zy)) = ((zx)z)y$
2. $x(z(yz)) = ((xz)y)z$
3. $(zx)(yz) = (z(xy))z = z((xy)z)$
<<tiddler HideTags>>
To incorporate triality, fermions need minimum of $4$ weight coordinates, $\{\om_t, \om_S, h, q\}$, with helicity, $h$, and $\om_t = 4 \om_S h q$.
Projective representation of charge, parity, time, and triality conjugations in these coords:
<html><center><table class="gtable">
<tr border=none>
<td border=none>
$$
\ba{cc}
C \sim \lb \ba{cccc}
- & & & \\[-4pt]
& + & & \\[-4pt]
& & + & \\[-4pt]
& & & - \\
\ea \rb
\;\; &
P \sim \lb \ba{cccc}
- & & & \\[-4pt]
& + & & \\[-4pt]
& & - & \\[-4pt]
& & & + \\
\ea \rb
\\[30pt]
T \sim \lb \ba{cccc}
- & & & \\[-4pt]
& - & & \\[-4pt]
& & + & \\[-4pt]
& & & + \\
\ea \rb
\;\; &
t \sim \ha \! \lb \ba{cccc}
+ & - & + & + \\[-4pt]
+ & - & - & - \\[-4pt]
+ & + & + & - \\[-4pt]
+ & + & - & + \\
\ea \rb
\ea
$$
</td>
<td>$\s \s$</td>
<td>
<img src="talks/ToE24/CPTt2.png" width="280px">
</td>
</tr>
</table></center></html>$CPTt$ produces 24-cell of three fermion generations, gen I cube red, gen II cube green, gen II cube blue, related by triality, black.
Note: gen II and gen III fermion charges only make physical sense after transformation by $t^2$ and $t$.
!Colors Used
*@@bgcolor(#8cf): #8cf - Background blue@@ -- popup
*@@bgcolor(#18f): #18f - Top blue@@ -- header top
*@@bgcolor(#04b): #04b - Mid blue@@ -- header
*@@bgcolor(#014):color(#fff): #014 - Bottom blue@@ -- commands
*@@bgcolor(#ffc): #ffc - Bright yellow@@ --
*@@bgcolor(#fe8): #fe8 - Highlight yellow@@ --
*@@bgcolor(#db4): #db4 - Background yellow@@ --
*@@bgcolor(#841): #841 - Border yellow@@ --
*@@bgcolor(#703):color(#fff): #703 - Title red@@ --
*@@bgcolor(#866): #866 - Subtitle grey@@
*@@bgcolor(#888): #888 - footer@@
*@@bgcolor(#999): #999 - dark grey@@
*@@bgcolor(#bbb): #bbb - tag2 grey@@
*@@bgcolor(#ccc): #ccc - grey@@
*@@bgcolor(#ddd): #ddd - tag1 grey@@
*@@bgcolor(#eee): #eee - light grey@@
*@@bgcolor(#ff0): #ff0 - error1@@
*@@bgcolor(#f00): #f00 - error2@@
*@@bgcolor(#eef): #ff0 - cascade1@@
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The ''Nieh-Yan density'' is an invariant closed 4-form,
$$N = \f{d} \lp \f{e} \cdot \ff{T} \rp = \ff{T} \cdot \ff{T} - \ff{R} \cdot \f{e} \f{e}$$
This may make for an interesting KK action term...
Another invariant cloese 4-form is the ''Pontryagin density'',
$$P = \li \ff{R} \ff{R} \ri$$
mentioned in http://arxiv.org/abs/gr-qc/0603134
''Symmetries'' of the [[action]] produce conserved currents.
If there exists a variation, $\Phi \mapsto \Phi + \de \Phi$, such that the [[action]] is invariant, $\de S = 0$, then the Lagrangian must vary by a [[divergence]], $\de{\cal L} = \mathrm{div} \lp \ve{\La} \de \Ph \rp$. Working this out, using integration by parts,
$$
\mathrm{div} \lp \ve{\La} \de \Ph \rp = \de{\cal L} = \fr{\pa {\cal L}}{\pa \Phi} \de \Ph + \fr{\pa {\cal L}}{\pa (\f{\pa} \Phi)} \f{\pa} \de \Ph
= \fr{\pa {\cal L}}{\pa \Phi} \de \Ph - \mathrm{div} \lp \fr{\pa {\cal L}}{\pa (\f{\pa} \Phi)} \rp \de \Ph + \mathrm{div} \lp \fr{\pa {\cal L}}{\pa (\f{\pa} \Phi)} \de \Ph \rp
$$
One can define the related ''current''
$$
\vec{J} = \fr{\pa {\cal L}}{\pa (\f{\pa} \Phi)} - \ve{\La}
$$
and use the [[Euler-Lagrange equation]] with the above to see that the current is conserved,
$$
\mathrm{div} \vec{J} = - \fr{\pa {\cal L}}{\pa \Phi} + \mathrm{div} \lp \fr{\pa {\cal L}}{\pa (\f{\pa} \Phi)} \rp = 0
$$
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*[[On a Covartiant Formulation of the Barbero-Immirzi Connection|papers/070134.pdf]]
**New paper by [[Carlo Rovelli]] et. al. on a cleaner way of getting a $su(2)$ gravity connection from a $spin(4)$ connection.
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<html><center>
<img SRC="talks/IfA11/images/Glyphtionary.png" height=300px>
</center></html>
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<img SRC="talks/IfA11/images/Glyphtionary.png" height=300px>
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<<tiddler HideTags>>@@display:block;text-align:center;<html><center>
<!-- <embed src="talks/Perimeter07/anim/e8tour (om up)/p1.png" width="608" height="609"></embed> -->
</center></html>@@
<<tiddler HideTags>>''Cartan subalgebra'': $\;\;\;
C = C^a T_a = \ha \om^S \, T^\om_{12} + \ha \om^T \, T^\om_{34} + W \,T^W_3 + Y \,T^Y + g^3 \, T^g_3 + g^8 \, T^g_8 \vp{A_{\big(}}$
$\s\s\s\s\s\s\;
\subset \; spin(1,3) \,\oplus\, su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \vp{A_{\big(}} $
(Spanned by a maximal commuting set of six gravitational and Standard Model generators.)
''Weight vectors'', $\ps_\al$ : $ \;\;\; C \, \ps_\al = i \al \, \ps_\al = i C^a \al_a \ps_\al \vp{A^{\big(}_{\big(}} \s $ ''Weights'': $\;\;\; \al_a = \{\om^S,\om^T,W,Y,g^3,g^8\} $
''Root vectors'', $V_\al$, are eigenvectors in the adjoint: $ \;\;\; [ C , V_\al ] = i \al \, V_\al = i C^a \al_a V_\al $
''Interactions''
$$
\begin{array}{rcl}
{\rm boson} \; + \; {\rm fermion} \!\!&\!\!=\!\!&\!\! {\rm fermion} \\
V_\al \, \ps_\be \!\!&\!\!=\!\!&\!\! \ps_\de \quad \Leftrightarrow \quad \al + \be = \de \\[.5em]
{\rm pf:} \;\;\;\; C \, V_\al \, \ps_\be \!\!&\!\!=\!\!&\!\! [C, V_\al ] \ps_\be + V_\al C \, \ps_\be
= i \al \, V_\al \ps_\be + i \be \, V_\al \ps_\be = i (\al + \be) V_\al \ps_\be \\[1em]
{\rm boson} \; + \; {\rm boson} \!\!&\!\!=\!\!&\!\! {\rm boson} \\
[ V_\al , V_\be ] \!\!&\!\!=\!\!&\!\! V_\de \quad \Leftrightarrow \quad \al + \be = \de \\[.5em]
{\rm pf:} \;\;\;\; [C , [ V_\al , V_\be ] ] \!\!&\!\!=\!\!&\!\!
- [V_\al,[V_\be,C]] - [V_\be,[C,V_\al]]
= i (\al + \be) [ V_\al , V_\be ]
\end{array}
$$
<html><center>
<table class="gtable">
<tr>
<td>
<table class="gtable">
<tr><td>
weight vectors
</td></tr><tr><td>
weights
</td></tr>
</table>
</td>
<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>
<td>
<table class="gtable">
<tr><td>
eigenvectors
</td></tr><tr><td>
eigenvalues
</td></tr>
</table>
</td>
<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>
<td>
<table class="gtable">
<tr><td>
states
</td></tr><tr><td>
quantum numbers
</td></tr>
</table>
</td>
<td><SPAN class="math">\;\; \longleftrightarrow \;\;</SPAN></td>
<td>
<table class="gtable">
<tr><td>
particles
</td></tr><tr><td>
charges
</td></tr>
</table>
</td>
</tr>
</table>
</center></html>
authors: [[Laurent Freidel]], J. Kowalski--Glikman, A. Starodubtsev
arxiv: http://arxiv.org/abs/gr-qc/0607014
locally: [[0607014|papers/0607014.pdf]]
abstract:
Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom.
*Nice summation of BF.
*Three topological terms: Euler, Pontryagin, Nieh-Yan
*Interesting treatment of Imirzi parameter
*Ahh, their main idea seems to be putting in a term and identifying it as a spinning particle. I guesss that's nice, but what's the big deal? Anyway, it's a cute way of putting in point particle matter, rather than QFT matter fields. The matter action is an integral along the parameterized path of the particle.
*Path integral to Wilson line correspondence
*matter as gravitational singularity.
This looks like the result you should get if you include the Dirac action and insert an arbitrarily boosted point particle solution for the Dirac field.
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Pati-Salam.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">spin(4) \,\oplus\, spin(6) \,\,\oplus\,\, 4 \otimes 4</SPAN>
</td></tr>
</table>
</center></html>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">so(6) + so(4) + 4 \!\times\!4 + \bar{"}</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/CSUF09/images/PSElectric.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">so(6) + so(4) + 4 \!\times\!4 + \bar{"}</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/CSUF09/images/PSE6Cox.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>@@display:block;text-align:center;[img[images/png/pati-salam table.png]]@@$$
\big( SO(3,1) + 4\times4 + SU(2)_L + SU(2)_R \big) + \big( U(1) + SU(3) \big)
$$
The three [[trace]]less, Hermitian, ''Pauli matrices'', $\si_A$, are
$$
\begin{array}{ccc}
\sigma_{1}=\left[\begin{array}{cc}
0 & 1\\
1 & 0\end{array}\right] & \sigma_{2}=\left[\begin{array}{cc}
0 & -i\\
i & 0\end{array}\right] & \sigma_{3}=\left[\begin{array}{cc}
1 & 0\\
0 & -1\end{array}\right]\end{array}
$$
The [[cross product|antisymmetric bracket]] of any two gives
$$
\si_A \times \si_B = \ha \lp \si_A \si_B - \si_B \si_A \rp = i \ep_{ABC} \si_C
$$
with $\ep_{ABC}$ the [[permutation symbol]]. The second Pauli matrix can be related to the [[skew]] operator, $\ep=-i\si_2$. The symmetric product gives
$$
\si_A \cdot \si_B = \ha \lp \si_A \si_B + \si_B \si_A \rp = \de_{AB} 1
$$
Thus the Pauli matrices are a basis [[Clifford matrix representation]] of [[Cl(3)]]. The product of the three Pauli matrices is the [[pseudoscalar]],
$$
\si = \si_1 \si_2 \si_3 = \left[\begin{array}{cc}
i & 0\\
0 & i\end{array}\right]
= i 1
$$
with the identity matrix, $1 = \si_0$, often referred to as the (non-traceless) ''zero-eth Pauli matrix''.
Using the zero-eth Pauli matrix we can define a set of four ''extended Pauli matrices'', $\si_\mu$, and the ''conjugate Pauli matrices'', $\bar{\si}_\mu = \si^\mu$, with the index raised using the $(1,3)$-signature [[Minkowski metric]]. Explicitly, we have $\si_\mu = (1,\si_1,\si_2,\si_3) = \bar{\si}^\mu$ and $\bar{\si}_\mu = (1,-\si_1,-\si_2,-\si_3) = \si^\mu$. Note that this goes against the most popular particle physics convention, which usually flips the signs of the $\si_A$ in the extended Pauli matrices. We could "fix" this by having the three Pauli matrices defined with up indices, or using different [[Dirac matrices]], but prefer to maintain index consistency.
A useful identity is $\bar{\si}_\mu{}^*= \si_2 \si_\mu \si_2 = - \ep \, \si_\mu \ep$, which is consistent with the definition $\bar{\si}_\mu = - \ep \, \si^T_\mu \ep$, of the [[2D matrix conjugate|determinant]].
A ''Pauli spinor'', $\ch$, is represented by a column of 2 complex (or complex [[Grassmann|Grassmann number]]) numbers -- the ''spin up'' and ''spin down'' components, multiplying eigenvectors of the [[spin operator]], the up and down ''Pauli basis spinors'',
$$
\ch^\wedge = \lb \ba{c}1 \\ 0 \ea \rb
\s
\ch^\vee = \lb \ba{c} 0 \\ 1 \ea \rb
\s
\ch = \lb \ba{c}a \\ b \ea \rb = a \ch^\wedge + b \ch^\vee
\s
\{ a, b \} \in \mathbb{C} \text{ or } \mathbb{G}
$$
A Pauli spinor is a two complex dimensional [[spinor]] of [[Cl(3)]] -- a unitary [[representation space]] of [[su(2)]]. Under a [[spatial rotation]], a ''rotated Pauli spinor'', using the [[Pauli matrices]] , $\si_\pi$, as the basis vector matrix representatives of $Cl(3)$ and the corresponding bivectors, is
$$
\ch' = U_a \, \ch =
\lp \si_0 \cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}} \rp
\ch
$$
in which $U_a \in$ [[SU(2)]] is [[unitary]].
As a unitary representation space, Pauli spinors are quaternionic: there is an [[antiunitary]] operator, $j = -i \si_2 K$, with $K$ complex conjugation, commuting with the $i \si_\pi \in su(2)$, that squares to negative the identity, $j^2 = -1$. The corresponding other [[quaternion]]ic operators, commuting with $su(2)$, are $i = i$ and $k = \si_2 K$.
<<tiddler HideTags>>
@@display:block;text-align:center;[img[images/png/standard model and gravity 4s.png]]@@
[>img[images/person/Peter Michor.jpg]]Homepage: http://www.mat.univie.ac.at/~michor/
*Location: Austria
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Michor_P/0/1/0/all/0/1
Big on [[natural]]ness.
I think I annoyed him with my questions.
Selected work:
*[[Topics in Differential Geometry|papers/Topics in Differential Geometry.pdf]]
**Great introductory book to the tough stuff, and seems to contain the best from his other work
**Haar measure on p124
**[[Hodge dual]] p209
**[[FuN derivative]] p215
**grab theorem from p226 on relationship between [[Lie algebra]] and holonomy algebra
**homogeneous space p230
**gauge transformations p240
**[[FuN curvature]] p245
**covariant derivative p248, p260
**[[holonomy]] p251
**characteristic classes p263 (Wow!)
**Hamiltonian mechanics p283
*[[The Frolicher-Nijenhuis Bracket|papers/The Frolicher-Nijenhuis Bracket.pdf]]
*[[Remarks on the Frolicher-Nijenhuis Bracket|papers/Remarks on the Frolicher-Nijenhuis Bracket.pdf]]
*[[Gauge Theory for Fiber Bundles|papers/Gauge Theory for Fiber Bundles.pdf]]
*[[Natural Operations in Differential Geometry|papers/Natural Operations in Differential Geometry.pdf]]
<<tiddler HideTags>>Pirated from GS&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]:
\begin{eqnarray}
E &=& B + \Ps = \ha b^{\al\be} \ga^{\small (16)+}_{\al\be} + \ps^a Q^+_a \\
&\in& so(16) + S^{\small (16)+} = {\rm Lie}(E8)
\end{eqnarray}
[[Lie brackets|Lie algebra]] between generators (structure constants):
$$
{\small
\begin{array}{rcl}
\big[ \ga^{\small (16)+}_{\al \be}, \ga^{\small (16)+}_{\ga \de} \big] &=& 2 \, \big\{ - \et_{\al \ga} \ga^{\small (16)+}_{\be \de} + \et_{\al \de} \ga^{\small (16)+}_{\be \ga} + \et_{\be \ga} \ga^{\small (16)+}_{\al \de} - \et_{\be \de} \ga^{\small (16)+}_{\al \ga} \big\}^{\p{(}} \\
\big[ \ga^{\small (16)+}_{\al \be}, Q^+_a \big] &=& \big( \ga^{\small (16)+}_{\al \be} \big)^b{}_c \big( Q^+_a \big)^c Q^+_b = \ga^{\small (16)+}_{\al \be} Q^+_a \\
\big[ Q^+_a, Q^+_b \big] &=& - \big( {\ga^{\small (16)+}}^{\al \be} \big)_{ab} \ga^{\small (16)+}_{\al \be}
\end{array}
}
$$
${\rm Lie}(E8)$ brackets act as multiplication between $120$ dimensional [[Cl(16)]] [[Clifford|Clifford algebra]] [[bivector|Clifford basis elements]]s, $B$, and positive [[chiral]], $128$ dim column [[spinor]]s, $\Ps$:
$$
\begin{array}{rcll}
\lb B_1, B_2 \rb \!\!&\!\!=\!\!&\!\! B_1 B_2 - B_2 B_1 & \in \; so(16) \\
\lb B, \Ps \rb \!\!&\!\!=\!\!&\!\! B^+ \, \Ps & \in \; S^{\small (16)+} \\
\lb \Ps_1, \Ps_2 \rb \!\!&\!\!=\!\!&\!\! -\Ps_1^\da \Ga^+ \Ps_2 & \in \; so(16)_{{\p{\big(}}_{\p{(}}}
\end{array}
$$
<<tiddler HideTags>>
The group of spacetime reflections, $R_u v = v_\perp - v_\parallel$, is the ''pin group'', $G = Pin(1,3) \subset Cl(1,3)$.
We choose chiral matrix representative basis vectors, $\ga_\mu$, of $Cl(1,3)$,
$$
u = u^\mu \ga_\mu =
{\small
\lb \begin{array}{cccc}
0 & 0 & u^0-u^3 & -u^1+iu^2 \\
0 & 0 & -u^1-iu^2 & u^0+u^3 \\
u^0+u^3 & u^1-iu^2 & 0 & 0 \\
u^1+iu^2 & u^0-u^3 & 0 & 0
\end{array} \rb
}
\s \s
\ga = \ga_0 \ga_1 \ga_2 \ga_3 =
{\small
\lb \begin{array}{cccc}
-i & 0 & 0 & 0 \\
0 & -i & 0 & 0 \\
0 & 0 & i & 0 \\
0 & 0 & 0 & i
\end{array} \rb
}
$$
A reflection, represented by $\Pi(R_u) = U = u \ga \in Cl(1,3)$, through a vector, $u$, acts on vectors (in the vector representation space, $v \in V$) via ''adjoint action'', and on ''spinors'' (in the spinor representation space) via ''left action'',
$$
v' = R_u \, v = U \, v \, U^- \!= (u \ga) \, v \, (u \ga)^- \!= - u v u^- \!= v_\perp - v_\parallel \s \s
\ps' = R_u \, \ps = U \, \ps = (u \ga) \, \ps
$$
Even numbers of reflections generate the ''spacetime spin group'', $Spin(1,3) \subset Pin(1,3)$, of Lorentz transformations, with $Spin^+(1,3) \subset Spin(1,3)$ the component connected to the identity. Explicitly:
$$
Pin(1,3) = Spin(1,3) \rtimes \{1,T \} = Spin^+(1,3) \rtimes \{1,P,T,PT \}
\s
Spin(1,3) = Spin^+(1,3) \rtimes \{1, PT \}
$$
$P$ and $T$ are distinguished reflections, ''parity reversal'' and ''unitary time reversal'',
$$
U'_P = -(\ga_1 \ga)(\ga_2 \ga)(\ga_3 \ga) = \ga_0
\s \s
U'_T = \ga_0 \ga = \ga_1 \ga_2 \ga_3
\s \s
U'_{PT} = U'_P U'_T = \ga
$$
$G_{PT} = \{ \pm 1, \pm P, \pm T, \pm PT \} = Q_8$ is a finite subgroup of $Pin(1,3)$.
<<tiddler HideTags>>Work forwards, guess the answer, then work backwards.
Work forwards towards unification:
#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].
#Calculate its [[curvature]] to get the interactions.
#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.
#Correct [[standard model]] and gravitational interactions and charges from the curvature.
Guess the answer:
*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.
*One very large [[Lie group]] is a match!
Work backwards:
#All interactions from the [[structure|Lie algebra]] of this group, after symmetry breaking.
#Explains exactly what and why [[spinor]]s are.
#Gives three generations.
#Calculating particle masses (CKM) is a possibility.
Format blocks of CSS definitions as:
{{{
/***
Description and comments go here, surrounded by comment brackets.
Followed by CSS code, which will be displayed in a code block.
***/
/*{{{*/
div {color: #ff0000;}
/*}}}*/
}}}
That way the code will be run without the wikitext messing it up, and it will still be displayed nicely.
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/GraviGUT E8.png" width="420" height="420">
</td></tr><tr><td>
<SPAN class="math">E_{8(-24)} = spin(12,4) \,\,\oplus\,\, 128^\mathbb{R}_{S+}</SPAN>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">
\,
</SPAN></td>
<td>
<table class="gtable">
<tr><td>
<SPAN class="math">
\begin{array}{l}
\mbox{120 generators in } spin(12,4) \\[.25em]
\s \mbox{91 in } spin(11,3) \\[.25em]
\s \s \mbox{6 for } \om \mbox{ in } spin(1,3) \\[.25em]
\s \s \mbox{45 in } spin(10) \\[.25em]
\s \s \s \mbox{12 for } W, B, g \\[.25em]
\s \s \s \mbox{3 for } W', Z' \\[.25em]
\s \s \s \mbox{30 for colored } X \mbox{ bosons} \\[.25em]
\s \s \mbox{40 for } e\ph \mbox{ frame 4} \times \mbox{Higgs 10 } \\[.25em]
\s \mbox{1 for Peccei-Quinn } w \mbox{ in } spin(1,1) \\[.25em]
\s \mbox{8 for } e\th \mbox{ frame 4} \times \mbox{''axion'' Higgs 2}\\[.25em]
\s \mbox{20 for more } X \mbox{ bosons} \\[.25em]
\mbox{128 generators in } 128^\mathbb{R}_{S^+} \mbox{ of } spin(12,4) \\[.25em]
\s \mbox{64 for SM fermions in } 64^\mathbb{R}_{S^+} \mbox{ of } spin(11,3) \\[.25em]
\s \mbox{64 for ''mirror fermions'' in } 64^\mathbb{R}_{S^-} \\[.25em]
\end{array}
</SPAN>
</td></tr>
</table>
</td>
</tr>
</table>
</center>
</html>
http://arxiv.org/abs/quant-ph/0610204
author: Rafael Sorkin
*primacy of path integral history formulation
decent summary to look at for quantizing perturbed BF
http://arxiv.org/pdf/hep-th/0610194
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Strong interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3} \vp{{\big(}^{\big(}}</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\[1em]
{\small
\lb \begin{array}{cccccc}
& \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
\!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
& & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
& & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & \\[-.5em]
& & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
& & & & - \fr{1}{\sqrt{3}} g^8 &
\end{array} \rb
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\!
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
} \\[-1.5em]
\end{array}
$$
<html><center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r} \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">\s\s\s</SPAN></td>
<td>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Strong interaction.png" width="310" height="310">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3}</SPAN>
</td></tr>
</table>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\[1em]
{\small
\lb \begin{array}{ccc}
\fr{i}{2} g^3 \!+\! \fr{i}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
& \!\! - \! \fr{i}{2} g^3 \!+\! \fr{i}{2 \sqrt{3}} g^8 \!\! & \\[-.5em]
& & - \fr{i}{\sqrt{3}} g^8
\end{array} \rb
\lb
\begin{array}{c}
1 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\!
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ 0 \\ 0
\end{array}
\rb
}
\end{array}
$$<html><center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r} \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">\s\s\s</SPAN></td>
<td>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Strong interaction.png" width="310" height="310">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3}</SPAN>
</td></tr>
</table>
</td>
</tr>
</table>
</center>
</html>
http://arxiv.org/abs/gr-qc/0404088
*looks to be an excellent treatment of issues with path integral treatment of GR
*justifies 3+1 dimensions from algebraic topology
<<tiddler HideTags>>
Quaternions, $e_a$, and $-1$, so $8$ elements:
$$
\{ \pm 1, \pm e_1, \pm e_2, \pm e_3 \} \; \in \; G = Q_8
$$
Quaternion multiplication does not necessarily commute,
$$
e_1 e_2 = e_3 = - e_2 e_1
$$
The ''center'' of a group is the subgroup of elements that commute with everything,
$$
\{ +1, -1 \} \; \in \; C \subset G
$$
Group multiplication table, $e_a e_b = M_{ab}{}^c e_c$, with further multiplications by $-1$ implied:
| $1$ | $e_1$ | $e_2$ | $e_3$ |
| $e_1$ | $-1$ | $e_3$ | $-e_2$ |
| $e_2$ | $-e_3$ | $-1$ | $e_1$ |
| $e_3$ | $e_2$ | $-e_1$ | $-1$ |
Group representation by Pauli matrices:
$$
1 = e_0 = \si_0 = {\small \lb \ba{cc} 1 & 0 \\ 0 & 1 \ea \rb} \s
e_1 = -i \si_1 = {\small \lb \ba{cc} 0 & -i \\ -i & 0 \ea \rb} \s
e_2 = -i \si_2 = {\small \lb \ba{cc} 0 & -1 \\ 1 & 0 \ea \rb} \s
e_ 3= -i \si_3 = {\small \lb \ba{cc} -i & 0 \\ 0 & i \ea \rb} \s
$$
<<tiddler HideTags>>
How to extend $G_{CPT}$ non-trivially to act on generation-triples of fermions? Introduce the quaternion triality generator:
$$
t = -\ha (1 + e_1 + e_2 + e_3) \s \s t^- = t^2 = \ha (-1 + e_1 + e_2 + e_3) \s \s t^3=1
$$
This can act via the adjoint to cycle imaginary quaternions,
$$
\text{ad}_t e_1 = t e_1 t^-\! = e_2 \s \s \text{ad}_t e_2 = t e_2 t^-\! = e_3 \s \s \text{ad}_t e_3 = t e_3 t^-\! = e_1
$$
Whether we include the adjoint generator, $\{\text{ad}_t, e_2, e_3\}$, or the $t$ generator itself, $\{t, e_2, e_3\}$, these generators produce the ''binary tetrahedral group'', $2T$, of order $24$, which is a semi-direct product of subgroups $Q_8$ and $\mathbb{Z}_3 = \{1,\text{ad}_t,\text{ad}_t^-\} = \{1,t,t^-\}$.
Combining this triality generator with $C \sim i e_1$, $P \sim -K e_3$, and $T \sim -K e_2$, we draw several conclusions:
*Triality and $CPT \sim i$ commute.
*The ''PTt Group'' generated by $\{\text{ad}_t, P, T\}$ is $G_{PTt} = 2T$.
*The ''CPTt Group'' generated by $\{\text{ad}_t, C, P, T\}$ is $G_{CPTt} = 2T \circ D_4$, of order $96$, the central product of the binary tetrahedral group, $2T$, and the dihedral group, $D_4 = \{\pm 1, \pm i , \pm K, \pm i K \}$, of order $8$, with a shared central $\mathbb{Z}_2 = \{1,-1\}$.
*Three generations of fermions can be described by three sets of triality-related biquaternionic spinors,
$$
\ps^I = \ps_1 \s \s \ps^{II} = \text{ad}_t \ps_2 = t \ps_2 t^- \s \s \ps^{III} = \text{ad}_t^2 \ps_2 = t^- \ps_3 t
$$
Note these imply the complex structure in our biquaternionic spinors is triality invariant.
| !rank | !group | !a.k.a. | !dim | !name |
| $r$ | $A_r$ | $SU(r+1)$ | $r(r+2)$ | [[special unitary group]] |
| $r$ | $B_r$ | $SO(2r+1)$ | $r(2r+1)$ | odd [[special orthogonal group]] |
| $r$ | $C_r$ | $Sp(2r)$ | $r(2r+1)$ | symplectic group |
| $r>2$ | $D_r$ | $SO(2r)$ | $r(2r-1)$ | even [[special orthogonal group]] |
| $2$ | $G_2$ | | $14$ | G2 |
| $4$ | $F_4$ | | $52$ | F4 |
| $6$ | $E_6$ | | $78$ | [[E6]] |
| $7$ | $E_7$ | | $133$ | E7 |
| $8$ | $E_8$ | | $248$ | [[E8]] |
<<tiddler HideTags>>
"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."
-- Hermann Nicolai
/***
|Name|RearrangeTiddlersPlugin|
|Source|http://www.TiddlyTools.com/#RearrangeTiddlersPlugin|
|Version|2.0.0|
|Author|Eric Shulman|
|OriginalAuthor|Joe Raii|
|License|http://www.TiddlyTools.com/#LegalStatements|
|~CoreVersion|2.1|
|Type|plugin|
|Description|drag tiddlers by title to re-order story column display|
adapted from: http://www.cs.utexas.edu/~joeraii/dragn/#Draggable
changes by ELS:
* hijack refreshTiddler() instead of overridding createTiddler()
* find title element by className instead of elementID
* set cursor style via code instead of stylesheet
* set tooltip help text
* set tiddler "position:relative" when starting drag event, restore saved value when drag ends
* update 2006.08.07: use getElementsByTagName("*") to find title element, even when it is 'buried' deep in tiddler DOM elements (due to custom template usage)
* update 2007.03.01: use apply() to invoke hijacked core function
* update 2008.01.13: only hijack core function once. (allows for dynamic loading of plugin via bookmarklet)
* update 2008.10.19: added onclick popup menu with 'move to top' and 'move to bottom' commands
* update 2010.11.30: use story.getTiddler()
***/
//{{{
if (Story.prototype.rearrangeTiddlersHijack_refreshTiddler===undefined) {
Story.prototype.rearrangeTiddlersHijack_refreshTiddler = Story.prototype.refreshTiddler;
Story.prototype.refreshTiddler = function(title,template)
{
this.rearrangeTiddlersHijack_refreshTiddler.apply(this,arguments);
var theTiddler = this.getTiddler(title); if (!theTiddler) return;
var theHandle;
var children=theTiddler.getElementsByTagName("*");
for (var i=0; i<children.length; i++) if (hasClass(children[i],"title")) { theHandle=children[i]; break; }
if (!theHandle) return theTiddler;
Drag.init(theHandle, theTiddler, 0, 0, null, null);
theHandle.style.cursor="move";
theHandle.title="drag title to re-arrange tiddlers, click for more options..."
theTiddler.onDrag = function(x,y,myElem) {
if (this.style.position!="relative")
{ this.savedstyle=this.style.position; this.style.position="relative"; }
y = myElem.offsetTop;
var next = myElem.nextSibling;
var prev = myElem.previousSibling;
if (next && y + myElem.offsetHeight > next.offsetTop + next.offsetHeight/2) {
myElem.parentNode.removeChild(myElem);
next.parentNode.insertBefore(myElem, next.nextSibling);//elems[pos+1]);
myElem.style["top"] = -next.offsetHeight/2+"px";
}
if (prev && y < prev.offsetTop + prev.offsetHeight/2) {
myElem.parentNode.removeChild(myElem);
prev.parentNode.insertBefore(myElem, prev);
myElem.style["top"] = prev.offsetHeight/2+"px";
}
};
theTiddler.onDragEnd = function(x,y,myElem) {
myElem.style["top"] = "0px";
if (this.savedstyle!=undefined)
this.style.position=this.savedstyle;
};
theHandle.onclick=function(ev) {
ev=ev||window.event;
var p=Popup.create(this); if (!p) return;
var b=createTiddlyButton(createTiddlyElement(p,"li"),
"\u25B2 move to top of column ","move this tiddler to the top of the story column",
function() {
var t=story.getTiddler(this.getAttribute("tid"));
t.parentNode.insertBefore(t,t.parentNode.firstChild); // move to top of column
window.scrollTo(0,ensureVisible(t));
return false;
});
b.setAttribute("tid",title);
var b=createTiddlyButton(createTiddlyElement(p,"li"),
"\u25BC move to bottom of column ","move this tiddler to the bottom of the story column",
function() {
var t=story.getTiddler(this.getAttribute("tid"));
t.parentNode.insertBefore(t,null); // move to bottom of column
window.scrollTo(0,ensureVisible(t));
return false;
});
b.setAttribute("tid",title);
Popup.show();
ev.cancelBubble=true; if (ev.stopPropagation) ev.stopPropagation(); return(false);
};
return theTiddler;
}
}
/**************************************************
* dom-drag.js
* 09.25.2001
* www.youngpup.net
**************************************************
* 10.28.2001 - fixed minor bug where events
* sometimes fired off the handle, not the root.
**************************************************/
var Drag = {
obj:null,
init:
function(o, oRoot, minX, maxX, minY, maxY) {
o.onmousedown = Drag.start;
o.root = oRoot && oRoot != null ? oRoot : o ;
if (isNaN(parseInt(o.root.style.left))) o.root.style.left="0px";
if (isNaN(parseInt(o.root.style.top))) o.root.style.top="0px";
o.minX = typeof minX != 'undefined' ? minX : null;
o.minY = typeof minY != 'undefined' ? minY : null;
o.maxX = typeof maxX != 'undefined' ? maxX : null;
o.maxY = typeof maxY != 'undefined' ? maxY : null;
o.root.onDragStart = new Function();
o.root.onDragEnd = new Function();
o.root.onDrag = new Function();
},
start:
function(e) {
var o = Drag.obj = this;
e = Drag.fixE(e);
var y = parseInt(o.root.style.top);
var x = parseInt(o.root.style.left);
o.root.onDragStart(x, y, Drag.obj.root);
o.lastMouseX = e.clientX;
o.lastMouseY = e.clientY;
if (o.minX != null) o.minMouseX = e.clientX - x + o.minX;
if (o.maxX != null) o.maxMouseX = o.minMouseX + o.maxX - o.minX;
if (o.minY != null) o.minMouseY = e.clientY - y + o.minY;
if (o.maxY != null) o.maxMouseY = o.minMouseY + o.maxY - o.minY;
document.onmousemove = Drag.drag;
document.onmouseup = Drag.end;
Drag.obj.root.style["z-index"] = "10";
return false;
},
drag:
function(e) {
e = Drag.fixE(e);
var o = Drag.obj;
var ey = e.clientY;
var ex = e.clientX;
var y = parseInt(o.root.style.top);
var x = parseInt(o.root.style.left);
var nx, ny;
if (o.minX != null) ex = Math.max(ex, o.minMouseX);
if (o.maxX != null) ex = Math.min(ex, o.maxMouseX);
if (o.minY != null) ey = Math.max(ey, o.minMouseY);
if (o.maxY != null) ey = Math.min(ey, o.maxMouseY);
nx = x + (ex - o.lastMouseX);
ny = y + (ey - o.lastMouseY);
Drag.obj.root.style["left"] = nx + "px";
Drag.obj.root.style["top"] = ny + "px";
Drag.obj.lastMouseX = ex;
Drag.obj.lastMouseY = ey;
Drag.obj.root.onDrag(nx, ny, Drag.obj.root);
return false;
},
end:
function() {
document.onmousemove = null;
document.onmouseup = null;
Drag.obj.root.style["z-index"] = "0";
Drag.obj.root.onDragEnd(parseInt(Drag.obj.root.style["left"]), parseInt(Drag.obj.root.style["top"]), Drag.obj.root);
Drag.obj = null;
},
fixE:
function(e) {
if (typeof e == 'undefined') e = window.event;
if (typeof e.layerX == 'undefined') e.layerX = e.offsetX;
if (typeof e.layerY == 'undefined') e.layerY = e.offsetY;
return e;
}
};
//}}}
//''Shows DefaultTiddlers + most recently modified tiddlers as default when any TiddlyWiki or adaptation is first loaded.''//
//To use, copy this tiddler's contents to a new tiddler on your site and tag it "systemConfig".//
{{{
var num = 3;
var ignore_tags = ['systemConfig', 'systemTiddlers', 'plugin', 'system'];
function in_array(item, arr){for(var i=0;i<arr.length;i++)if(item==arr[i])return true};
function get_parent(tiddler){while(tiddler && in_array('comments', tiddler.tags)) tiddler=store.fetchTiddler(tiddler.tags[0]);return tiddler};
function unique_list(list){var l=[];for(i=0;i<list.length;i++)if(!in_array(list[i], l))l.push(list[i]);return l};
function get_recent_tiddlers(){
var tiddlers = store.getTiddlers('modified');
var names = store.getTiddlerText("DefaultTiddlers").readBracketedList();
var ignore_tiddlers = [];
for(var i=0; i<ignore_tags.length; i++)
ignore_tiddlers=ignore_tiddlers.concat(store.getTaggedTiddlers(ignore_tags[i]));
for(var i=tiddlers.length-1; i>=0; i--) {
if(in_array('comments', tiddlers[i].tags)) {
var t = get_parent(tiddlers[i]);
if(t)names.push(t.title)
}
else if(!in_array(tiddlers[i], ignore_tiddlers))
names.push(tiddlers[i].title);
}
return unique_list(names).slice(0, num);
}
var names = get_recent_tiddlers();
_restart = restart
restart = function() {
if(window.location.hash) _restart();
else story.displayTiddlers(null,names);
}
}}}
<<tiddler HideTags>>
One particularly interesting way $e8$ can be broken down:
\begin{eqnarray}
e8 &=& e6 + su(3) + 54 \! \times \! 3 \\
&=& so(1,9) + u(1) + 32 + su(3) + 54 \! \times \! 3\\
&=& so(1,3) + su(2) + su(2) + u(1) + 4 \! \times \! 8 + u(1) + 32 + su(3) + 54 \! \times \! 3 \\
&\to& {\scriptsize \ha} \om + W + B + \fr{1}{4} e \ph + G + 3 \! \times \! \ps + X
\end{eqnarray}
How does this $e8$ breakdown relate to [[e8 triality decomposition]]?
\begin{eqnarray}
e8 &=& so(1,7) + so(8) + 3 \! \times \! 8 \! \times \! 8 \\
&=& so(1,3) + so(4) + 4 \! \times \! 4 + so(6) + so(2) + 6 \! \times \! 2 + 3 \! \times \! 8 \! \times \! 8 \\
&=& so(1,3) + su(2) + su(2) + 4 \! \times \! 4 + su(4) + u(1) + 6 \! \times \! 2 + 3 \! \times \! 8 \! \times \! 8
\end{eqnarray}
/***
''Name:'' ReferencesPlugin
''Author:'' Garrett Lisi
''Description:'' Places a comma separated list of referring tiddlers at the bottom of each tiddler -- replacing the "references" command bar button.
''Installation:'' Copy this tiddler, change the [[StyleSheet]] to set the references class style, and add a line in the [[ViewTemplate]].
''Code:''
***/
/*{{{*/
config.macros.references = {};
config.macros.references.handler = function(place,macroName,params,wikifier,paramString,tiddler)
{
var references = store.getReferringTiddlers(tiddler.title);
if(references.length>0)
{
// createTiddlyText(place,"\xAB ");
createTiddlyLink(place,references[0].title,true);
}
for(var r=1; r<references.length; r++)
if(references[r].title != tiddler.title)
{
createTiddlyText(place,", ");
createTiddlyLink(place,references[r].title,true);
}
}
/*}}}*/
<<tiddler HideTags>>''Cartan subalgebra'': $\quad C=C^a T_a \;\; \subset \; {\rm Lie}(G) \vp{|_(}$
Built from a maximal commuting set of $R$ generators,
$$
\big[ T_a, T_b \big] = T_a T_b - T_b T_a = 0 \qquad \forall \quad 1 \le a,b \le R
$$
''Root vectors'', $V_\be$, are eigenvectors of $C$ in the Lie bracket,
$$
[ C , V_\be ] = \al_\be V_\be = \sum_a i C^a\al_{a\be} V_\be
$$
''Roots'', $\al_{a\be}$, are the eigenvalue coefficients. The pattern of roots in $R$ dimensions corresponds to the Lie algebra,
$$
[ V_\be , V_\ga ] = V_\de \quad \Leftrightarrow \quad \al_{\be} + \al_{\ga} = \al_{\de}
$$
''Weight vectors'' and ''weights'' are eigenvectors and eigenvalue coefficients of $C$ acting on some representation space,
$$
C \, V_\be = \al_\be V_\be
$$
Weight vectors are particles, weights are their quantum numbers.
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Riemann curvature]] gives the ''Ricci curvature'',
$$
\f{R}{}_m = \ve{\pa_k} \ff{R}^k{}_m = \ve{\pa_k} \f{dx^i} \f{dx}^j \ha R_{ij}{}^k{}_m
= \f{dx}^j R_{ij}{}^i{}_m
$$
with the components of the ''Ricci curvature tensor'' equaling a partial contraction of the Riemann curvature tensor,
$$
R_{jm} = R_{ij}{}^i{}_m = 2 \pa_{\lb i \rd} \Ga^i{}_{\ld j \rb m} + 2 \Ga^i{}_{\lb i \rd l} \Ga^l{}_{\ld j \rb m}
$$
This tensor is symmetric if the [[torsion]] vanishes, $R_{jm}=R_{mj}$. In terms of the [[tangent bundle spin connection|tangent bundle connection]], the Ricci curvature is
$$
\f{R}{}_\al = \f{e^\de} R_{\de \al} = \ve{e_\be} \ff{R}^\be{}_\al
= \f{dx^j} 2 \lp e_\be \rp^i \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp
$$
with coefficients $R_{\de \al} = R_{\be \al}{}^\be{}_\de$. If the spin connection is torsionless, the Ricci curvature tensor may also be written as
\begin{eqnarray}
R_{\ga \al} &=& R_{\be \ga}{}^\be{}_\al = 2 \pa_{\lb \be \rd} w_{\ld \ga \rb}{}^\be{}_\al + 2 w_{\lb \be \ga \rb}{}^\ep w_\ep{}^\be{}_\al - 2 w_{\lb \be \rd}{}^{\ep \be} w_{\ld \ga \rb}{}_{\ep \al} \\
&=& 2 \pa_{\lb \be \rd} w_{\ld \ga \rb}{}^\be{}_\al + w_{\be \ga}{}^\ep w_\ep{}^\be{}_\al - w_\be{}^{\ep \be} w_{\ga \ep \al}
\end{eqnarray}
The [[vector bundle curvature]] for a [[tangent bundle]] describes the local geometry of the base manifold. Applying the [[tangent bundle covariant derivative|tangent bundle connection]] twice, and taking the [[antisymmetric|index bracket]] part, gives the tangent bundle curvature,
$$
\na_{\lb i \rd} \na_{\ld j \rb} \ve{v} = \na_{\lb i \rd} \lp \pa_{\ld j \rb} v^k + \Ga^k{}_{\ld j \rb l} v^l \rp \ve{\pa_k}
= \lp \pa_{\lb i \rd} \Ga^k{}_{\ld j \rb l} + \Ga^k{}_{\lb i \rd m} \Ga^m{}_{\ld j \rb l} \rp v^l \ve{\pa_k}
= \ha R_{ij}{}^k{}_l v^l \ve{\pa_k}
$$
The components of the ''Riemann curvature'' (//''tangent bundle curvature''//), $\ff{R}^k{}_l = \ha \f{dx^i} \f{dx^j} R_{ij}{}^k{}_l$, are the components of the conventional Riemann curvature tensor after rearrangement, $R_{ij}{}^k{}_l \leftrightarrow R^k{}_{lij}$. (//The non-conventional Riemann index placement used here instead follows the conventional index placement for curvature tensors.//) The components are:
$$
R_{ij}{}^k{}_l = 2 \pa_{\lb i \rd} \Ga^k{}_{\ld j \rb l} + 2 \Ga^k{}_{\lb i \rd m} \Ga^m{}_{\ld j \rb l}
$$
Written with fewer indices, this is:
$$
\ff{R}^k{}_l = \f{d} \f{\Ga}^k{}_l + \f{\Ga}^k{}_m \f{\Ga}^m{}_l
$$
A different expression for the tangent bundle curvature, $\ff{R}^\be{}_\al = \ha \f{dx^i} \f{dx^j} R_{ij}{}^\be{}_\al$, may also be written in terms of the [[tangent bundle spin connection|tangent bundle connection]],
$$
\na_{\lb i \rd} \na_{\ld j \rb} \ve{v} = \na_{\lb i \rd} \lp \pa_{\ld j \rb} v^\be + w_{\ld j \rb}{}^\be{}_\al v^\al \rp \ve{e_\be}
= \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp v^\al \ve{e_\be}
= \ha R_{ij}{}^\be{}_\al v^\al \ve{e_\be}
$$
with components:
$$
R_{ij}{}^\be{}_\al = 2 \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + 2 w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al
$$
Or, with fewer indices:
$$
\ff{R}^\be{}_\al = \ff{F}^\be{}_\al = \f{d} \f{w}^\be{}_\al + \f{w}^\be{}_\ga \f{w}^\ga{}_\al
$$
If the spin connection is [[torsion]]less, the Riemann curvature tensor may also be written, using the [[frame]], as
$$
R_{ \de \ga}{}^\be{}_\al = \lp e_\de\rp^i \lp e_\ga\rp^j R_{ij}{}^\be{}_\al
= 2 \pa_{\lb \de \rd} w_{\ld \ga \rb}{}^\be{}_\al + 2 w_{\lb \de \ga \rb}{}^\ep w_\ep{}^\be{}_\al - 2 w_{\lb \de \rd}{}^{\ep \be} w_{\ld \ga \rb}{}_{\ep \al}
$$
in which $\pa_\al = \lp e_\al \rp^i \pa_i$ and $w_\al{}^\ga{}_\be = \lp e_\al \rp^i w_i{}^\ga{}_\be$.
The Riemann curvature may alternatively be obtained from the [[tangent bundle holonomy]].
<<tiddler HideTags>>''Cartan subalgebra'': $\quad C=C^a T_a \;\; \subset \; {\rm Lie}(G) \vp{|_(}$
Built from a maximal commuting set of $R$ generators,
$$
\big[ T_a, T_b \big] = T_a T_b - T_b T_a = 0 \qquad \forall \quad 1 \le a,b \le R
$$
''Root vectors'', $T_\be$, are eigenvectors of $C$ in the Lie bracket,
$$
[ C , T_\be ] = \al_\be T_\be = \sum_a i C^a\al_{a\be} T_\be
$$
''Roots'', $\al_{a\be}$, are the eigenvalue coefficients. The pattern of roots in $R$ dimensions corresponds to the Lie algebra,
$$
[ T_\be , T_\ga ] = T_\de \quad \Leftrightarrow \quad \al_{\be} + \al_{\ga} = \al_{\de}
$$
''Weight vectors'' and ''weights'' are eigenvectors and eigenvalue coefficients of $C$ acting on some representation space,
$$
C \, \ps_\be = \al_\be \ps_\be
$$
Weight vectors are particle states, weights are their charge quantum numbers.
GUT unification can be done using the $SU(5)$ subalgebra of $SO(10)$. $SU(5)$ embeds in $SO(10)$ by multiplying the real part of a $SU(5)$ matrix by a $2 \times 2$ identity matrix and multiplying the imaginary part by the [[skew]] matrix, which acts as the [[complex structure]]. But there is a (probably) better way. $SO(10)$ has $SU(2) \times SU(2) \times SU(4)$ as a maximal subalgebra. The $SU(2) \times SU(2)$ is $SO(4)$ and the $SU(4)$ is $SO(6)$, so the $SO(4) \times SO(6)$ are diagonal blocks of the $SO(10)$. The Dynkin diagram surgery for this reduction is the removal of the central $SU(2)$ node.
Ref:
*Howard Georgi's book, p283:
**http://www.amazon.com/gp/reader/0738202339/ref=sib_dp_pop_toc/104-9709999-3726336?ie=UTF8&p=S00E#
**Also see p169 of his recent talk on GUT's for the 16 complex dim spinor rep:
***[[GUTs|papers/yt100sym_georgi.pdf]]
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">so(10) + 16 + \bar{16}</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/CSUF09/images/E6Electric.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
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The ''three dimensional [[special unitary group]]'' (//''special unitary group of order two''//), $G = SU(2)$, is the [[Lie group]] of [[unitary]] $2 \times 2$ complex matrices with unit [[determinant]]. Its elements, $g \in G$, may be parameterized and obtained by [[exponentiating|exponentiation]] [[su(2)]] [[Lie algebra]] generators,
$$
g(x) = e^{x^A T_A} = e^X
$$
with $X=x^A T_A \in su(2)$. It is possible to carry out this exponentiation explicitly, and do calculations in these coordinates. However, it is more instructive to convert to [[spherical coordinates]], with
$$
X = a^1 \sin(a^2) \cos(a^3) T_1 + a^1 \sin(a^2) \sin(a^3) T_2 + a^1 \cos(a^2) T_3
= \left[\begin{array}{cc}
i a^1 \cos(a^2) & i a^1 e^{i a^3} \sin(a^2)\\
i a^1 e^{-i a^3} \sin(a^2) & -i a^1 \cos(a^2)\end{array}\right]
$$
and perform the [[spectral decomposition|eigen]],
$$
X = U \La U^-
= \left[\begin{array}{cc}
- e^{i a^3} \sin(\fr{a^2}{2}) & e^{i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
\left[\begin{array}{cc}
- i a^1 & 0 \\
0 & i a^1 \end{array}\right]
\left[\begin{array}{cc}
- e^{-i a^3} \sin(\fr{a^2}{2}) & e^{-i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
$$
in order to exponentiate and get:
\begin{eqnarray}
g(a) &=& e^X = U e^\La U^-
= \left[\begin{array}{cc}
- e^{i a^3} \sin(\fr{a^2}{2}) & e^{i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right]
\left[\begin{array}{cc}
e^{- i a^1} & 0 \\
0 & e^{i a^1} \end{array}\right]
\left[\begin{array}{cc}
- e^{-i a^3} \sin(\fr{a^2}{2}) & e^{-i a^3} \cos(\fr{a^2}{2}) \\
\cos(\fr{a^2}{2}) & \sin(\fr{a^2}{2}) \end{array}\right] \\
&=&
\left[\begin{array}{cc}
\cos(a^1) + i \sin(a^1) \cos(a^2) & i \sin(a^1) e^{i a^3} \sin(a^2) \\
i \sin(a^1) e^{-i a^3} \sin(a^2) & \cos(a^1) - i \sin(a^1) \cos(a^2) \end{array}\right] \\
&=& \cos(a^1) \, 1 + \fr{\sin(a^1)}{a^1} X
\end{eqnarray}
This could have been found more easily by using $XX = - (a^1)^2$ to compute the exponential -- but this won't be true for general Lie groups, while the above method generalizes nicely.
The [[Lie group geometry]] is described by the left and right acting (right and left invariant) vector fields, and their dual 1-form fields. Over most of the group manifold, the [[Maurer-Cartan form]],
$$
\f{\cal I}(a) = g^-(a) \f{d} g(a) = \f{da^i} \lp\xi^R_i\rp^A T_A = \f{da^i} \lp e_i\rp^A T_A
$$
has components (best computed using Mathematica or something):
\begin{eqnarray}
\lp e_i\rp^A &=& \lp T^A, g^-(a) \pa_i g(a) \rp \\
&=& \ha
\left[\begin{array}{ccc}
\sin(a^2) \cos(a^3) & - \sin(a^2) \sin(a^3) & \cos(a^2) \\
\sin(a^1) \lp \cos(a^1) \cos(a^2) \cos(a^3) + \sin(a^1) \sin(a^3) \rp & \sin(a^1) \lp \sin(a^1) \cos(a^3) - \cos(a^1) \cos(a^2) \sin(a^3) \rp & - \ha \sin(2 a^1) \sin(a^2) \\
\ha \lp - \sin^2(a^1) \sin(2 a^2) \cos(a^3) + \sin(2 a^1) \sin(a^2) \sin(a^3) \rp & \sin(a^1) \sin(a^2) \lp \cos(a^1) \cos(a^3) + \sin(a^1) \cos(a^2) \sin(a^3) \rp & \sin^2(a^1) \sin^2(a^2)
\end{array}\right]
\end{eqnarray}
Identifying these as the [[frame]] components for the [[Lie group tangent bundle geometry]], using the su(2) [[Killing form]], $g_{AB} = -2 \de_{AB}$, gives the metric for the Lie group geometry,
$$
g_{ij}(a) = \lp e_i\rp^A g_{AB} \lp e_j \rp^B
=
\left[\begin{array}{ccc}
-8 & 0 & 0 \\
0 & -8 \sin^2(a^1) & 0 \\
0 & 0 & -8 \sin^2(a^1) \sin^2(a^2)
\end{array}\right]
$$
$SU(2) = Spin(3)$ may also be thought of as the group generated by the bivectors of the three dimensional Clifford algebra, [[Cl(3)]]. Under this representation, each group element, $g$, is a $Cl(3)$ scalar plus a bivector. This is also equivalent to representation by [[quaternion]]s. The ''unitary group'', $\left\{ U\in GL(C)\mid UU^{\da}=1\right\} $, corresponds to the unitary [[subgroup]] of the Clifford Algebra, $\left\{ U\in Cl\mid U\gamma_{0}\widetilde{U}=\gamma_{0}\right\}$, with $\widetilde{U}$ the [[Clifford reverse|Clifford conjugate]].
The ''eight dimensional [[special unitary group]]'' (//''special unitary group of order three''//), $G = SU(3)$, is the [[Lie group]] of [[unitary]] $3 \times 3$ complex matrices with unit [[determinant]]. Its elements, $g \in G$, may be parameterized and obtained by [[exponentiating|exponentiation]] the [[su(3)]] [[Lie algebra]] generators,
$$
g(x) = e^{x^i T_i}
$$
For [[loops|vector-form algebra]] and higher grade multivectors.
http://www.mimuw.edu.pl/~pwit/TOK/sem4/online/node9.html
*[[John Baez]] has a nice recent writeup from his course on quantization:
**[[path integrals|papers/w07week08a.pdf]]
The ''Schwarzschild solution'' gives the unique geometry of [[spacetime]] in the vicinity of an uncharged, non-rotating, spherically symmetric mass, $M$. This approximately describes spacetime around the sun, earth, or black holes. The solution is most concisely expressed by the [[frame]],
$$
\f{e} = \f{dt} \lp 1 - \fr{R_s}{r} \rp^\ha \ga_0 + \f{dr} \fr{1}{c} \lp 1 - \fr{R_s}{r} \rp^{-\ha} \ga_1
+ \f{d\th} \fr{r}{c} \ga_2 + \f{d\ph} \fr{r \sin{\th}}{c} \ga_3
$$
having diagonal frame matrix. The coordinates are $(x^0,x^1,x^2,x^3)=(t,r,\th,\ph)$ and have [[units]] $(T,L,0,0)$. The solution has a coordinate singularity at $r=R_S=\fr{2GM}{c^2}$, corresponding to the ''Schwarzchild radius'' -- the horizon beyond which light cannot escape.
A coordinate singularity can be avoided by using freely falling coordinates, such as those of Gullstrand-Painleve, in which the frame is
$$
\f{e} = \f{dt} \ga_0 + \f{dt} \sqrt{\fr{R_S}{r}} \ga_1 + \f{dr} \fr{1}{c} \ga_1 + \f{d\th} \fr{r}{c} \ga_2 + \f{d\ph} \fr{r \sin{\th}}{c} \ga_3
$$
The angular [[spherical coordinates]], $\th$ and $\ph$, range from $0$ to $\pi$ and from $0$ to $2\pi$. The radial coordinate, $r$, is scaled so the area of the 2D surface at $r=R$ is
$$
A = c^2 \int_{r=R} \f{e^2} \f{e^3} = \int \f{d\th}\f{d\ph} R^2 \sin{\th}=4\pi R^2
$$
for any time, $t$. The coframe is
$$
\ve{e} = \ga^0 \ve{\pa_t} - \ga^0 c \sqrt{\fr{R_S}{r}} \ve{\pa_r} + \ga^1 c \ve{\pa_r} + \ga^2 \fr{c}{r} \ve{\pa_\th} + \ga^3 \fr{c}{r \sin(\th)} \ve{\pa_\ph}
$$
The [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& - \f{d t} \fr{c R_S}{2 r^2} \ga_{01} - \f{d r} \fr{1}{2 r} \sqrt{\fr{R_S}{r}} \ga_{01}
+ \f{d \th} \sqrt{\fr{R_S}{r}} \ga_{02} + \f{d \th} \ga_{12}
+ \f{d \ph} \sqrt{\fr{R_S}{r}} \sin(\th) \ga_{03} + \f{d \ph} \sin(\th) \ga_{13} + \f{d \ph} \cos(\th) \ga_{23}
\end{eqnarray}
The [[Clifford-Riemann curvature]] is
\begin{eqnarray}
\ff{R} &=& \f{d} \f{\om} + \ha \f{\om} \f{\om} \\
&=& - \f{d t} \f{d r} \fr{c R_S}{r^3} \ga_{01}
+ \f{d t} \f{d \th} \fr{c R_S}{2 r^2} \ga_{02}
+ \f{d t} \f{d \th} \fr{c R_S}{2r^2} \sqrt{\fr{R_S}{r}} \ga_{12}
+ \f{d t} \f{d \ph} \fr{c R_S \sin{\th}}{2r^2} \ga_{03} \\
&+& \f{d t} \f{d \ph} \fr{c R_S \sin{\th}}{2r^2} \sqrt{\fr{R_S}{r}} \ga_{13}
+ \f{d r} \f{d \th} \fr{R_S}{2r^2} \ga_{12}
+ \f{d r} \f{d \ph} \fr{R_S \sin{\th}}{2r^2} \ga_{13}
- \f{d \th} \f{d \ph} \fr{R_S \sin{\th}}{r} \ga_{23}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
\begin{eqnarray}
\f{R} &=& \ve{e} \times \ff{R} = 0
\end{eqnarray}
showing that the Schwarzschild solution satisfies the vacuum [[Einstein's equation]] away from the curvature singularity at $r=0$.
Ref:
*http://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates
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http://deferentialgeometry.org
$$
\begin{array}{rcl}
\udf{A} \!\!&\!\!=\!\!&\!\! \f{H}{}_1 + \f{H}{}_2 + \ud{\Ps} = {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{W} + \f{B}{}_1 + \f{w} + \f{B}{}_2 + \f{x} \Ph + \f{g} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d}
\; \in \; \udf{\rm Lie}(E8) = \udf{e8}
\\[.5em]
\!\!&\!\!=\!\!&\!\!
{\small
\lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om}{}_L \!+\! i \f{W}{}^3 \!&\! i \f{W}{}^1 \!+\! \f{W}{}^2 \!&\! - \! \frac{1}{4} \f{e}{}_R \ph_1 \!&\! \frac{1}{4} \f{e}{}_R \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\
i \f{W}{}^1 \!-\! \f{W}{}^2 \!&\! \frac{1}{2} \f{\om}{}_L \!-\! i \f{W}{}^3 \!&\! \p{-} \frac{1}{4} \f{e}{}_R \ph_- \!&\! \frac{1}{4} \f{e}{}_R \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\
-\frac{1}{4} \f{e}{}_L \ph_0 & \frac{1}{4} \f{e}{}_L \ph_+ & \! \frac{1}{2} \f{\om}{}_R \!+\! i \f{B}{}_1^3 \! \!&\! i \f{B}{}_1^1 \!+\! \f{B}{}_1^2 \!&
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\
\p{-}\frac{1}{4} \f{e}{}_L \ph_- & \frac{1}{4} \f{e}{}_L \ph_1 &\! i \f{B}{}_1^1 \!-\! \f{B}{}_1^2 \!&\! \! \frac{1}{2} \f{\om}{}_R \!-\! i \f{B}{}_1^3 \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\
& & & & \; i \f{B}{}_2 &\!\! \!\!&\!\! \!\!&\!\! \\
& & & & &\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!+\! i \f{g}{}^{3+8} \!\!\!&\!\!\! i\f{g}{}^1 \!-\! \f{g}{}^2 \!\!\!&\!\!\! i\f{g}{}^4 \!-\! \f{g}{}^5 \\
& & & & &\!\!\! i\f{g}{}^1 \!+\! \f{g}{}^2 \!\!\!&\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!-\! i \f{g}{}^{3+8} \!\!\!&\!\!\! i\f{g}{}^6 \!-\! \f{g}{}^7 \\
& & & & &\!\!\! i\f{g}{}^4 \!+\! \f{g}{}^5 \!\!\!&\!\!\! i\f{g}{}^6 \!+\! \f{g}{}^7 \!\!\!&\!\!\! \frac{-i}{3} \! \f{B}{}_2 \!-\!\! \frac{2i}{\sqrt{3}}\f{g}{}^8
\end{array} \rb
}
\\[1.5em]
\udff{F} \!\!&\!\!=\!\!&\!\! \f{d} \udf{A} + \udf{A} \udf{A} =
( \f{d} \f{H}{}_1 + \f{H}{}_1 \f{H}{}_1 ) + ( \f{d} \f{H}{}_2 + \f{H}{}_2 \f{H}{}_2 ) + ( \f{d} \ud{\Ps} + \f{H}{}_1 \ud{\Ps} - \ud{\Ps} \f{H}{}_2 ) \; \in \; \udff{e8} \\[.5em]
\!\!&\!\!=\!\!&\!\!
\ha \big( (\f{d} \f{\om} + \ha \f{\om} \f{\om}) - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( ( \f{d} \f{e} \!+\! \ha [ \f{\om}, \f{e} ] ) \ph - \f{e} ( \f{d} \ph \!+\! [ \f{W} \!+\! \f{B}{}_1, \ph ] ) \big)
+ (\f{d} \f{W} + \f{W} \f{W})
\\
&&
\!\!+\, (\f{d} \f{B}{}_1 + \f{B}{}_1 \f{B}{}_1) + \f{d} \f{w} + \f{d} \f{B}{}_2 + \f{x}\Ph\f{x}\Ph
+ \big( ( \f{d} \f{x} \!+\! [ \f{w} \!+\! \f{B}{}_2, \! \f{x} ] ) \Ph \!-\! \f{x} ( \f{d} \Ph \!+\! [ \f{g}, \! \Ph ] ) \big)
+ (\f{d} \f{g} + \f{g} \f{g}) \\
&&
\!\!+\, \big( ( \f{d} + {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e}\ph ) \ud{\Ps}
+ \f{W} \ud{\Ps}{}_L + \f{B}{}_1 \ud{\Ps}{}_R - \ud{\Ps} ( \f{w} + \f{B}{}_2 + \f{x} \Ph ) - \ud{\Ps}{}_q \, \f{g} \big) \\[.5em]
\!\!&\!\!=\!\!&\!\!
\ha \big( \ff{R} - \fr{1}{8} \f{e} \f{e} \ph^2 \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \ff{F}{}_W + \ff{F}{}_{B_1}
+ \ff{F}{}_{w} + \ff{F}{}_{B_2} + \f{x}\Ph\f{x}\Ph
+ \big( (\f{D} \f{x}) \Ph - \f{x} \f{D} \Ph \big)
+ \ff{F}{}_{g}
+ \f{D} \ud{\Psi}
\end{array}
$$
$$
S \,= \int \big< \ff{\od{B}} \udff{F}
+ {\scriptsize \frac{\pi G}{4}} \ff{B}{}_G \ff{B}{}_G \ga - \ff{B'} \ff{*B'} \big>
= \int \big< \fff{\od{B}} \f{D} \ud{\Ps}
+ \nf{e} {\scriptsize \frac{1}{16 \pi G}} \ph^2 \big( R - \fr{3}{2} \ph^2 \big) - \fr{1}{4} \ff{F'} \ff{*F'} \big>
\vp{{\Big(}_{\Big(}^{\Big(}}
\s\s\s\s\;\;\;\;
$$
$$
Z = \int D A \, e^{\fr{i}{\hbar} S[A]} \s\s\s\;\; p[A] = \frac{1}{Z} \, e^{\fr{i}{\hbar} S[A]} \s\s\;\;\;\;
$$
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<SPAN class="math">$spin(1,3) \,\,\oplus\,\, 4_S^\mathbb{C} \,\,\oplus\,\, 4_V$</SPAN>
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<SPAN class="math">spin(1,3) \,\,\oplus\,\, 4_S^\mathbb{C} \,\,\oplus\,\, 4_V</SPAN>
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$Pin(1,3)$ is a $6$ dimensional Lie group, with spatial rotation and Lorentz boost generators, $J_\pi = \fr{1}{4} \ep_{\pi \rh \si} \ga_\rh \ga_\si$ and $K_\pi = \ha \ga_0 \ga_\pi$. Two commuting generators span the ''Cartan subalgebra'',
$$
J_3 = \fr{1}{4} \ga_1 \ga_2 =
{\small
\lb \begin{array}{cccc}
-\fr{i}{2} & 0 & 0 & 0 \\
0 & +\fr{i}{2} & 0 & 0 \\
0 & 0 & -\fr{i}{2} & 0 \\
0 & 0 & 0 & +\fr{i}{2}
\end{array} \rb
}
\s \s \s \s
K_3 = \fr{1}{2} \ga_0 \ga_3 =
{\small
\lb \begin{array}{cccc}
+\fr{1}{2} & 0 & 0 & 0 \\
0 & -\fr{1}{2} & 0 & 0 \\
0 & 0 & -\fr{1}{2} & 0 \\
0 & 0 & 0 & +\fr{1}{2}
\end{array} \rb
}
$$
These act on vectors, spinors, and the Lie algebra itself. Eigenvectors (''weight vectors'') are particle states and eigenvalues (''weights'' or ''charges''), ''spin'' and ''boost'', $\om_S$ and $\om_T$, are conserved in interactions.
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$$
\ba{rclcrcl}
J_3 \times v^{\wedge/\vee}_{S/T} \ae -i \, \om_S \, v^{\wedge/\vee}_{S/T} &\;&
K_3 \times v^{\wedge/\vee}_{S/T} \ae - \om_T \, v^{\wedge/\vee}_{S/T} \\[20px]
J_3 \, \ps^{\wedge/\vee}_{L/R} \ae -i \, \om_S \, \ps^{\wedge/\vee}_{L/R} &\;&
K_3 \, \ps^{\wedge/\vee}_{L/R} \ae - \om_T \, \ps^{\wedge/\vee}_{L/R} \\[20px]
\lb J_3 , E^{\wedge/\vee}_{L/R} \rb \ae -i \, \om_S \, E^{\wedge/\vee}_{L/R} &\;&
\lb K_3 , E^{\wedge/\vee}_{L/R} \rb \ae - \om_T \, E^{\wedge/\vee}_{L/R} \\
& & & & & &
\ea
$$
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<img src="talks/RW12/images/Spin(10).png" width="480" height="480">
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<SPAN class="math">$spin(10) \,\,\oplus\,\, 16^\mathbb{C}_{S+}$</SPAN>
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<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
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<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
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<SPAN class="math">spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}</SPAN>
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[[Spin(11,3), particles and octonions|papers/2104.01786.pdf]]
Authors: Kirill Krasnov
The fermionic fields of one generation of the Standard Model, including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S of the group Spin(11,3). We describe an octonionic model for Spin(11,3) in which the semi-spinor representation gets identified with S=OxO', where O,O' are the usual and split octonions respectively. It is then well-known that choosing a unit imaginary octonion u in Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion u' in Im(O') equips O' with a complex structure J', except that there are now two inequivalent complex structures, one parametrised by a choice of a timelike and the other of a spacelike unit u'. In either case, the identification S=OxO' implies that there are two natural commuting complex structures J, J' on S. Our main new observation is that the subgroup of Spin(11,3) that commutes with both J, J' on S is the direct product Spin(6) x Spin(4) x Spin(1,3) of the Pati-Salam and Lorentz groups, when u' is chosen to be timelike. The splitting of S into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S into eigenspaces of J' corresponds to splitting of Lorentz Dirac spinors into two different chiralities. We also study the simplest possible symmetry breaking scenario with the "Higgs" field taking values in the representation that corresponds to 3-forms in R^{11,3}. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM, and thus breaks Spin(11,3) down to the product of the SM, Lorentz and U(1)_{B-L} groups, with the last one remaining unbroken. This 3-form Higgs field also produces the Dirac mass terms for all the particles.
Notes:
He has ferms in $8 \otimes 8' = 64_+$ of $spin(11,3)$... Those $8$'s are of $spin(7)$ and $spin(4,3)$... which are rank 3 Lie algebras, with Cartan coords $\{x,y,z\}$ and $\{\om_S,U,V\}$... which must complete to $\{\om_T,\om_S,U,V,x,y,z\}$ under $spin(11,3)$, with the $\om_T$ bivector made from one (space) vec from $Cl(4,3)$ and one (time) from $Cl(7)$, which works... but that is not a $8_+ \otimes 8'_+$ under $spin(8) + spin(4,4) \subset spin(12,4)$ using $\om_t$ that's from $spin(11,3)$. It's half in $8_+ \otimes 8'_+$ and half in $8_- \otimes 8'_-$. To get this to work right, use $\Ga_t$ from $spin(7)$ with extra vector in $spin(8)$ to make a $p$ "particle vs anti-particle" quantum number. And the $\Ga_3$ in $spin(4,3)$ can combine with the extra vector in $spin(4,4)$ to make a $p\om_T$ operator. Then you have a $8_+ \otimes 8'_+$ for one gen. But can't triality rotate that without running in to $\om$ components in $8_v \otimes 8'_v$.
Krasnov put $\Ga_t$ in $spin(4,3)$ though, and thus didn't get the $su(2)_W$ in there... though that wasn't especially clear in the paper. His $J$ is $xyz$ and $J'$ is $\ga=\om_T \om_S$. Maybe he's using $\{x,y,z\}$ and $\{\om_T, \om_S, h\}$ to get $8 \otimes 8'$. Or could be $\{\om_T, \om_S, U\}$ or $\{\om_T, \om_S, V\}$.
<<tiddler HideTags>>$Cl^1(12,4)$ basis vector matrices, $\Ga'_x$, using the $\Ga_i$ from $Cl^1(11,3)$ :
$$
\Ga'_i = \si_1 \otimes \Ga_i \s\;\; \Ga'_{15} = \si_1 \otimes \Ga \s\;\; \Ga'_{16} = - i \, \si_2 \otimes 1
$$
$Cl^2(12,4)$ basis bivectors, $\Ga'_{xy}$ :
$$
\Ga'_{ij} = 1 \otimes \Ga_{ij} =
\lb \begin{array}{cc}
\Ga'^+_{ij} & 0 \cr
0 & \Ga'^-_{ij} \cr
\end{array} \rb
\s\;\;
\Ga'_{i \, 15} = 1 \otimes \Ga_i \Ga
\s\;\;
\Ga'_{i \, 16} = \si_3 \otimes \Ga_i
\s\;\;
\Ga'_{15 \, 16} = \si_3 \otimes \Ga
$$
$spin(12,4)$ positive chiral basis bivector matrices:
$$
\Ga'^+_{ij} = \Ga_{ij}
\s\;\;
\Ga'^+_{i \, 15} = \Ga_i \Ga
\s\;\;
\Ga'^+_{i \, 16} = \Ga_i
\s\;\;
\Ga'^+_{15 \, 16} = \Ga
$$
A $G = \ha G^{xy} \Ga'^+_{xy} \in spin(12,4)$ acting on a $128^{\mathbb{R}}_{S+}$ spinor, comprised of $spin(11,3)$ spinors:
$$
G \, \ps^{128}_{S+} =
\lb \begin{array}{cc}
\ha G^{ij} \Ga^+_{ij} + G^{15 \, 16} & -G^{i \, 15} \Ga^+_i + G^{i \, 16} \Ga^+_i \cr
G^{i \, 15} \Ga^+_ i + G^{i \, 16} \Ga^+_i & \ha G^{ij} \Ga^-_{ij} - G^{15 \, 16} \cr
\end{array} \rb
\lb \begin{array}{c}
\ps^{64}_{S+} \cr
\ps^{64}_{S-} \cr
\end{array} \rb
$$
Embedding of $spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}$ GraviGUT in $spin(12,4) \,\oplus\, 128^\mathbb{R}_{S+}$ GraviGUT:
$$
\Ga^+_{ij} = \Ga'^{++}_{ij} \s\;\; Q_{\io} = Q'^+_{\io}
$$
Invariant bilinear form for spinors:
$$
\ch^\da A \ps = (\ch, \ps) = (g \ch, g \ps) \;\; \Rightarrow \;\; (\ch, G \ps) = - (G \ch, \ps) \;\; \Leftrightarrow \;\;
\ch^\da A G \ps = - \ch^\da G^\da A \ps
$$
$$
A G = - G^\da A \;\; \forall \;\; G = \ha G^{xy} \Ga'^+_{xy} \;\;\; \Leftarrow \;\;\; A = (\Ga'_1 \Ga'_2 \Ga'_3 \Ga'_{16})^+ = - \si_1 \otimes 1 \otimes 1 \otimes \si_2 \otimes \si_2 \otimes 1 \otimes 1
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Standard Model Q.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\,\oplus\,\, (2_L \!\,\oplus\,\! 2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Standard Model Q.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(2)_L \,\oplus\, u(1)_Y \,\oplus\, su(3) \,\,\oplus\,\, (2_L \!\,\oplus\,\! 2_R) \!\otimes\! (1\!\,\oplus\,\!3)</SPAN>
</td></tr>
</table>
</center></html>
$$
-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu}
-g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu} \\
-\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu}
+\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu}
+\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu} \\
-\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu}
-\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}}
M^{2}Z^{0}_{\mu}Z^{0}_{\mu}
-\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu}
-\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2}
-\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-} \\
-\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0}
-\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-%%@
})]+\frac{2M^{4}}{g^{2}}\alpha_{h}
-igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})
-Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu})
+Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] \\
-igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})
-A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu})
+A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})]
-\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2}
W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu} \\
+g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu}
W^{-}_{\nu})
+g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu}
W^{-}_{\nu})
+g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})-%%@
2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}] \\
-g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}]
-\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2}
\phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}]
-gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H \\
-\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0})
-W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})]
+\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H)
-W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)]
+\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H)
-ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) \\
+igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
-ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-%%@
}\partial_{\mu}\phi^{+})
+igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+})
-\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}] \\
-\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}-%%@
1)^{2}\phi^{+}\phi^{-}]
-\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-%%@
}_{\mu}\phi^{+})
-\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+})
+\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+})
+\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) \\
-g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}-%%@
g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-}
-\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda}
-\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda}
-\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j}
-\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j}
+igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu} \\
e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} %%@
u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu}
d^{\lambda}_{j})]
+\frac{ig}{4c_{w}}Z^{0}_{\mu}
[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+
(\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+
(\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+
(\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})] \\
+\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda})
+(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})]
+\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})
+(\bar{d}^{\kappa}_{j}C^{\da}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})]
+\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M}
[-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda})
+\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})]
-\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda}) \\
+i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})]
+\frac{ig}{2M\sqrt{2}}\phi^{+}
[-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j})
+m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}]
+\frac{ig}{2M\sqrt{2}}\phi^{-}
[m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\da}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j})
-m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\da}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}] \\
-\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j})
-\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j})
+\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j})
-\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j}) \\
+\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-}
+\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y
+igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0})
+igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y)
+igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+})
+igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+})
+igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-})
+igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) \\
-\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H]
+\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}]
+\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}]
+igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}]
+\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}]
$$
<<tiddler HideTags>>$${\small \begin{array}{rcrccccccccccc}
T^\om_{23} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{13} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{12} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{14} \!\!&\!\! = \!\!&\!\! \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{24} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{34} \!\!&\!\! = \!\!&\!\! \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\vp{A^{\Big(}} T^W_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^W_2 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^W_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\vp{A^{\Big(}} T^Y \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\end{array}
\s\;\;\;\;\;
\begin{array}{rcrccccccccccc}
\vp{A^{\Big(}} T^g_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_4 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_6 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_8 \!\!&\!\! = \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{2 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2
\end{array}
}
\s\;\;\;
\ps=
\lb \!\!
\begin{array}{c}
\nu \cr e \cr u^r \cr d^r \cr u^g \cr d^g \cr u^b \cr d^b \cr
\end{array}
\!\! \rb
\begin{array}{c}
\left\{
\;
\lb \!\!
\begin{array}{c}
e_{Lr}^{\wedge} \cr e_{Li}^{\wedge} \cr e_{Lr}^{\vee} \cr e_{Li}^{\vee} \cr
e_{Rr}^{\wedge} \cr e_{Ri}^{\wedge} \cr e_{Rr}^{\vee} \cr e_{Ri}^{\vee} \\
\end{array}
\!\! \rb
\right.
\\[6.5em]
\end{array}
$$
<<tiddler HideTags>>$${\small \begin{array}{rcrccccccccccc}
T^\om_{23} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{13} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{12} \!\!&\!\! = \!\!&\!\! i \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{14} \!\!&\!\! = \!\!&\!\! \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^\om_{24} \!\!&\!\! = \!\!&\!\! - \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^\om_{34} \!\!&\!\! = \!\!&\!\! \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\vp{A^{\Big(}} T^W_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^W_2 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^W_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\vp{A^{\Big(}} T^Y \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{6} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\end{array}
\s\;\;\;\;\;
\begin{array}{rcrccccccccccc}
\vp{A^{\Big(}} T^g_1 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_2 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_3 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_4 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_5 \!\!&\!\! = \!\!&\!\! \fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_6 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
T^g_7 \!\!&\!\! = \!\!&\!\! -\fr{i}{4} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{4} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \cr
T^g_8 \!\!&\!\! = \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! -\fr{i}{4 \sqrt{3}} \!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2 \cr
\!\!&\!\! \!\!&\!\! +\fr{i}{2 \sqrt{3}} \!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_3 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! 1 \!\!\!&\!\! \otimes \!\!\!&\!\! \si_2
\end{array}
}
\s\;\;\;
\ps=
\lb \!\!
\begin{array}{c}
\nu \cr e \cr u^r \cr d^r \cr u^g \cr d^g \cr u^b \cr d^b \cr
\end{array}
\!\! \rb
\begin{array}{c}
\left\{
\;
\lb \!\!
\begin{array}{c}
e_{Lr}^{\wedge} \cr e_{Li}^{\wedge} \cr e_{Lr}^{\vee} \cr e_{Li}^{\vee} \cr
e_{Rr}^{\wedge} \cr e_{Ri}^{\wedge} \cr e_{Rr}^{\vee} \cr e_{Ri}^{\vee} \\
\end{array}
\!\! \rb
\right.
\\[6.5em]
\end{array}
$$
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">su(3) + su(2)_L + u(1)_Y \,+\, (1\!+\!3) \!\times\! 2_L + (1\!+\!3) \!\times\! (1\!+\!1)_R + \bar{"}</SPAN>
</td>
</tr>
<tr>
<td COLSPAN="3">
<img SRC="talks/CSUF09/images/SMElectric.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">\big( so(1,3) + su(2)_L + u(1)_R + 4 \!\times\! (2\!+\!2) + u(1)_B + su(3) \big) + 8 \!\times\! 8</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/CSUF09/images/SMGPSsG.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">E8 = \big( so(1,7) + so(7,1) \big) + 8_+ \!\times\! 8_+ + 8_v \!\times\! 8_v + 8_- \!\times\! 8_-</SPAN>
</td>
</tr>
<tr>
<td>
<img SRC="talks/CSUF09/images/E8PSsG.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\udf{A} = \f{H} + \f{G} + \ud{\ps}
=
{\small
\lb \begin{array}{cc}
\f{H^+} & \ud{\ps}^- \\
& \f{G^-}
\end{array} \rb
}
\s\s\s \in \;\; \f{so}(1,7) + \f{so}(8) + \ud{\mathbb{C}}(8 \times 8)
$$
$$
{\small
\!\! = \!\! \lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!&\! \frac{1}{4} \f{e_R} \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\
i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!&\! \frac{1}{4} \f{e_R} \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\
-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \! \!& &
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\
\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\
& & & & \; i \f{B} &\!\! \!\!&\!\! \!\!&\!\! \\
& & & & &\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
& & & & &\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
& & & & &\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
Correct interactions and charges from [[curvature]]:
$$\begin{array}{rcl}
\udff{F} \!\!&\!\!=\!\!&\!\! \f{d} \udf{A} + \udf{A} \udf{A} \\
\!\!&\!\!=\!\!&\!\! ( \f{d} \f{H} + \f{H} \f{H} ) + ( \f{d} \f{G} + \f{G} \f{G} ) + ( \f{d} \ud{\ps} + \f{H} \ud{\ps} + \ud{\ps} \f{G} )
\end{array}$$
<html>
<center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="ptable">
<tr>
<th COLSPAN="2"><SPAN class="math">G2</SPAN></th>
<th></th>
<th><SPAN class="math">V_\be</SPAN></th>
<th></th>
<th><SPAN class="math">g^3</SPAN></th>
<th><SPAN class="math">g^8</SPAN></th>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{g}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_2 - i T_1)</SPAN></td>
<td></td>
<td><SPAN class="math">1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}g}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_2 - i T_1)</SPAN></td>
<td></td>
<td><SPAN class="math">-1</SPAN></td>
<td><SPAN class="math">0</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{r\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_5-i T_4)</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{r}b}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_5-i T_4)</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{\bar{g}b}</SPAN></td>
<td></td>
<td><SPAN class="math">(-T_7-i T_6)</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">-\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr class="butt">
<td><SPAN class="math">\mcir{#6666FF} </SPAN></td>
<td><SPAN class="math">g^{g\bar{b}}</SPAN></td>
<td></td>
<td><SPAN class="math">(T_7-i T_6)</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\fr{\sqrt{3}}{2}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#D90000} </SPAN></td>
<td><SPAN class="math">q^r</SPAN></td>
<td></td>
<td><SPAN class="math">[1,0,0]</SPAN></td>
<td></td>
<td><SPAN class="math">\ha</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#00BF00} </SPAN></td>
<td><SPAN class="math">q^g</SPAN></td>
<td></td>
<td><SPAN class="math">[0,1,0]</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\btri{#0000F7} </SPAN></td>
<td><SPAN class="math">q^b</SPAN></td>
<td></td>
<td><SPAN class="math">[0,0,1]</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#D90000} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^r</SPAN></td>
<td></td>
<td><SPAN class="math">[1,0,0]</SPAN></td>
<td></td>
<td><SPAN class="math">-\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#00BF00} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^g</SPAN></td>
<td></td>
<td><SPAN class="math">[0,1,0]</SPAN></td>
<td></td>
<td><SPAN class="math">\fr{1}{2}</SPAN></td>
<td><SPAN class="math">\smash{-\fr{1}{2\sqrt{3}}}</SPAN></td>
</tr>
<tr>
<td><SPAN class="math">\butr{#0000F7} </SPAN></td>
<td><SPAN class="math">\bar{q}{}^b</SPAN></td>
<td></td>
<td><SPAN class="math">[0,0,1]</SPAN></td>
<td></td>
<td><SPAN class="math">0</SPAN></td>
<td><SPAN class="math">{\fr{1}{\sqrt{3}}}</SPAN></td>
</tr>
</table>
</td>
<td> </td>
<td>
<embed src="images/svg/g2.svg" type="image/svg+xml" width="450px" height="450px" />
<br><br>
<img SRC="images/png/dynkin g2.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/IfA11/images/Strong interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">su(3) \,\,\oplus\,\, 3 \,\,\oplus\,\, \bar{3} \vp{{\big(}^{\big(}}</SPAN>
</td></tr>
</table>
</center></html>
<html>
<center>
<table class="gtable">
<tr>
<td COLSPAN="3">
<SPAN class="math">su(3) + 3 + \bar{3}</SPAN>
</td>
</tr>
<tr>
<td>
<SPAN class="math">g^{r \bar{g}}+q^g \to q^r</SPAN>
<br><br>
<img SRC="images/png/quark gluon vertex.png" height=160px>
<br><br>
<SPAN class="math">T_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r}</SPAN>
<br><br>
<SPAN class="math">\al_{g^{r \bar{g}}} + \al_{q^g} = \al_{q^r}</SPAN>
</td>
<td> </td>
<td>
<img SRC="talks/CSUF09/images/InteractionG2.png">
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>
<<tiddler HideTags>>$$
\begin{array}{rclclc}
\f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^k} \om_k^{\p{k}\mu\nu} \ha \ga_{\mu\nu} \!&\!\! \in \!\!&\! \f{spin}(3,1) = \f{Cl}^2(3,1)
&
\f{e} = \f{dx^k} (e_k)^\mu \ga_\mu \, \in \, \f{Cl}^1(3,1) \vp{|_{(}} \\
\f{W} \!\!&\!\!=\!\!&\!\! \f{dx^k} W_k^{\p{i}\pi} \fr{i}{2} \si_\pi \!&\!\! \in \!\!&\! \f{su}(2)_L
\qquad
\frac{i}{2}
\!
\lb
\matrix{
\f{W}^3 & \!\!\! \f{W}^1 \!\!-\! i \f{W}^2 \! \\
\! \f{W}^1 \!\!+\! i \f{W}^2 \!\!\! & -\f{W}^3
}
\rb
&
\quad
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\qquad
\lb \matrix{
u_L \\ d_L
} \rb
\\
\f{B} \!\!&\!\!=\!\!&\!\! \f{dx^k} B_k i \!&\!\! \in \!\!&\! \f{u}(1)_Y
&
\quad
\\
\f{g} \!\!&\!\!=\!\!&\!\! \f{dx^k} g_k^{\p{k}A} \fr{i}{2} \la_A \!&\!\! \in \!\!&\! \f{su}(3)
&
\quad
\lb u^r, u^g, u^b \rb \vp{|^{(^(}_{(}}
\end{array}
\begin{array}{c}
\quad
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
\; \\
\; \\
\end{array}
$$
<html><center>
<img src="talks/Zuck09/images/Periodic table.png" width="440" height="340">
</center></html>
<<tiddler HideTags>>$$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\! \f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab} \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]
\mbox{weak:} \!\!&\!\! \f{W} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} W_\mu^{\p{\mu}I} \fr{i}{2} \si_I \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]
\mbox{hyper:} \!\!&\!\! \f{B} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} B^Y_\mu \,\, i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]
\mbox{strong:} \!\!&\!\! \f{g} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} g_\mu^{\p{\mu}A} \fr{i}{2} \la_A \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;
\begin{array}{ll}
\mbox{frame:} & \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]
\mbox{Higgs:} & \ph =
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \\
\p{A} & \p{B{{\Big(}^(}}
\end{array}
\;\;\;
\begin{array}{c}
\mbox{fermions:} \\
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\lb \matrix{
u_L \\ d_L
} \rb
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\
\in 4{}_S^\mathbb{C},\vp{A^{\big(}} \s 2^\mathbb{C}, \s
3^\mathbb{C}
\end{array}
$$
<html><center>
<img src="talks/Zuck09/images/Periodic table.png" width="414" height="320">
</center></html>
<<tiddler HideTags>>$$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\! \f{\om} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab} \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]
\mbox{weak:} \!\!&\!\! \f{W} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} W_\mu^{\p{i}B} \fr{i}{2} \si_B \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]
\mbox{hyper:} \!\!&\!\! \f{B} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} B_\mu i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]
\mbox{strong:} \!\!&\!\! \f{g} \!\!&\!\!=\!\!&\!\! \f{dx^\mu} g_\mu^{\p{k}A} \fr{i}{2} \la_A \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;
\begin{array}{ll}
\mbox{frame:} & \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]
\mbox{Higgs:} & \ph =
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \\
\p{A} & \p{B{{\Big(}^(}}
\end{array}
\;\;\;
\begin{array}{c}
\mbox{fermions:} \\
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\lb \matrix{
u_L \\ d_L
} \rb
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\
\in 4{}_S^\mathbb{C},\vp{A^{\big(}} \s 2^\mathbb{C}, \s
3^\mathbb{C}
\end{array}
$$
<html><center>
<img src="images/png/standard model and gravity 3.png" width="414" height="320">
</center></html>
<<tiddler HideTags>>$$
S_\ps = \int \nf{e} \bar{\ps} \ve{e} \f{D} \ps =
\int \nf{d^4x} |e| \left\{ \bar{\ps} \ga^a \lp e_a \rp^\mu \big(
\pa_\mu
+ {\tiny \frac{1}{4}} \om_\mu^{\p{\mu}bc} \ga_{bc}
+ W_\mu^{\p{\mu}\pi} T^W_\pi
+ B_\mu T^Y
+ g_\mu^{\p{\mu}A} T^g_A
\big) \ud{\ps}
+ \bar{\ps} \ph \ud{\ps} \right\}
$$ $$
\begin{array}{lrclcl}
\mbox{grav:} \!\!&\!\!\! \f{\om} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} \om_\mu^{\p{\mu}ab} \ha \ga_{ab} \!\!&\!\! \in \!\!\!&\!\! spin(1,3) \\[.25em]
\mbox{weak:} \!\!&\!\!\! \f{W} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} W_\mu^{\p{i}B} \fr{i}{2} \si_B \!\!&\!\! \in \!\!\!&\!\! su(2)_L \\[.25em]
\mbox{hyper:} \!\!&\!\!\! \f{B} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} B_\mu i \!\!&\!\! \in \!\!\!&\!\! u(1)_Y \\[.25em]
\mbox{strong:} \!\!&\!\!\! \f{g} \!\!&\!\!\!=\!\!\!&\!\! \f{dx^\mu} g_\mu^{\p{k}A} \fr{i}{2} \la_A \!\!&\!\! \in \!\!\!&\!\! su(3)
\end{array}
\;\;\;\;\;
\begin{array}{ll}
\mbox{frame:} \!\!&\!\! \f{e} = \f{dx^\mu} (e_\mu)^a \ga_a \, \in 4 \\[.5em]
\mbox{Higgs:} \!\!&\!\! \ph =
\lb \matrix{
\ph_+ \\ \ph_0
} \rb
\,
\in 2^\mathbb{C} \s\;\; \ud{\ps} = \!\!\! \\
\p{A} \!\!&\!\! \p{B{{\Big(}^(}} \s\s\s \mbox{fermions:} \!\!\!\!
\end{array}
\begin{array}{c}
\p{a}
\lb \matrix{
u_L^\wedge \\ u_L^\vee \\ u_R^\wedge \\ u_R^\vee
} \rb
,
\matrix{
\lb \matrix{
u_L \\ d_L
} \rb
\\[.5em]
\lb \matrix{
\nu_{L} \\ e_L
} \rb
}
,
\lb \matrix{
u^r \\ u^g \\ u^b
} \rb
\\
\in \, 4{}_S^\mathbb{C} \;\; , \vp{A^{\big(}} \;\;\;\; 2^\mathbb{C} \;\;\; , \;\;\; 3^\mathbb{C}
\end{array}
$$
<html>
<table class="gtable">
<tr>
<td>
<table class="gtable">
<tr><td>
<img src="images/png/Wvertex.png" width="181" height="149">
</tr></td>
</table>
</td>
<td>
</td>
<td>
<img src="images/png/standard model and gravity 3.png" width="414" height="320">
</td>
</tr>
</table>
</html>
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A carriage return ends a paragraph, so a slightly larger (this should be made bigger) space appears between paragraphs. But the beginning of a new paragraph is not indented, even if tabs or spaces are inserted. So, for now, use extra carriage returns to separate paragraphs.
And maybe implement tab indentation in the future.
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="images/png/subatomic.png" height="450">
</td></tr><tr><td>
<SPAN class="math"></SPAN>
</td></tr>
</table>
</center></html>
http://arxiv.org/abs/hep-th/0610039
Super coset spaces play an important role in the formulation of supersymmetric theories. The aim of this paper is to review and discuss the geometry of super coset spaces with particular focus on the way the geometrical structures of the super coset space G/H are inherited from the super Lie group G. The isometries of the super coset space are discussed and a definition of Killing supervectors - the supervectors associated with infinitesimal isometries - is given that can be easily extended to spaces other than coset spaces.
<<tiddler HideTags>>Bosons and fermions in separate parts of one Lie algebra (or Lie superalgebra),
$$
\big( so(1,7) + so(7,1) \big) + S
$$
''Superconnection'':
\begin{eqnarray}
\udf{A} &=& \f{H} + \ud{\Ps} \\
&& \f{H} = \f{dx^i} H_i^{\p{i}A} T_A \\
&& \ud{\Ps} = \ud{\Ps^\al} T_\al
\end{eqnarray}
''Supercurvature'':
\begin{eqnarray}
\udff{F} &=& \f{d} \udf{A} + \ha \big[ \udf{A}, \udf{A} \big] \\
&=& \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\Ps} + \ha \big[ \f{H}, \ud{\Ps} \big] \big)
+ \ha \big[ \ud{\Ps}, \ud{\Ps} \big] \\
&=& \ff{F^H} + \f{D} \ud{\Ps} + \ud{\Ps} \ud{\Ps}
\end{eqnarray}
Geometric description of fermions: Lie algebra valued Grassmann fields.
<<tiddler HideTags>>Geometric description of fermion fibers as Lie algebra valued anticommuting fields.
<html><center><table class="gtable">
<tr>
<td><img src="talks/Zuck09/images/Twist.png" width="250" height="250"></td>
<td><SPAN class="math">\s\;\; \longleftrightarrow \s\;\;</SPAN></td>
<td><img src="talks/Zuck09/images/Circle twist.png" width="250" height="250"></td>
</tr>
</table></center></html>For a principal $G$-bundle, with $H$ a reductive subalgebra in $G=H \oplus K$, define the ''superconnection'' to be the direct sum of an $H$ connection and a $K$ valued anticommuting field:
$$
\begin{array}{rclcrclcrcl}
\udf{A} \!\!&\!\!=\!\!&\!\! \f{H} + \ud{\ps} & \;\; & \in \!\!&\!\! G \!\!& \; & [H,H] \!\!&\!\! \subset \!\!&\!\! H \\
& &\!\! \f{H} = \f{dx^\mu} H_\mu^{\p{\mu}A} T_A \!\!& \;\; & \in \!\!&\!\! H \!\!& \s\;\;\;\; & [H,K] \!\!&\!\! \subset \!\!&\!\! K \\
& &\!\! \ud{\ps} = \ud{\ps}^{\, \io} T_\io \!\!& \;\; & \in \!\!&\!\! K \!\!& \; & [K,K] \!\!&\!\! \subset \!\!&\!\! G
\end{array}
$$
''Supercurvature'':
$$
\udff{F} = \f{d} \udf{A} + {\textstyle \ha} [\udf{A},\udf{A}]
= (\f{d} \f{H} + {\textstyle \ha} [\f{H},\f{H}] ) + (\f{d} \ud{\ps} + [\f{H},\ud{\ps}] ) + {\textstyle \ha} [\ud{\ps},\ud{\ps}]
= \ff{F}{}_H + \f{D} \ud{\ps} + \ud{\ps}\ud{\ps}
$$
<<tiddler HideTags>>
\begin{eqnarray}
\udf{A} &=& \f{H} + \ud{\ps} \s\;\;\; \in \; spin(11,3) \,\oplus\, 64^\mathbb{R}_{S+}\\
&& \f{H} = \f{dx^i} H_i^{\p{i}A} T_A \\
&& \ud{\ps} = \ud{\ps^\io} Q_\io
\end{eqnarray}
''Supercurvature'':
\begin{eqnarray}
\udff{F} &=& \f{d} \udf{A} + \ha \big[ \udf{A}, \udf{A} \big] \\
&=& \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \ud{\ps} + {\textstyle \ha} \big[ \f{H}, \ud{\ps} \big] \big)
+ {\textstyle \ha} \big[ \ud{\ps}, \ud{\ps} \big] \\
&=& \ff{F^H} + \f{D} \ud{\ps} + \ud{\ps} \ud{\ps}
\end{eqnarray}
|!Symbol|![[LaTeX]]|!Use|
| $\mathbb{R} \;\; \mathbb{C} \;\; n \;\; \ud{a}$ | {{{ \mathbb{R} \mathbb{C} n \ud{a} }}} |[[real numbers|http://en.wikipedia.org/wiki/Real_numbers]], [[complex number]]s, dimension, [[Grassmann number]] |
| $M \; \; T_p M \; \; T_p^* M$ | {{{M T_p M T_p^* M }}} |[[manifold]], [[tangent space to M at point p|coordinate basis vectors]], [[cotangent space to M at point p|coordinate basis 1-forms]] |
| $x^i \; \; \ve{\pa_i} \; \; \ve{v} \; \; \vv{l}$ | {{{x^i \ve{\pa_i} \ve{v} \vv{l} }}} |[[coordinates|manifold]] and [[coordinate basis vectors]] with coordinate [[indices]], [[tangent vector]], [[loop|vector-form algebra]] |
| $t \; \; \ta$ | {{{ t \ta }}} |parameter time, [[proper time]] |
| $\f{dx^i} \;\; \f{a} \;\; \ff{b} \;\; \fff{c} \;\; \nf{f}$ | {{{\f{dx^i} \f{a} \ff{b} \fff{c} \nf{f} }}} |[[coordinate basis 1-forms]], [[1-form]], [[2-form|differential form]], 3-form, [[differential form]] of high or unspecified form grade |
| $\pa_i \;\; \f{\pa} \;\; \f{d}$ | {{{\pa_i \f{\pa} \f{d} }}} |[[partial derivative]], partial derivative, [[exterior derivative]] |
| $\ph \; \; \ph^* \; \; \ph_*$ | {{{\ph \ph^* \ph_* }}} |[[diffeomorphism]], [[pullback]], pushforward |
| ${\cal L}_{\ve{v}} \;\; \lb\ve{v},\ve{u}\rb_L \;\; \ve{\De}$ | {{{{\cal L}_{\ve{v}} \lb\ve{v},\ve{u}\rb_L \ve{\De} }}} |[[Lie derivative]], [[Lie bracket|Lie derivative]] of two [[vector fields|tangent bundle]], [[distribution]] |
| $\nf{\ve{A}} \;\; {\cal L}_{\nf{\ve{K}}} \;\; \lb\nf{\ve{K}},\nf{\ve{L}}\rb_L \;\; \f{\ve{P}}$ | {{{ \f{\ve{A}} {\cal L}_{ \nf{\ve{K}} } \lb\nf{\ve{K}},\nf{\ve{L}}\rb_L \f{\ve{P}} }}} |[[vector valued form]], [[FuN derivative]], FuN bracket, [[vector projection]] |
| $\f{\ve{\cal A}} \;\; \ff{\ve{\cal F}} \;\; \f{\cal D}$ | {{{ \f{\ve{\cal A}} \ff{\ve{\cal F}} \f{\cal D} }}} |[[Ehresmann connection]], [[FuN curvature]], [[Ehresmann covariant derivative]] |
| $\de_i^j \;\; \et_{\al \be} \; \; \ep_{\al \dots \be} \; \; \otimes$ | {{{ \de_i^j \et_{\al \be} \ep_{\al \dots \be} \otimes }}} |[[Kronecker delta|http://en.wikipedia.org/wiki/Kronecker_delta]], [[Minkowski metric]], [[permutation symbol]], [[Kronecker product]] |
| $G \;\; {\frak g} \;\; g^- \;\; T_A \;\; \lb{T_A,T_B}\rb$ | {{{G {\frak g} g^- T_A \lb{T_A,T_B}\rb }}} |[[Lie group]], [[Lie algebra]], [[inverse]] of a group element, [[Lie algebra]] generators, [[commutator]] bracket |
| $\f{\na} \;\; \f{A} \;\; \ff{F}$ | {{{\f{\na} \f{A} \ff{F} }}} |[[covariant derivative]], [[connection]], [[curvature]] |
| $\ve{\de} \;\; \ve{\na}$ | {{{\ve{\de} \ve{\na} }}} |[[codifferential]], [[covariant vector derivative|codifferential]] |
| $\f{\cal I} \;\; \f{\ve{\cal I}} \;\; \ve{\xi^L_A} \;\; \ve{\xi^R_A}$ | {{{\f{\cal I} \f{\ve{\cal I}} \ve{\xi^L_A} \ve{\xi^R_A} }}} |[[Maurer-Cartan form]], Ehresmann-Maurer-Cartan form, [[left and right action vector fields|Lie group geometry]] |
| $Cl \; \; Cl^*$ | {{{Cl Cl^* }}} |[[Clifford algebra]], [[Clifford group]] |
| $\ga_\al \; \; \ga_{\al \dots \be} \; \; \ga$ | {{{\ga_\al \ga_{\al \dots \be} \ga }}} |[[Clifford basis vectors]], [[Clifford basis elements]], Clifford [[pseudoscalar]] |
| $\hat{A} \; \; \tilde{A} \; \; \bar{A} \; \; A^\da \; \; \overline{A}$ | {{{\hat{ \tilde{ \bar{ \overline{A}^\da } } } }}} |[[Clifford involution, reverse, conjugate|Clifford conjugate]], [[Hermitian]] conjugate, [[Dirac adjoint]] |
| $\cdot \; \; \times$ | {{{\cdot \times }}} |symmetric and antisymmetric [[Clifford algebra]] product |
| $\lb{A,\dots,B}\rb_A \;\; a_{\lb{\al\dots\be}\rb}$ | {{{ \lb{A,\dots,B}\rb_A a_{\lb{\al\dots\be}\rb} }}} |[[antisymmetric bracket]], [[index bracket]] |
| $\li{A}\ri_q \; \; \li{A}\ri$ | {{{ \li{A}\ri_q \li{A}\ri }}} |[[Clifford grade]] $q$ part, [[scalar part]] |
| $\f{A} \; \; \ff{b} \; \; \ve{e}$ | {{{ \f{A} \ff{b} \ve{e} }}} |[[Lieform]]s or [[Clifform]]s |
| $\lp{e_i}\rp^\al \;\; \lp{e_\al}\rp^i \;\; g_{ij} \;\; \lp\ve{u},\ve{v}\rp$ | {{{ \lp{e_i}\rp^\al \lp{e_\al}\rp^i g_{ij} \lp\ve{u},\ve{v}\rp }}} |co[[frame]] matrix, frame matrix, [[metric]], scalar product |
| $\f{e^\al} \;\; \ve{e_\al}$ | {{{ \f{e^\al} \ve{e_\al} }}} |co[[frame]] 1-forms, orthonormal basis vectors |
| $\f{e} \;\; \ve{e}$ | {{{ \f{e} \ve{e} }}} |co[[frame]], frame |
| $\nf{e} \;\; \ll{e}\rl$ | {{{ \nf{e} \ll{e}\rl }}} |[[volume form]], frame [[determinant]] |
| $\nf{*f} \;\; \ff{\vv{\ep}}$ | {{{ \nf{*f} \ff{ \vv{\ep} } }}} |[[Hodge dual]], Hodge dual projector |
| $\f{e^s} \;\; \lp{e^s_i}\rp^\al \;\; s$ | {{{ \f{e^s} \;\; \lp{e^s_i}\rp^\al \;\; s }}} |[[special frame]], special coframe matrix, conformal scalar |
| $TM \;\; T^*M$ | {{{ TM T^*M }}} |[[tangent bundle]], [[cotangent bundle]] |
| $\Ga^k{}_{ij} \;\; \f{\Ga}^k{}_j \;\; \ff{R}^k{}_j$ | {{{ \Ga^k{}_{ij} \f{\Ga}^k{}_j \ff{R}^k{}_j }}} |[[Christoffel symbols]], [[tangent bundle connection]], [[Riemann curvature]] |
| $\f{R}{}_j \;\; R$ | {{{ \f{R}{}_j R }}} |[[Ricci curvature]], [[curvature scalar]] |
| $L^\be{}_\al \;\; \f{w}^\be{}_\al \;\; \ff{F}^\be{}_\al$ | {{{ L^\be{}_\al \f{w}^\be{}_\al \ff{F}^\be{}_\al }}} |[[Lorentz rotation]], [[tangent bundle spin connection|tangent bundle connection]], [[Riemann curvature]] |
| $ ClM \;\; Cl^1M$ | {{{ ClM Cl^1M }}} |[[Clifford bundle]], [[Clifford vector bundle]] |
| $\f{A} \;\; \f{\om} \;\; \ff{R}$ | {{{ \f{A} \f{\om} \ff{R} }}} |[[Clifford connection]], [[spin connection]], [[Clifford-Riemann curvature]] |
| $\f{R} \;\; R$ | {{{ \f{R} R }}} |[[Clifford-Ricci curvature]], [[Clifford curvature scalar]] |
| $\ff{T} \;\; \f{\ka}$ | {{{ \ff{T} \f{\ka} }}} |[[torsion]], contorsion |
| $\ud{C} \;\; \nf{\od{B}} \;\; \udf{A} \;\; \udff{F}$ | {{{ \ud{C} \nf{\od{B}} \udf{A} \udff{F} }}} |[[BRST|BRST technique]] ghost, anti-ghost, extended connection, extended curvature |
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*<<slider chkSliderphysicsF physicsF 'physics >' 'physics stuff, what this site is all about'>>
*<<slider chkSlidermetaF metaF 'meta >' 'describe the operation of this site'>>
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*<<tag physics>>- physics stuff, what this site is all about
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***<<tag pb>>- principal bundles
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****<<tag cartan>>- Cartan geometry
***<<tag brst>>- BRST formalism
***<<tag kk>>- Kaluza-Klein theory, Killing vector fields
****<<tag ss>>- homogeneous spaces
****<<tag cartan>>- Cartan geometry
***<<tag lie>>- Lie algebras, Lie groups
***<<tag sm>>- the standard model of particles
****<<tag gut>>- SU(5), SO(10), TSmith, etc.
****<<tag higgs>>- Higgs scalar and symmetry breaking
***<<tag clifford>>- Clifford algebra
****<<tag dirac>>- Dirac operators, Dirac equation
*****<<tag spin>>- spin odds and ends (more theoretical then dirac)
**<<tag gr>>- General Relativity
***<<tag nat>>- natural operators, vectors, forms
***<<tag kk>>- Kaluza-Klein theory, Killing vector fields
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****<<tag cartan>>- Cartan geometry
***<<tag cosmo>>- cosmology
***<<tag grscal>>- gr plus a scalar field, Brans-Dicke theories, conformal transformations
***<<tag lqg>>- loop quantum gravity, loops, spin foams, spin networks
***<<tag tors>>- torsion, teleparallel gravity
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**<<tag qm>>- quantum mechanics
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The tags group collections of notes into sets -- they're adjectives, directories, or folders. By selecting a tag, visible in the upper right of each note, you can jump to any note labeled with that tag -- a navigational convenience. It's efficacious to build a flexible hierarchy of tagged content. Each note should be labeled by its most appropriate, "lowest" leaf tags -- more than one as appropriate. Click on the tag to see a popup menu of notes with that tag. Alternatively, these folders and their contents may be selectively displayed in the "Contents" tab to the left.
//implementing hierarchical tagging seems to be a bit of a mess right now... wait and it will get better.//
[>img[images/person/Thanu Padmanabhan.jpg]]Homepage: http://www.iucaa.ernet.in/~paddy/
*Location: Pune, India
*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Padmanabhan_T/0/1/0/all/0/1
Selected work:
*[[Holographic Gravity and the Surface term in the Einstein-Hilbert Action|http://arxiv.org/abs/gr-qc/0412068]]
**Einstein's equation from the extrinsic curvature surface term alone!
**related paper by Sotiriou: http://arxiv.org/abs/gr-qc/0603096
*http://arxiv.org/abs/gr-qc/0309053
**Horizons, for collections (congruences) of observers, at coordinate singularities for the corresponding coords.
**Observers only have access to fields and dynamics inside this region, bounded by horizons.
**Upon a Wick rotation, the horizon and region beyond disappears into the origin of Euclidean space -- a conical singularity.
***The resulting Euclidean space region is naturally periodic in $t$ (in example, it resembles angular coord on a cone).
**Boundary area quantization in order to avoid quantum entanglement across the boundary.
**Action/Entropy is related to the degrees of freedom hidden behind the boundary/horizon.
**Lots of good stuff in appendices. explicit calculations.
***extrinsic curvature and boundary term
***derivation of unique EH action
*http://arxiv.org/abs/gr-qc/0204019
**"action is the free energy of the horizon"
*http://arxiv.org/abs/gr-qc/0311036
**How to relate stat mech, QM, and GR
**Introductory level, good survey
**conical singulartity regularization (again)
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[[Garrett Lisi]] Singularity University 2010@@
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[[Garrett Lisi]] Mindshare, 2011@@
Wow, I'm [[This Week's Find in Mathematical Physics|http://math.ucr.edu/home/baez/week253.html]]! Here's a quick tiddler I wrote that week when I found out:
[[John Baez]] discusses my work on describing all fields of the standard model and gravity as parts of a [[superconnection]] for the [[E8]] [[principal bundle]] over a four dimensional base manifold. And he discusses a LOT of other, related mathematics that will be keeping me busy over the next year, at least.
On 6/25/08 I delivered my [[talk for Loops 07]] in Morelia. (That links to a wiki tiddler including links to all my talk slides (printed out from this web page), as well as supporting links, AND the audio files for the talk as I practiced it (high bandwidth recommended). The slides are also available as [[this pdf file|talks/Loops07/Loops2007.pdf]]. The talk as I actually gave it is available at the [[Loops '07|http://www.matmor.unam.mx/eventos/loops07/cont_abs.html]] site under my name. It includes many excellent questions asked by Lee Smolin and others afterwards. (An excellent question being one a speaker has already thought through and wants to talk about anyway.) This talk was VERY well received -- I've had so many physics conversations with LQG people in the past two days that my head is ready to explode. They are such great and friendly people! Sadly, on the morning of the third day I started having flu symptoms, and pulled myself away from the discussion and into hotel room quarantine -- I didn't want to infect new friends with viruses, just ideas. I'm here in my room trying to get better, and being sad, when up pops a link to This Week's Finds.... happy and sick is a bizarre combination.
!Summary of the proposed [[E8]] [[T.o.E.|theory of everything]] for physicists, from the top down:
The universe is described by a Yang-Mills theory, with [[E8]] as the gauge group over a four dimensional base [[manifold]]. The field is a non-compact [[e8]] valued [[connection]] [[1-form]] which breaks up into different parts of the [[Lie algebra]],
\begin{eqnarray}
\f{A} &=& \f{H} + \f{G} + \f{\Ps}{}_I + \f{\Ps}{}_{II} + \f{\Ps}{}_{III} \\
&& \in \f{so}(1,7) \oplus \f{so}(1,7) \oplus \f{End}(V^{(1,7)}) \oplus \f{End}(S^{(1,7)+}) \oplus \f{End}(S^{(1,7)-})
\end{eqnarray}
The action for this connection is presumed to be invariant under gauge transformations taking the $\Ps$'s to zero -- we gauge away the $\Ps$'s. The [[BRST technique]] (Faddeev-Popov) of standard QFT is used to replace the $\Ps$'s with blocks of [[Grassmann|Grassmann number]] valued ghost fields in the same part of the Lie agebra, giving the BRST extended connection,
$$
\udf{A} = \f{H} + \f{G} + \ud{\Ps}{}_I + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III}
$$
Computing the [[curvature]] of this extended connection gives
$$
\udff{F} = \big( \f{d} \f{H} + \f{H} \f{H} \big) + \big( \f{d} \f{G} + \f{G} \f{G} \big) + \big( \f{d} \ud{\Ps}{}_{I-III} + \f{H} \ud{\Ps}{}_{I-III} + \ud{\Ps}{}_{I-III} \f{G} \big)
$$
Here, in the beautiful structure of $E8$, the $so(1,7)$ parts of the connection act as [[Clifford bivectors|Clifford algebra]] multiplying a set of three blocks of Grassmann valued [[spinor]]s from the left and from the right. These ghost fields have precisely the transformations and charges of the fermions -- so I go ahead and interpret them as the physical fermions. (If you don't like this derivation, you're welcome to just start with a superconnection and go from there.) The first $so(1,7)$ part of the connection, $\f{H}$, is broken up into the gravitational [[spin connection]], $\f{\om} \in \f{\rm so}(1,3)$, the electroweak fields, $\f{W} + \f{B} \in \f{\rm su}(2) \oplus \f{\rm u}(1) \subset \f{\rm su}(2) \oplus \f{\rm su}(2) = \f{\rm so}(4)$, and the combined [[frame]] and Higgs are assigned to the rest of $\f{H}$, giving
$$
\f{H} = \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W}
$$
The $\f{G_{\rm strong}} \in \f{\rm su}(3) \subset \f{\rm su}(4) = \f{\rm so}(6)$ gluons and part of the $\f{B}$ go in the second $\f{\rm so}(1,7)$ part of the connection, $\f{G}$, with $19$ of $28$ generators left unused. (We could just as well have started with $\f{\rm so}(8)$ for this part of $E8$ -- I'm not sure which is better yet.) The curvature, $\udff{F}$, gives all the correct interactions of the standard model and [[modified BF gravity]]. This is very beautiful -- the entire structure of the standard model and gravity, including interactions, fits snugly in $E8$, considered by some to be the most beautiful structure in mathematics. Exactly what you want for a T.O.E. But it doesn't work perfectly yet -- I haven't figured out if the Higgs can correctly mix the generations; ideally, I want to get the CKMPMNS (mass) matrix out of $E8$. This will be what I work on from here until I get it or find it doesn't work -- and it will very distinctly be one or the other. Since we have room (unassigned generators) in $\f{G}$, we can steal some, reducing that block to $\f{G} \in \f{\rm so}(6)$ while still fitting the gluons, and using those stolen generators to have more Higgs fields in a larger $\f{H} \in \f{\rm so}(1,9)$. These Higgs terms mix between the generations, and I will be trying to use this new gauge field reassignment to fit the fermions in what's left over and get the mass matrix out. Depite how well this picture has come together so far, it may just not work, but it's what I'm after.
From my perspective, fitting the standard model into the structure of $E8$ came as a complete shock: I did not initially build this theory from the top down, but from the bottom up -- by spending years massaging the standard model and gravity into the most elegant (but weird!) and minimalistic mathematical framework possible. If you look at [[this paper|http://arxiv.org/abs/gr-qc/0511120]], you will find this strange Lie algebra block structure of $\udf{A} = \f{H} + \f{G} + \ud{\Ps}$, with a big empty space I couldn't explain. (Note that E8 is not mentioned in this paper!) When I bumped into $E8$, and found this same algebraic block structure but with two more generations of fermions... what can I say, you get only one or two moments like that in life, if you're very lucky.
!!Summary of the summary:
Everything is described by a broken [[e8]] valued [[superconnection]] over our four dimensional base manifold,
\begin{eqnarray}
\udf{A} &=& {\small \frac{1}{2}} \f{\om} + {\small \frac{1}{4}} \f{e} \ph + \f{B} + \f{W} + \f{G} + \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d} \\
&& + \ud{\nu^\mu} + \ud{\mu} + \ud{c} + \ud{s}
+ \ud{\nu^\ta} + \ud{\ta} + \ud{t} + \ud{b}
\end{eqnarray}
All standard model interactions (and gravity) come from the curvature of this connection.
For the current version of how this work is playing out, check out [[the big picture]].
To hide text within a tiddler so that it is not displayed you can wrap it in {{{/%}}} and {{{%/}}}. It can be a useful trick for hiding drafts or annotating complex markup. Edit this tiddler to see an example.
/%This text is not displayed
until you try to edit %/
What to do next.
New tiddlers and changes:
*[[Cl(3,1)]]
*[[energy-momentum tensor]]
*If $so(8)$ and [[su(3)]] are embedded in [[E8]] as in [[the big picture]], then the coupling constants for GR and EW at that ToE scale should be the same, and the [[su(3)]] coupling constant should be larger by a factor of $\sqrt{2}$ because of how the $su(3)$ root hexagon is scaled compared to the [[Gell-Mann matrices]].
**Maybe pull running SM coupling constants from Frank Wilczek's [[paper|http://arxiv.org/abs/hep-th/9803075]].
*clean up [[the big picture]]
**improve Higgs and torsion part in action
*[[SO(1,7)]] or [[SO(7,1)]] modeled on [[SO(8)]]
*[[Cl(1,15)]] modeled on [[Cl(16)]]
*[[CP2]] from [[broken SU(3)]]
*[[doubly homogeneous space]] -- [[normalizer]] $H \triangleleft N_G(H) \subset G$
**Baez TWF on double coset space
*[[calculus of variations]]
*[[Hamiltonian]]
New [[Tags]] and hierarchy adjustment:
*[[e8]] under sym
*[[toe]] under gr and sm
*[[conf]] conferences and talks, under meta
New Illustrations:
*[[submanifold]]
*[[Killing vector]]
*[[Ehresmann Cartan geometry]]
Papers to read:
*Frederick Witt, on [[triality]] (includes spinor valued 1-forms)
**[[Special metrics and Triality|papers/0602414.pdf]]
**[[Special metric structures and closed forms|papers/0502443.pdf]]
***thesis
Change referenced paper files to "author - title|author - title.pdf"?
New features:
*Have Google searches navigate to search results?
And<<slider chkSliderTDM [[To Do Maybe]] 'Maybe do these >' 'things I maybe want to do'>> ([[To Do Maybe]])
New notes:
*might need to change signs for all [[Clifford rotation]]s, to make counter-clockwise positive instead of negative, and change coefficient order in the [[spin connection]].
New [[Tags]]
*nat is awfully full
Content to add:
*from FQXi proposal
*from BF paper
*from physics notes
*summary/presentation collection of notes
*from slides to content
Features to add:
*have search return a stack of collapsed notes
*remove system tags and tiddlers from [[TabContents]] and [[Tags]]
*Contents lists hierarchical tags + descriptions (folder slider tooltips)
**any way to do this automatically?
***maybe with Udo's data in folders
*web public
**Halo Scan comments?
**"Comments" for collecting comments. solicit "new note" requests
**"Sandbox" is one public note for people to fool around in
**Ziddlywiki http://ziddlywiki.org/forum/ pretty cool discussion format.
**comments via a form at the end of a note? some plugin allows this... Udo?
***http://tiddlywiki.abego-software.de/
***won't currently work with pytw. also tidddler->tiddler problem...
*papers under their author, and independently under their tags (this way, don't need to include all people)
**tiddlers like "Clifford papers" to collect refs
**ref papers or outside links as needed
*venn tag grouping... intersections, exclusions, etc...
*tagged templates? journal, paper, comment. http://www.gensoft.revhost.net/TaggedTemplating.html
**nah, just copy existing table layout
*script to edit in LyX
**needs to translate back and forth via intermediate files
*add editing command toolbar, http://aiddlywiki.sourceforge.net/wikibar_demo_2.html
**or tinyMCE wysiwyg editor
**tool Palettes
***LaTeX palettes in edit view that let you click on buttons to insert (latex) text. These should be customizable.
*pwd tags meta/system/plugin
*more/change [[Keyboard Shortcuts]]
**keyboard commands to edit note
http://arxiv.org/abs/gr-qc/0608135
Authors: Etera Livine
We review the canonical analysis of the Palatini action without going to the time gauge as in the standard derivation of Loop Quantum Gravity. This allows to keep track of the Lorentz gauge symmetry and leads to a theory of Covariant Loop Quantum Gravity. This new formulation does not suffer from the Immirzi ambiguity, it has a continuous area spectrum and uses spin networks for the Lorentz group. Finally, its dynamics can easily be related to Barrett-Crane like spin foam models.
*looks like a good intro to recent developments
<<tiddler HideTags>>[[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:
<<<
we now look at the vector space
$$
so(8) + so(8) + end(V) + end(S^+) + end(S^-)
$$
...Since $so(8)$ has a representation as linear transformations of $V$, it has two representations on $end(V)$, corresponding to ''left and right matrix multiplication''; glomming these two together we get a representation of $so(8) + so(8)$ on $end(V)$. Similarly we have representations of $so(8) + so(8)$ on $end(S^+)$ and $end(S^-)$. Putting all this stuff together we get a Lie algebra, if we do it right - and it's $E8$.
<<<
$$
E = H + G + \Ps_I + \Ps_{II} + \Ps_{III} \;\;\;\; \in {\rm Lie}(E8)\p{{}_{(}}
$$
$$
\begin{array}{rclcrcl}
[ H , \Ps_I ] \!\!&\!\!=\!\!&\!\! H \, \Ps_I & & [ H , \Ps_{II} ] \!\!&\!\!=\!\!&\!\! H^+ \, \Ps_{II} & & [ H , \Ps_{III} ] \!\!&\!\!=\!\!&\!\! H^- \, \Ps_{III} \\
[ G , \Ps_I ] \!\!&\!\!=\!\!&\!\! \Ps_I \, G & & [ G , \Ps_{II} ] \!\!&\!\!=\!\!&\!\! \Ps_{II} \, G^+ & & [ G , \Ps_{III} ] \!\!&\!\!=\!\!&\!\! \Ps_{III} \, G^-\p{{}_{(}}
\end{array}
$$
<<tiddler HideTags>>
<html><center>
<img src="talks/Zuck09/images/Twist.png" width="300" height="300">
</center></html>
easy to invert
Use a complex frame, Kronecker delta instead of Minkowski, and the Cholesky decomposition of the metric.
<<tiddler HideTags>>
<html><center><u>Groups and representation spaces inside larger groups and representation spaces</u></center></html>
GUT,
$$
\ba{l}
su(3)_S + su(2)_W + u(1)_Y \\[3pt]
+ \; (3,2)_{+\fr{1}{3}} + (1,2)_{-1} + (3,1)_{+\fr{4}{3}} + (3,1)_{-\fr{2}{3}} + (1,1)_{-2} \\
+ \; (\bar{3},2)_{-\fr{1}{3}} + (1,2)_{+1} + (\bar{3},1)_{-\fr{4}{3}} + (\bar{3},1)_{+\fr{2}{3}} + (1,1)_{+2}
\ea
\s \subset \s
so(10) + 16_{s+} + 16_{s-}
\s \subset \s
e_6
$$
ToE or GraviGUT, includes gravity (and fermions),
$$
sl(2,\mathbb{C}) + so(10) + 4 \times 10 + 2 \times 16_{s+} + 2 \times 16_{s-}
\s \subset \s
so(11,3) + 64_{s+}
\s \subset \s
e_{8(-24)}
$$
Should also include three generations of each kind of Dirac fermion, acted on by a finite group,
$$
G_{CPTt} + 8_f \times 3
\s \subset \s
?
$$
<html><center><u>Also, unification of fields and field equations is nice</u></center></html>
$$
\udf{A} = \ha \f{\om} + \f{e} \ph + \f{A} + \ud{\Ps}
\s \s \udff{F} = \f{d} \udf{A} + \udf{A} \udf{A} = \ha \ff{R} - \f{e} \f{e} \ph^2 + \ff{T} \ph - \f{e} \f{D} \ph + \ff{F}^A + \f{D} \ud{\Ps}
$$
$$
S = \int \ha \udff{F} \star \udff{F} \sim
\int d^4x \, |e| \, \{
R + \La + \ha (D \ph)(D \ph) + V(\ph) -\fr{1}{4} F^A F^A + \bar{\Ps} D \!\!\!\! / \, \Ps
\}
$$
<<tiddler HideTags>>Work forwards, guess the answer, then work backwards.
Work forwards:
#[[Gauge fields|principal bundle]], [[gravity|spacetime]] and Higgs in one [[connection]].
#Calculate its [[curvature]] to get the interactions.
#Join fermions as ([[Grassmann|Grassmann number]] valued) [[BRST ghosts|BRST technique]] of a larger connection.
#Correct [[standard model]] and gravitational interactions and charges from the curvature.
Guess the answer:
*Pure [[geometry of a principal bundle|Ehresmann principal bundle connection]] -- just vector fields.
*One very large [[Lie group]] is a match!
Work backwards:
#All interactions from the group [[structure|Lie algebra]], after symmetry breaking.
#Explains exactly what and why [[spinor]]s are.
#Gives three generations.
#Lots still to do, but do-able.~~ ~~
<<tiddler HideTags>>@@display:block;text-align:center;
<html><center>
<img src="talks/StAnth09/images/bubblechamber3.png" width="585" height="440">
</center></html>
$\p{{}_{\small (}^{(}}$
[[Garrett Lisi]] [[Structure and representations of exceptional groups|http://www.birs.ca/birspages.php?task=displayevent&event_id=10w5039]] Banff 2010@@
<<tiddler HideTags>>$$\begin{array}{rcl}
L_D \!\!&\!\!=\!\!&\!\! \bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
\big) \ps
+ \bar{\ps} \ph \ps \\
\!\!&\!\!=\!\!&\!\!
\bar{\ps} \ga^\mu \lp e_\mu\rp^i \big(
\pa_i
+ {\small \frac{1}{4}} \om_i^{\p{i}\nu\rh} \ga_{\nu\rh}
+ G_i^{\p{i}A} T_A
+ {\small \frac{1}{4}} (e_i)^\nu \ga_\nu \ph
\big) \ps \\
\!\!&\!\!=\!\!&\!\!
\bar{\ps} \ve{e} \big( \f{\pa} + \f{H} \big) \ps \vp{|_{\big(}} \\
\!\!&\!\!\p{=}\!\!&\!\! \s\s\; \f{H} = \ha \f{\om} + \f{G} + {\small \frac{1}{4}} \f{e} \ph
\end{array}$$
| $\; \f{e} = \f{dx^i} (e_i)^\mu \ga_\mu \;$ |$\in \f{Cl}^1(1,3)$ |gravitational [[frame]] |
| $\; \ve{e} = \ga^\mu (e_\mu)^i \ve{\pa_i} \;$ |$\in \ve{Cl}{}^1(1,3)$ |inverse [[frame]] |
| $\; \f{\om} = \f{dx^i} \ha \om_i^{\p{i}\mu\nu} \ga_{\mu\nu} \;$ |$\in \f{Cl}^2(1,3)$ |[[spin connection]] |
| $\; \f{G} = \f{dx^i }G_i^{\p{i}A} T_A \;$ |$\in \f{su}(3) \!+\! \f{su}(2) \!+\! \f{u}(1)$ |[[gauge fields|principal bundle]] |
$$
\begin{array}{rcl}
\ff{F} \!\!&\!\!=\!\!&\!\! \f{d} \f{H} + \f{H} \f{H} \\
\!\!&\!\!=\!\!&\!\! \ha ( \f{d} \f{\om} + \ha \f{\om} \f{\om} ) + \fr{1}{16} m^2 \f{e} \f{e}
+ ( \f{d} \f{G} + \f{G} \f{G} ) \\
\!\!&\!\!\p{=}\!\!&\!\! + \fr{1}{4} ( \f{d} \f{e} + \ha [ \f{\om}, \f{e} ] ) \ph - \fr{1}{4} \f{e} ( \f{d} \ph + [ \f{G}, \ph ] ) \\
\!\!&\!\!=\!\!&\!\! \ha \big( \ff{R} + \fr{1}{8} m^2 \f{e} \f{e} \big)
+ \ff{F^G}
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
\end{array}
$$
<<tiddler HideTags>>\begin{eqnarray}
\lp D \!\!\!\! / + \ph \rp \ud{\ps} &=& \ve{e} \, \f{\na} \ud{\ps}
= \ve{e} \lp \f{d} + \f{H} \rp \ud{\ps} \\
\f{H} &=& \ha \f{\om} + \fr{1}{4}\f{e}\ph + \f{B} + \f{W} \\
&\in& \f{\rm Lie}(H) = \f{Cl}^2(1,7) = \f{so}(1,7) \subset \f{\mathbb{C}}(8\times8)
\end{eqnarray}
$$
\begin{array}{ccc}
\begin{array}{rcl}
\ve{e} \!\! &\!=\!& \!\! \ga^\mu \lp e_\mu \rp^a \ve{\pa_a} \\
\f{e} \!\! &\!=\!& \!\! \f{dx^a} \lp e_a \rp^\mu \ga_\mu \\
\f{\om} \!\! &\!=\!& \!\! \f{dx^a} \ha \om_a^{\p{a}\nu \rh} \ga_{\nu \rh} \\
\f{B},\f{W} \!\! &\!=\!& \!\! \f{dx^a} \ha W_a^{\p{a}\ph\ps} \ga_{\ph\ps} \\
\ph \!\! &\!=\!& \!\! \ph^\ph \ga_\ph \\
\fr{1}{4} \f{e} \ph \!\! &\!=\!& \!\! \fr{1}{4} \f{dx^a} \lp e_a \rp^\mu \ph^\ph \ga_{\mu \ph}
\end{array}
&
\;\;\;\;
&
\begin{array}{l}
{\rm My \; funny \; notation \! :} \\
\ve{\pa_a} \, \f{dx^b} = {\bf i}_{\pa_a} dx^b = \de_a^b \\
\ve{e} \f{e} = \ga^\mu \lp e_\mu \rp^a \ve{\pa_a} \, \f{dx^b} \lp e_b \rp^\nu \ga_\nu = 4 \\
\; \\
{\rm Higgs \; "vector" \; constrained \! :} \\
\ph \cdot \ph = \ph^\ph \ph^\ps \et_{\ph\ps} = -M^2
\end{array}
\end{array}
$$
@@display:block;text-align:center;[[indices]]: ^^ ^^spacetime coordinates: $0 \le a,b \le 3$
Clifford labels: (lower, spacetime) $0 \le \mu,\nu,\rh \le 3$, (higher) $5 \le \ph,\ps \le 8$@@
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<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">d_L \to W^- + u_L</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction 2.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">W^- \to e_L + \bar{\nu}_R</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<img src="talks/RW12/images/Weak interaction n.png" width="480" height="480">
</td></tr><tr><td>
<SPAN class="math">n = u_R^r + d_L^g + d_R^b</SPAN>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\begin{array}{rcl}
(g^3 T^g_3 + g^8 T^g_8) \; \ps_{q_r} \!\!&\!\!=\!\!&\!\! i ( g^3 \al^{q_r}_3 + g^8 \al^{q_r}_8 ) \; \ps_{q_r} \\
{\small
\lb \begin{array}{cccccc}
& \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
\!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
& & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
& & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & \\[-.5em]
& & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
& & & & - \fr{1}{\sqrt{3}} g^8 &
\end{array} \rb
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb }
\!\!&\!\!=\!\!&\!\!
{\small
i \lp g^3 \big( \fr{1}{2} \big) + g^8 \big( \fr{1}{2 \sqrt{3}} \big) \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
}
\end{array}
$$
<html><center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="gtable">
<tr><td>
<SPAN class="math">V_{g^{r \bar{g}}} \ps_{q^g} = \ps_{q^r} \vp{A_{\big(}}</SPAN>
</td></tr>
<tr><td>
<SPAN class="math">\al^{g^{r \bar{g}}} + \al^{q^g} = \al^{q^r} \vp{A_{\Big(}}</SPAN>
</td></tr><tr><td>
<img SRC="images/png/quark gluon vertex.png" height=160px>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">\s\s\s</SPAN></td>
<td>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Strong interaction.png" width="324" height="280">
</td></tr><tr><td>
<SPAN class="math">su(3) + 3 + \bar{3} \vp{A^{\big(}}</SPAN>
</td></tr>
</table>
</td>
</tr>
</table>
</center>
</html>
<<tiddler HideTags>>6D Cartan subalgebra: $\;\; C = \om_S \, T^\om_{12} + \om_T \, T^\om_{34} + W \, T^W_3 + Y \, T^Y + g^3 \, T^g_3 + g^8 \, T^g_8 \vp{A_{\big(}}$
$$
{\small
\lb \begin{array}{cccccc}
& \!\! - \! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & \\[-.5em]
\!\! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & & & \\[-.5em]
& & & \!\! \fr{1}{2} g^3 \!-\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & \\[-.5em]
& & \!\! - \! \fr{1}{2} g^3 \!+\! \fr{1}{2 \sqrt{3}} g^8 \!\! & & & \\[-.5em]
& & & & & \fr{1}{\sqrt{3}} g^8 \\[-.5em]
& & & & - \fr{1}{\sqrt{3}} g^8 &
\end{array} \rb
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
=
i \lp \big( \fr{1}{2} \big) g^3 + \big( \fr{1}{2 \sqrt{3}} \big) g^8 \rp
\lb
\begin{array}{c}
1 \\ -i \\ 0 \\ 0 \\ 0 \\ 0
\end{array}
\rb
}
$$
<html><center>
<table class="gtable">
<tr border=none>
<td border=none>
<table class="gtable">
<tr><td>
<table class="gtable">
<tr><td>
weight vectors
</td></tr><tr><td>
weights
</td></tr>
</table>
</td><td><SPAN class="math">\longleftrightarrow</SPAN></td><td>
<table class="gtable">
<tr><td>
states
</td></tr><tr><td>
quantum numbers
</td></tr>
</table>
</td></tr>
<tr><td><SPAN class="math">\updownarrow</SPAN></td><td></td><td><SPAN class="math">\updownarrow</SPAN></td></tr>
<tr><td>
<table class="gtable">
<tr><td>
eigenvectors
</td></tr><tr><td>
eigenvalues
</td></tr>
</table>
</td><td><SPAN class="math">\longleftrightarrow</SPAN></td><td>
<table class="gtable">
<tr><td>
particles
</td></tr><tr><td>
charges
</td></tr>
</table>
</td></tr>
</table>
</td>
<td>
<SPAN class="math">\s\s</SPAN></td>
<td>
<table class="gtable">
<tr><td>
<img src="talks/Zuck09/images/Strong interaction.png" width="324" height="280">
</td></tr><tr><td>
<SPAN class="math">su(3) + 3 + \bar{3} \vp{A^{\big(}}</SPAN>
</td></tr>
</table>
</td>
</tr>
</table>
</center>
</html>
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</script>This is [[Garrett Lisi]]'s personal wiki notebook in theoretical physics.
Each note describes some bit of mathematical physics, while linking it to everything else. The notes are organized by [[Tags]] in an expandable list to the left of this window. Or you can look up mathematical objects by [[symbol|Symbols]], or type any phrase to search for in the field to the left. From any note you can follow links to others or click in the bottom list of notes that link to it. (These are wiki-links, so tabbed browsing won't work in the usual way.) You can also see which notes have been edited recently, under "[[Latest|TabTimeline]]." That's about it for basic orientation — you can read more [[About]] what you're looking at, or pick it up as you go.
Welcome to my brain, have fun looking around.
The last two notes I edited are below.
The left-chiral [[Weyl equation|Dirac equation]] in a [[rest frame]],
$$
0 = \lp i \pa_0 - i \si_\va \pa_\va \rp \psi_{L}
$$
has a positive energy solution, $\ps_L = \ch e^{- i p_\mu x^\mu}$, and a negative energy solution, $\ps_L = \ch e^{+ i p_\mu x^\mu}$, with $p$ the [[momentum]]. These, respectively, reduce the left-chiral Weyl equation to
$$
0 = \pm \lp E + p^\va \si_\va \rp \ch
$$
Since the [[helicity state]]s, $\ch_\pm$, satisfy $p_u^\va \si_\va \, \ch_\pm = \pm \ch_\pm$ the left-chiral Weyl equation is solved by the left-handed state, $\ch = \ch_-$, for both positive and negative energy solutions,
$$
\ps_L = \ch_- e^{- i p_\mu x^\mu}
\s \s
\ps_L = \ch_- e^{+ i p_\mu x^\mu}
$$
The right-chiral Weyl equation,
$$
0 = \lp i \pa_0 + i \si_\va \pa_\va \rp \psi_{R}
$$
is similarly solved by positive and negative energy right-handed helicity states,
$$
\ps_R = \ch_+ e^{- i p_\mu x^\mu}
\s \s
\ps_R = \ch_+ e^{+ i p_\mu x^\mu}
$$
To match [[massless Dirac solutions]], we use the [[charge conjugate]] helicity states or phase-related helicity states,
$$
\xi_\pm = \mp \ep \ch_\mp^* = - \ch_\pm
\s \s
\ch_-^{\wedge/\vee} = \lp 1 - p_u \rp \ch^{\wedge/\vee}
\s \s
\xi_-^{\wedge/\vee} = \lp 1 - p_u \rp \xi^{\wedge/\vee}
$$
// more... //
A ''Weyl [[spinor]]'', $\ps_{L/R}$, is the left or right [[chiral]] part of a Dirac spinor, using the [[Weyl representation|Dirac matrices]] of the [[spacetime]] [[Cl(1,3)]] [[Clifford algebra]]. These parts are the ''left-handed Weyl spinor'' and ''right-handed Weyl spinor'',
$$
\ps_L
= \ps_L^\wedge \ch^\wedge + \ps_L^\vee \ch^\vee
= \lb \begin{array}{c}
\ps_L^\wedge \\
\ps_L^\vee
\end {array} \rb
\s \s
\ps_R
= \ps_R^\wedge \ch^\wedge + \ps_R^\vee \ch^\vee
= \lb \begin{array}{c}
\ps_R^\wedge \\
\ps_R^\vee
\end {array} \rb
$$
each a [[Pauli spinor]]. Instead of using spin eigenvectors, Weyl spinors can also be written in terms of a [[helicity]] eigenspinor basis, $\ps_\pm$, for any chosen [[momentum]].
Under a [[spatial rotation]] or [[Lorentz boost]], a Weyl spinor transforms via the left or right chiral half of a [[Clifford rotation]], $\ps'_{L/R}(x') = U_{L/R} \, \ps_{L/R} = e^{\ha B_{L/R}} \ps_{L/R} (x)$, with rotations and boosts corresponding to
\begin{eqnarray}
U_{L/R}^\text{rotation} &=& \si_0 \cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}} \\
U_{L/R}^\text{boost} &=& \si_0 \cosh{\fr{\ze}{2}} \mp n^\pi \si_\pi \sinh{\fr{\ze}{2}}
\end{eqnarray}
using [[Pauli matrices]]. A Weyl spinor is thus an element of the chiral spinor [[representation space]] of [[spin(1,3)]] represented as $sl(2,\mathbb{C})$.
A ''Weyl spinor field'' over spacetime, $\ud{\ps}_{L/R}(x)$, is a [[Grassmann number]] field or operator over spacetime, which we often conflate with a Weyl spinor.
<<tiddler HideTags>>
$$
L_D = \bar{\ps} \lp \ga^\mu \pa_\mu + i m \rp \ps
$$
$$
\ga^\mu \pa_\mu =
\lb \begin{array}{cccc}
0 & 0 & \pa_0+\pa_3 & \pa_1-i\pa_2 \\
0 & 0 & \pa_1+i\pa_2 & \pa_0-\pa_3 \\
\pa_0-\pa_3 & -\pa_1+i\pa_2 & 0 & 0 \\
-\pa_1-i\pa_2 & \pa_0+\pa_3 & 0 & 0
\end{array} \rb
$$
$$
\ps =
\lb \matrix{
e_L^\wedge \\ e_L^\vee \\ e_R^\wedge \\ e_R^\vee
} \rb
\in \ud{\mathbb{C}}^4
$$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
<b>abstruse goose</b>
</td></tr>
<tr><td><SPAN class="math">$\p{a}$</SPAN></td></tr>
<tr><td>
<img SRC="talks/IfA11/images/wis.png" height=480px>
</td></tr>
</table>
</center></html>
<<tiddler HideTags>>$$
\udf{A} = \f{H} + \f{G} + \ud{\ps}
=
{\small
\lb \begin{array}{cc}
\f{H^+} & \ud{\ps}^- \\
& \f{G^-}
\end{array} \rb
}
\s\s\s\s\s\s\s\s\s\;\;\;
\s\;\;\;
$$
$$
{\small
\!\! = \!\! \lb \begin{array}{cccccccc}
\frac{1}{2} \f{\om_L} \!+\! i \f{W^3} \!&\! i \f{W^1} \!+\! \f{W^2} \!&\! - \! \frac{1}{4} \f{e_R} \ph_0^* \!&\! \frac{1}{4} \f{e_R} \ph_+ \!&
\; \ud{\nu}{}_L &\!\! \ud{u}{}_L^r \!\!&\!\! \ud{u}{}_L^g \!\!&\!\! \ud{u}{}_L^b \\
i \f{W^1} \!-\! \f{W^2} \!&\! \frac{1}{2} \f{\om_L} \!-\! i \f{W^3} \!&\! \p{-} \frac{1}{4} \f{e_R} \ph_+^* \!&\! \frac{1}{4} \f{e_R} \ph_0 \!&
\; \ud{e}{}_L &\!\! \ud{d}{}_L^r \!\!&\!\! \ud{d}{}_L^g \!\!&\!\! \ud{d}{}_L^b \\
-\frac{1}{4} \f{e_L} \ph_0 & \frac{1}{4} \f{e_L} \ph_+ & \! \frac{1}{2} \f{\om_R} \!+\! i \f{B} \! \!& &
\; \ud{\nu}{}_R &\!\! \ud{u}{}_R^r \!\!&\!\! \ud{u}{}_R^g \!\!&\!\! \ud{u}{}_R^b \\
\p{-}\frac{1}{4} \f{e_L} \ph_+^* & \frac{1}{4} \f{e_L} \ph_0^* & &\! \! \frac{1}{2} \f{\om_R} \!-\! i \f{B} \! &
\; \ud{e}{}_R &\!\! \ud{d}{}_R^r \!\!&\!\! \ud{d}{}_R^g \!\!&\!\! \ud{d}{}_R^b \\
& & & & \; i \f{B} &\!\! \!\!&\!\! \!\!&\!\! \\
& & & & &\!\!\! \frac{-i}{3} \! \f{B} \!+\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^1} \!-\! \f{G^2} \!\!\!&\!\!\! i\f{G^4} \!-\! \f{G^5} \\
& & & & &\!\!\! i\f{G^1} \!+\! \f{G^2} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\! i \f{G^{3+8}} \!\!\!&\!\!\! i\f{G^6} \!-\! \f{G^7} \\
& & & & &\!\!\! i\f{G^4} \!+\! \f{G^5} \!\!\!&\!\!\! i\f{G^6} \!+\! \f{G^7} \!\!\!&\!\!\! \frac{-i}{3} \! \f{B} \!-\!\! \frac{2i}{\sqrt{3}}\f{G^8}
\end{array} \rb
}
$$
Note: Only one generation, and fermion masses not quite right.${\p{\big(}}_{\p{(}}$
For three generations: $\udf{A} \; \in \; \f{so}(1,7) + \f{so}(8) + 3 * \ud{\mathbb{R}}(8 \times 8) \; = \; \;\;\; ?{\p{\Big(}}$
BIG Lie algebra: $n \;\, = \;\;\;\;\;\, 28 \;\;\, + \;\; 28 \;\;\, + \;\;3 \; * \; 64 \s = \; 248{\p{\big(}}$
<<tiddler HideTags>><html><center>
<table class="gtable">
<tr><td>
</td></tr>
<tr><td><SPAN class="math">$\p{a}$</SPAN></td></tr>
<tr><td>
<img SRC="talks/ToE24/Legend.png" height=260px>
</td></tr></table>
</center></html>
<html><center>
1936, Isidor Isaac Rabi upon discovery of the muon: <br><br>
"Who ordered that?" <br><br><br>
1976, James Bjorken upon discovery of the tau lepton: <br><br>
"What, another one?" <br><br>
(apocryphal)
</center></html>
A ''Wick rotation'' of fields and algebraic elements is a way of consistently transforming between Lorentzian and Euclidean [[spacetime]], based on the idea of allowing a Lorentzian time coordinate to become complex, as an active [[diffeomorphism]],
$$
t \to t' = e^{i \th} t \in \mathbb{C}
$$
in which $e^{i \th}$ is the Wick phase, with $0 \le \th \le \fr{\pi}{2}$. For a full Wick rotation, $\th = \fr{\pi}{2}$, we have a ''Euclidean time coordinate'', $\ta \in \mathbb{R}$, such that $t' = i \ta$. From the diffeomorphism we have a linear operator, $\f{\ve{L}} = \f{dx^i} L_i{}^j \ve{\pa'_j}$, with $L_i{}^j = \pa_i x'^j = \mathrm{diag}(e^{i \th},1,1,1)$. The [[Dirac Lagrangian]] is covariant under this ''Wick diffeomorphism'' via [[Dirac Lagrangian diffeomorphism covariance]], but this is not quite what we want, since the resulting Lagrangian is still [[spin(1,3)]] invariant and not $spin(4)$ invariant as we wish for Euclidean spacetime.
We can combine the Wick diffeomorphism with a [[Cl(1,3)]] Clifford algebra [[similarity transformation|Clifford adjoint]],
$$
\ga_\mu \to \ga_\mu^\th = S \ga_\mu S^- \s \s S = e^{i \fr{\th}{2} \ga^0 \ga} = \cos \fr{\th}{2} + i \ga^0 \ga \, \sin \fr{\th}{2} \s \s S_E = e^{i \fr{\pi}{4} \ga^0 \ga} = \fr{1}{\sqrt{2}} (1 + i \ga^0 \ga)
$$
in which $\ga = \ga_0 \ga_1 \ga_2 \ga_3$ is the spacetime [[pseudoscalar]], and the operator $S$ is not a [[rotor|Clifford rotation]] but is [[unitary]], $S^\da = S^- = e^{-i \fr{\th}{2} \ga^0 \ga}$. This "rotates" the temporal Clifford vector and psuedoscalar into each other, along with a multiplication by $i$,
$$
\ba{rccclcrcl}
\ga^0_\th \!\!&\!\!=\!\!&\!\! S \ga^0 S^- \ae \ga^0 \cos \th - i \ga \sin \th & \s \s & \ga^0_E \ae -i \ga \\
\ga_\th \!\!&\!\!=\!\!&\!\! S \ga S^- \ae \ga \cos \th - i \ga^0 \sin \th & & \ga^0_E \ae - i \ga_0 \\
\ea
$$
and leaves $\ga^\pi_\th = \ga^\pi$. For $\th = \fr{\pi}{2}$ the Wick transformation of the gamma matrices, via $S_E$, is identical to the [[similarity transformation|Dirac matrices]] relating the Weyl representation to the Dirac representation.
Under this ''Wick similarity transformation'' a [[Dirac spinor]] transforms as $\Ps \to \Ps_\th = S \Ps$, and the [[Dirac adjoint]] transforms as
$$
\bar{\Ps} \xrightarrow{?} \bar{\Ps}_\th = (S \Ps)^\da \ga^0_\th = \Ps^\da \ga^0 S^- = \bar{\Ps} S^-
$$
The Dirac Lagrangian, which is covariant under Wick diffeomorphism, is invariant under Wick similarity transformation, and is covariant under their combination,
$$
\ba{rcl}
\mathcal{L}_\Ps \xrightarrow{?} \mathcal{L}'^\th_\Ps(t,x) \ae \bar{\Psi}'_\th \lp i \ga^\mu_\th (e'_\mu)^i \lp \pa_i + \ha \om'^\th_i + A'_i \rp - m \rp \Ps'_\th(t,x) \\
\ae (\bar{\Psi} S^-) \lp i (S \ga^\mu (e_\mu)^j L^-_j{}^i S^-) L_i{}^k \lp \pa'_k + \ha S \om_k S^- + A_k \rp - m \rp S \Psi (t'(t),x) \\
\ae \bar{\Psi} \lp i \ga^\mu (e_\mu)^k \lp \pa'_k + \ha \om_k + A_k \rp - m \rp \Psi(t'(t),x) = \mathcal{L}_\Ps (t'(t),x)
\ea
$$
in which the [[frame]] transforms as $\ve{e} \to \ve{e}'(t,x) = S \ve{e}(t'(t),x) \f{\ve{L}}^- S^-$. However, again, since this Lagrangian is still not $spin(4)$ invariant, this is not quite what we want. What we //do// want is a different transformation of the the Dirac adjoint such that we get a ''Euclidean adjoint'',
$$
\bar{\Ps} \to \bar{\Ps}_\th = \bar{\Ps} S = \Ps^\da \ga^0 S = \Ps^\da S^- \ga^0 = \Ps_\th^\da \ga^0
$$
This, combined with the Wick diffeomorphism and transformations of $\Ps$ and $\ga^\mu$ by $S$, gives a ''general Wick transformation'' to a ''complex Dirac Lagrangian'',
$$
\ba{rcl}
\mathcal{L}_\Ps \to \mathcal{L}'^\th_\Ps(t,x) \ae \bar{\Psi}'_\th \lp i \ga^\mu_\th (e'_\mu)^i \lp \pa_i + \ha \om'^\th_i + A'_i \rp - m \rp \Psi'_\th(t,x) \\
\ae (\bar{\Psi} S) \lp i (S \ga^\mu (e_\mu)^j L^-_j{}^i S^-) L_i{}^k \lp \pa'_k + \ha S \om_k S^- + A_k \rp - m \rp S \Psi(t'(t),x) \\
\ae \bar{\Psi} S^2 \lp i \ga^\mu (e_\mu)^k \lp \pa'_k + \ha \om_k + A_k \rp - m \rp \Psi(t'(t),x)
\ea
$$
in which $S^2 = \cos \th + i \ga^0 \ga \sin \th$. For $\th = \fr{\pi}{2}$ we have the ''Wick transformation'', giving a ''Euclidean Dirac Lagrangian'',
$$
\ba{rcl}
\mathcal{L}_\Ps \to \mathcal{L}^E_\Ps(t,x) \ae \Ps'^\da_E \ga^0 \lp i \ga^\mu_E (e'_\mu)^j \lp \pa_j + \ha \om'^E_j + A'_j \rp - m \rp \Psi'_E(t,x) \\
\ae \Ps^\da i \ga \lp i \ga^\mu (e_\mu)^k \lp \pa'_k + \ha \om_k + A_k \rp - m \rp \Psi(t'(t),x)
\ea
$$
invariant under $spin(4)$ [[Clifford rotation]]s.
The [[Lagrangian form|action]] for a [[principal bundle]] [[connection]], $\f{A}$, in a curved [[spacetime]] of any dimension, is
$$
\nf{{\cal L}} = \ha \lp \ff{F}, * \ff{F} \rp
$$
in which $\ff{F} = \f{d} \f{A} + \f{A} \f{A}$ is the [[curvature]], $*\ff{F}$ is its [[Hodge dual]], and $\lp,\rp$ is the [[Lie algebra]] [[Killing form]]. Over a local four dimensional spacetime patch, in coordinates, this corresponds to a Lagrangian density, ${\cal L} = \fr{1}{4} \lp F_{\mu \nu}, F^{\mu \nu} \rp$. Extremizing the [[action]] with respect to $\f{A}$,
\begin{eqnarray}
0 &=& \de S = \de \int \ha \lp \ff{F}, * \ff{F} \rp = \int \lp \de \ff{F}, * \ff{F} \rp \\
&=& \int \lp \f{d} \, \de \f{A} + \de \f{A} \, \f{A} + \f{A} \, \de \f{A} , *\ff{F} \rp \\
&=& \int \lp \de \f{A}, \f{d} * \ff{F} + \f{A} * \ff{F} - * \ff{F} \f{A} \rp
\end{eqnarray}
we get the [[Euler-Lagrange equation]] -- the ''Yang-Mills equation'' in curved spacetime,
$$
0 = \f{d}{}_A * \ff{F} = \f{d} * \ff{F} + \lb \f{A}, * \ff{F} \rb
$$
For the $U(1)$ Lie group, in a flat spacetime [[rest frame]], the Lagrangian reduces to
$$
{\cal L} = - \fr{1}{4} F_{\mu \nu} F^{\mu \nu} = - \ha \lp | \f{E} |^2 - | \f{B} |^2 \rp
$$
and the Yang-Mills equation is one of [[Maxwell's equations]].
The ''action'', $S$, of a dynamical system is a specific map, a functional, from the system's configuration space to the real numbers. The action for a field theory is usually an [[integral|integration]] over spacetime,
$$
S = \int \nf{{\cal L}} = \int \nf{e} \, {\cal L} (\Ph,\f{\pa} \Ph) = \int dt \, L
$$
in which $\nf{e}$ is the spacetime [[volume form]], $L$ is the ''Lagrangian'', $\cal{L}$ is the ''Lagrangian density'' which is also usually just called the "Lagrangian", and $\nf{{\cal L}}$ is the ''Lagrangian form''.
#[[Real simple compact Lie groups]]
#[[The Coleman-Mandula theorem]]
#[[BRST gauge fixing]]
##[[BRST technique]]
#[[BRST extended connection]]
#[[E8 connection]]
#[[E8 curvature]]
#[[Action for everything]]
#[[Pati-Salam model plus gravity]]
#[[Gravitational SO(3,1)]]
#[[Electroweak SU(2) and U(1)]]
#[[Graviweak SO(7,1)]]
#[[Graviweak F4]]
#[[E8 periodic table]]
#[[E8 Theory summary]]
#[[E8 Theory discussion]]
##[[principal bundle]]
##[[vector-form algebra]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
##[[Clifford algebra]]
##[[Cl(1,7)|Cl(8)]]
##[[connection]]
##[[Bosonic part of the connection]]
##[[spacetime frame]]
##[[spacetime spin connection]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
##[[Gravitational part of the action]]
##[[Fermionic part of the action]]
#[[Gauge theory geometry]]
#[[Real simple compact Lie groups]]
#[[The Coleman-Mandula theorem]]
#[[BRST gauge fixing]]
##[[BRST technique]]
#[[BRST extended connection]]
#[[E8 Theory discussion]]
##[[principal bundle]]
##[[vector-form algebra]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
##[[Clifford algebra]]
##[[connection]]
##[[Bosonic part of the connection]]
##[[spacetime frame]]
##[[spacetime spin connection]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
##[[Gravitational part of the action]]
##[[Fermionic part of the action]]
The ''adjoint representation'' of a [[Lie group]] is a distinguished [[representation]] whereby the $N$-dimensional representation space is the group's [[Lie algebra]], acted on as
$$
{\rm Ad}_g X = g X g^-
$$
This produces a corresponding adjoint representation of the Lie algebra,
$$
{\rm Ad}_A B = \fr{d}{dt} \lp e^{tA} B e^{-tA} \rp |_0= \lb A, B \rb = A B - B A
$$
An ''affine Lie algebra'', ${\mathfrak{g}}^+$, is an infinite-dimensional [[Lie algebra]] that presents with an infinite tower of basis elements, $T^m_A$, with $N$ basis elements of $\mathfrak{g}$ in each ''level'', $m \in {\mathbb Z}$, and a ''central extension basis element'' ([[central element]]), $C \in {\mathbb R}$. Adding a ''derivation basis element'' (''scaling element''), $D$, produces the corresponding ''affine Kac-Moody algebra''. The nonzero Lie brackets are
$$
\begin{array}{rcl}
\lb T^m_A , T^n_B \rb \!\!&\!\!=\!\!&\!\! C_{AB}{}^C T^{m+n}_C + m \, g_{AB} \de^{m+n,0} C \\
\lb D, T^m_A \rb \!\!&\!\!=\!\!&\!\! m \, T^m_A \\
\end{array}
$$
The $m$ times the [[Killing form]] in the second term above is necessary to satisfy the [[Jacobi identity|Lie algebra]]. There is a $R+1$ dimensional [[Cartan subalgebra|Lie algebra structure]] spanned by the $T^0_a$ in the Cartan subalgebra of the ''level zero subalgebra'', $\mathfrak{g}$, and $C$ (and $D$ if it's Kac-Moody, in which case it's a $R+2$ dimensional Cartan subalgebra).
Another way of presenting this affine Lie (or affine KM) algebra is as
$$
{\mathfrak{g}}^+ = \mathfrak{g} \otimes {\mathbb C}[t,t^-] \oplus {\mathbb C} C \oplus {\mathbb C} t \fr{d}{dt}
$$
in which ${\mathbb C}[t,t^-]$ is the space of Laurent polynomials. The correspondence with the description above is via $T^m_A \sim T_A \otimes t^m$ and $D \sim t \fr{d}{dt}$.
This affine Lie or KM algebra can be expressed in a [[Cartan-Weyl-like basis|Lie algebra structure]], $\{ H^m_a, C, D, V^m_\al \}$, with Lie brackets
\begin{eqnarray}
\big[ H^m_a , H^{-m}_b \big] &=& m \, g_{ab} C \\
\big[ H^m_a , V^n_\al \big] &=& \al_a V^{m+n}_\al \\
\big[ V^m_\al , V^{-m}_{-\al} \big] &=& H^0_\al + m C \\
\big[ V^m_\al , V^n_\be \big] &=& N_{\al \be} \, V^{m+n}_{\al+\be} \\
\big[ D , H^m_a \big] &=& m H^m_a \\
\big[ D , V^m_\al \big] &=& m V^m_\al
\end{eqnarray}
The Cartan-subalgebra is spanned by $\{ H^0_a, C, D \}$ and the resulting [[root system]] is an infinite-in-both-directions tower of root systems of $\mathfrak{g}$, with each level corresponding to $m$. The simple ''affine root'' is $\al_0 = \de - \th$, in which $\th$ is usually the highest root of $\mathfrak{g}$ and $\de$ is a ''null root'', $<\! \de,\de \!>\, = 0$, [[dual|dual space]] to $C$ and orthogonal to all the $\mathfrak{g}$ roots, $<\! \de,\al_a \!>\, = 0$. (The ''real root''s have $<\!\! \al,\al \!\!>\, >0$, while the ''imaginary root''s, including the null roots, have $<\!\! \al,\al \!\!>\, \le 0$.) The other null roots, $n \de$, with $n \in {\mathbb Z} - \{ 0 \}$, have multiplicity $R$, while the remaining real roots have multiplicity $1$. All real roots are specified by $\al + n \de$, with $n \in {\mathbb Z} - \{ 0 \}$. For an affine Kac-Moody algebra, the determinate of its $R+1$ dimensional [[Cartan matrix|root system]] (the normalized metric in the root space spanned by $\al_0$ and the $\al_a$) is zero. This ''generalized Cartan matrix'' typically looks like
$$
\lb
\begin{array}{ccc}
A_{ab} & & 0 \\
& & -2 \\
0 & -2 & 2 \\
\end{array}
\rb
$$
which agrees with the Chevalley-Serre relations and the Cartan-Weyl-like basis above if
$$
H_a = H_a^0 \s E_a^\pm = V_{\pm\al_a}^0 \s H_0 = C - H_\th \s E_0^\pm = V^{\pm1}_{\mp\th}
$$
Since the Cartan subalgebra is $R+2$ dimensional, there is also an ''extendend Cartan matrix'', which typically looks like
$$
\lb
\begin{array}{cccc}
A_{ab} & & 0 & 0 \\
& & -2 & 0 \\
0 & -2 & 2 & 1 \\
0 & 0 & 1 & 0 \\
\end{array}
\rb
$$
and has nonvanishing determinate. This describes a $R+2$ dimensional ''extended Lorentzian root space'', spanned by the ''extended simple roots'' $\{ \al_a, \al_0, \De \}$, metric-dual to $\{ H_a, H_0 , D \}$, or spanned by $\{ \al_a, \de, \De \}$, metric-dual to $\{ H^0_a, C , D \}$. The affine root system is in a $N+1$ dimensional subspace, spanned by the $\al_a$ and $\al_0$ or $\de$. The null-root, $\de = \ha(\ch_+ + \ch_-)$, is an equal combination of orthogonal space-like and time-like basis vectors, $\ch_\pm$, in this extended Lorentzian root space (which is also spanned by $\{ \al_a, \ch_+, \ch_- \}$). The ''null extended root'' (which is not a root, much less a simple root) is then $\De = \ha(\ch_+ - \ch_-)$. For completeness and clarity, the table of metric-dualities and non-zero inner products between this cast of characters is typically
$$
\begin{array}{cc|cccccc}
(,\!) & * & H_{b} & H_\th & H_0 & D & C & H_+ & H_- \\
* & <,\!> & \al_{b} & \th & \al_0 & \De & \de & \ch_+ & \ch_- \\
\hline
H_{a} & \al_{a} & <\! \al_a,\al_b \!> & <\! \al_a,\th \!> & & & & & \\
H_{\th} & \th & <\! \th,\al_b \!> & 2 & -2 & & & & \\
H_0 & \al_0 & & -2 & 2 & 1 & & 1 & -1 \\
D & \De & & & 1 & & 1 & 1 & 1 \\
C & \de & & & & 1 & & 1 & -1 \\
H_+ & \ch_+ & & & 1 & 1 & 1 & 2 & \\
H_- & \ch_- & & & -1 & 1 & -1 & & -2 \\
\end{array}
$$
To summarize, the finite lie algebra root system in $R$ dimensional root space is a subset of the affine root system tower in $R+1$ dimensions which is a subspace of $R+2$ dimensional extended Lorentzian root space, with a $2$ dimensional subspace, orthogonal to the original $R$ dimensional root space, that is spanned by complementary null vectors, $\de$ (a null-root in the affine root system) and $\De$, or by complementary space and time vectors, $\ch_+$ and $\ch_-$. Note that, unlike for a finite dimensional Lie algebra, which has a positive definite Cartan matrix, an affine algebra has a singular Cartan matrix, and the Cartan matrix of an infinite-dimensional Lie algebra is generally indefinite, with compact generators, $(H,H) < 0$, sometimes having real eigenvalues.
An ''almost complex structure'', $\f{\ve{J}}$, on a [[manifold]] is a [[vector projection]] that satisfies,
$$
\f{\ve{J}} \f{\ve{J}} = - \f{\ve{I}}
$$
The product of an almost complex structure with vectors is equivalent to multiplication by $i=\sqrt{-1}$. The manifold is ''complex'', admitting a complex coordinatization, iff the [[FuN curvature]] of the almost complex structure vanishes,
$$
\lb \f{\ve{J}} , \f{\ve{J}} \rb_L = 0
$$
and $\f{\ve{J}}$ is then a [[complex structure]].
Refs:
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***anomalies from the particle physics point of view
An ''antilinear involution'' on a [[Lie algebra]] is a map, $\om : \mathfrak{g} \to \mathfrak{g}$, such that
$$
\om(\om(A)) = A \;\;\;\; {\rm for} \;\; A \in \mathfrak{g}
$$
$$
\om(a A) = a^* \om(A) \;\;\;\; {\rm for} \;\; a \in \mathbb{C}
$$
$$
\om(\lb A, B \rb) = \lb \om(B), \om(A) \rb
$$
The ''antisymmetric bracket'' is an operation on a list of arbitrary [[Lie algebra]] generators or [[Clifford element]]s. The antisymmetric bracket of two elements,
\[ \lb A, B \rb_A = A \times B = \ha \lb A, B \rb = \ha \lp A B - B A \rp \]
equivalent to the ''cross product'', is equivalent to the [[commutator]] bracket with a multiplier of $\ha$. The antisymmetric bracket of three elements is
\[ \lb A, B, C \rb_A = \fr{1}{3!} \lp ABC + BCA + CAB - ACB - CBA - BAC \rp \]
and so on for more elements. An antisymmetric bracket changes sign under the exchange of any two neighboring elements.
The antisymmetric bracket does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the cross product of two [[Clifform]]s,
$$
\f{A} \ti \f{B} = \lb \f{A},\f{B} \rb_A = \f{dx^i} \f{dx^j} \lb A_i, B_j \rb_A
= \f{dx^i} \f{dx^j} \ha \lp A_i B_j - B_j A_i \rp = \ha \lp \f{A} \f{B} + \f{B} \f{A} \rp
$$
In general, for two $p$ and $k$ forms,
$$
\nf{A} \ti \nf{B} = \lb \nf{A}, \nf{B} \rb_A = \ha \lp \nf{A} \nf{B} - \lp -1 \rp^{pk} \nf{B} \nf{A} \rp
$$
The complex scalar product, using the [[Hermitian form]], of two complex vectors (in a [[unitary representation]] space) is anti-invariant under an ''antiunitary'' transformation,
$$
\langle \hat{U}' u | \hat{U}' v \rangle = \langle u | v \rangle ^* = \langle v | u \rangle
$$
A [[unitary]] operator combined with the [[complex conjugation|complex structure]] operation is an ''antiunitary operator'', $\hat{U}' = \hat{U} \hat{K}$. Alternatively, an operator, $\hat{U}'\!\!$, as a matrix, is antiunitary if its [[inverse]], $\hat{U}'^{-} = \hat{U}'^{T} = \hat{U}'^{ \da *}$, is its [[transpose]].
If a vector $v \in V$ is transformed by a Lie group element in an antiunitary representation to $v' = \hat{U}' v$, then linear operators, $\hat{A}$, on $V$ must transform as $\hat{A}' = \hat{U}' \hat{A} \, \hat{U}'{}^-$, such that
$$
\hat{A}' v' = \hat{U}' \hat{A} \, \hat{U}'^- \, \hat{U}' \, v = \hat{U}' \hat{A} \, v
$$
If $\hat{A}$ is multiplication by a complex number, $a \in \mathbb{C}$, then
$$
\hat{U}' \hat{A} \, v = \hat{U}' a \, v = a^* \hat{U}' v
$$
so we have $a' = \hat{U}' a \, \hat{U}'^- = a^*$, and specifically $i' = \hat{U}' i \, \hat{U}'^- = -i$.
Two [[fiber bundle]]s are ''associated'' if they have the same structure group and the same transition functions.
An ''automorphism'' is a structure preserving map from an object to itself.
If the object is a [[group]], with elements satisfying $g_1 g_2 = g_3$, then after a ''group automorphism'', $Aut:G \to G, g \mapsto g' = \ph(g)$, the new group elements must satisfy $g'_1 g'_2 = g'_3$ -- that's what's meant by "structure preserving". The group of all automorphisms of a group, $G$, is called the ''automorphism group'' of $G$, $Aut(G)$. The typical group automorphism is an ''inner automorphism'',
$$
g' = \ph_h(g) = A_h g = h g h^-
$$
with an element $h \in G$ acting on $G$ itself through conjugation (the [[adjoint action|group]]). These form the ''inner automorphism group'', $Inn(G)$. In some cases there may be group automorphisms that are not inner automorphisms. The automorphisms of $G$ which are not inner are called ''outer automorphism''s, and the [[coset]] is labeled $Out(G)=Aut(G)/Inn(G)$.
An ''automorphism bundle'' is a [[fiber bundle]] with a [[Lie group]], $G$, or algebra as the typical fiber and the [[automorphism]] group, $Aut(G)$, acting on $G$ as the structure group.
For most Lie groups the automorphism group is the same as the group itself, $Aut(G)=Inn(G)=G$, with all automorphisms represented by inner automorphisms, and the group action given by the [[adjoint action|group]] of $G$ on the $G$ fiber:
$$
g' = A_h g = h g h^-
$$
for any $h \in G$ in the structure group and $g \in G$ in the fiber. When not all automorphisms of $G$ are inner automorphisms things can get more interesting! But we will first handle the cases for when they are. Note that this bundle is different than a [[principal bundle]], for which the structure group action is the left action -- but there are many similar expressions.
For a section, $g(x)$, transforming under the adjoint action [[gauge transformation]], $g \mapsto g'=h g h^-$, the [[covariant derivative]] is
$$
\f{\na} g = \f{d} g + \f{A} g - g \f{A} = \f{d} g + \lb \f{A} , g \rb
$$
with the ''automorphism bundle [[connection]]'', $\f{A} = \f{dx^i} A_i{}^B T_B$, a 1-form over M valued in the [[Lie algebra]] of $G$.
Any fiber element, $g$, at $t=0$ may be [[parallel transport]]ed to $g(t)=h(t)gh^-$ along a path on the base by a parameter dependent element, $h \in G$, the path holonomy, $h(t) = Pe^{-\int_0^t \f{A}}$, satisfying the [[path holonomy]] equation,
$$
\fr{d}{dt} h(t) = - \ve{v} \f{A} h
$$
Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
\begin{eqnarray}
\f{\na} \f{\na} g &=& \f{d} \lp \f{d} g + \f{A} g - g \f{A} \rp + \f{A} \lp \f{d} g + \f{A} g - g \f{A} \rp + \lp \f{d} g + \f{A} g - g \f{A} \rp \f{A} \\
&=& \lp \f{d} \f{A} \rp g - \f{A} \f{d} g - \lp \f{d} g \rp \f{A} - g \f{d} \f{A}
+ \f{A} \lp \f{d} g + \f{A} g - g \f{A} \rp + \lp \f{d} g + \f{A} g - g \f{A} \rp \f{A} \\
&=& \lb \ff{F} , g \rb
\end{eqnarray}
gives the ''automorphism bundle [[curvature]]'',
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A}
$$
a Lie algebra valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]].
Under a gauge transformation, $g(x) \mapsto g'(x) = h(x) g(x) h^-(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} g' &=& h \lp \f{\na} g \rp h^-\\
\f{d} \lp h g h^- \rp + \f{A'} h g h^- - h g h^- \f{A'} &=& h \lp \f{d} g \rp h^- + h \f{A} g h^- - h g \f{A} h^-
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = h \f{A} h^- - \lp \f{d} h \rp h^- = h \f{A} h^- + h \lp \f{d} h^- \rp
$$
An infinitesimal transformation, $h \simeq 1 + H$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} H - \f{A} H + H \f{A} = \f{A} - \f{\na} H
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \f{A'} \f{A'} = h \ff{F} h^- \simeq \ff{F} + \lb H , \ff{F} \rb
$$
The covariant derivative acting on a Lie algebra valued [[differential form]] such as the curvature, transforming under the adjoint action, $\ff{F'} = h \ff{F} h^-$, is still
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \lb \f{A} , \ff{F} \rb
$$
It is worth repeating that when the automorphism group for $G$ includes automorphisms that are not inner, things are going to get more complicated...
//I'm starting to think that constructing this bundle just doesn't work. For one thing, I don't think I can make an [[Ehresmann connection]] for it since I haven't been able to build automorphism invariant vector fields. For another thing, automorphisms leave the identity point in the same place -- and singling out a point in the fiber like that would be an odd thing to do. I'll shelve the idea for now.//
review:
http://arxiv.org/abs/hep-ph/9512245
A ''bi-split-quaternion'', $q = q^a e'_a \in \mathbb{C} \otimes \mathbb{H}'$, is a [[complex|complex structure]] [[split-quaternion]], with $q^a = q^a_\mathbb{R} + i \, q^a_\mathbb{I}$. It is the analog of the [[biquaternion]]s for split-quaternions. This [[derision algebra|division algebra]] is represented by complex $2 \times 2$ matrices, $\mathbb{C}(2)$, with the split-quaternion basis elements expressed as real matrices, using generalized [[Pauli matrices]] and [[skew]], as
$$
e'_0 = \si_0 \;\;\;\;\; e'_1 = \si_1 \;\;\;\;\; e'_2 = - i \, \si_2 = \ep \;\;\;\;\; e'_3 = \si_3
$$
There are three conjugations: complex, $q^* = q^{a *} e'_a$, quaternionic, $\tilde{q} = q^{a} \tilde{e}{}'_a$, and [[Hermitian]], $q^\da = q^{a *} e'_a{}^\da$, which can be combined and commute. Note that since the basis elements are represented by real matrices, complex conjugation works for matrix representatives, $q^* = q^{a *} e'_a = q^{a *} e'^*_a$, and Hermitian conjugation of the basis elements is the [[transpose]], $e'_a{}^\da = e'_a{}^T$. Explicitly, the representation and a bunch of conjugations are
\begin{eqnarray}
q &=& q_\mathbb{R}^0 \si_0 + i q_\mathbb{I}^0 \si_0
+ q_\mathbb{R}^1 \si_1 + i q_\mathbb{I}^1 \si_1
-i q_\mathbb{R}^2 \si_2 + q_\mathbb{I}^2 \si_2
+ q_\mathbb{R}^3 \si_3 + i q_\mathbb{I}^3 \si_3 \\
\os{q} &=& q_\mathbb{R}^0 \si_0 + i q_\mathbb{I}^0 \si_0
- q_\mathbb{R}^1 \si_1 - i q_\mathbb{I}^1 \si_1
+i q_\mathbb{R}^2 \si_2 - q_\mathbb{I}^2 \si_2
- q_\mathbb{R}^3 \si_3 - i q_\mathbb{I}^3 \si_3 \\
q^* &=& q_\mathbb{R}^0 \si_0 - i q_\mathbb{I}^0 \si_0
+ q_\mathbb{R}^1 \si_1 - i q_\mathbb{I}^1 \si_1
-i q_\mathbb{R}^2 \si_2 - q_\mathbb{I}^2 \si_2
+ q_\mathbb{R}^3 \si_3 - i q_\mathbb{I}^3 \si_3 \\
q^\da &=& q_\mathbb{R}^0 \si_0 - i q_\mathbb{I}^0 \si_0
+ q_\mathbb{R}^1 \si_1 - i q_\mathbb{I}^1 \si_1
+i q_\mathbb{R}^2 \si_2 + q_\mathbb{I}^2 \si_2
+ q_\mathbb{R}^3 \si_3 - i q_\mathbb{I}^3 \si_3 \\
\os{q}^* &=& q_\mathbb{R}^0 \si_0 - i q_\mathbb{I}^0 \si_0
- q_\mathbb{R}^1 \si_1 + i q_\mathbb{I}^1 \si_1
+i q_\mathbb{R}^2 \si_2 + q_\mathbb{I}^2 \si_2
- q_\mathbb{R}^3 \si_3 + i q_\mathbb{I}^3 \si_3 \\
-\ep q^* \ep = \os{q}^\da &=& q_\mathbb{R}^0 \si_0 - i q_\mathbb{I}^0 \si_0
- q_\mathbb{R}^1 \si_1 + i q_\mathbb{I}^1 \si_1
-i q_\mathbb{R}^2 \si_2 - q_\mathbb{I}^2 \si_2
- q_\mathbb{R}^3 \si_3 + i q_\mathbb{I}^3 \si_3 \\
-\ep \os{q} \ep = q^T = q^{*\da} &=& q_\mathbb{R}^0 \si_0 + i q_\mathbb{I}^0 \si_0
+ q_\mathbb{R}^1 \si_1 + i q_\mathbb{I}^1 \si_1
+i q_\mathbb{R}^2 \si_2 - q_\mathbb{I}^2 \si_2
+ q_\mathbb{R}^3 \si_3 + i q_\mathbb{I}^3 \si_3 \\
-\ep q \ep = \os{q}^{*\da} &=& q_\mathbb{R}^0 \si_0 + i q_\mathbb{I}^0 \si_0
- q_\mathbb{R}^1 \si_1 - i q_\mathbb{I}^1 \si_1
-i q_\mathbb{R}^2 \si_2 + q_\mathbb{I}^2 \si_2
- q_\mathbb{R}^3 \si_3 - i q_\mathbb{I}^3 \si_3 \\
\end{eqnarray}
The split-quaternions are the real linear subspace of the bi-split-quaternions, $q^a \in \mathbb{R}$, so satisfying $q^* = q$, corresponding to a flat split-Euclidean spacetime. Another four dimensional linear ''[[Hermitian]] subspace'', $v_M^\da=v_M$, corresponds to positions or vectors in a Minkowski spacetime [[rest frame]],
$$
v_M = v_M^\mu \si_\mu = v^0_{M\mathbb{R}} e_0 + v^1_{M\mathbb{R}} e'_1 + v^2_{M\mathbb{I}} i e'_2 + v^3_{M\mathbb{R}} e'_3 \in M
$$
This is a [[right chiral vector|Dirac matrices]], $v_M=v_R$, in [[Cl(1,3)]]. Its quaternionic conjugate is
$$
\os{v}_M = v_M^0 \si_0 - v_M^\va \si_\va = v_M^\mu \bar{\si}_\mu = v_L
$$
The conventional semi-norm on the bi-split-quaternions is
$$
|v|^2 = \os{v} v = v^0 v^0 - v^1 v^1 + v^2 v^2 - v^3 v^3 \; \in \; \mathbb{C}
$$
but this is complex, and treats the complex and split-quaternionic parts unequally. An alternative norm, of signature $(4,4)$, is
$$
||v||^2 = \os{v}^* v = v^{0*} v^0 - v^{1*} v^1 + v^{2*} v^2 - v^{3*} v^3 \; \in \; \mathbb{R}
$$
The conventional norm on the Hermitian subspace is
$$
|v|^2 = \os{v} v = (v^0_\mathbb{R})^2 - (v^1_\mathbb{I})^2 - (v^2_\mathbb{R})^2 - (v^3_\mathbb{R})^2 \; \in \; \mathbb{R}
$$
A [[Lorentz rotation]], as a [[Clifford rotation]], is carried out by any ''unit bi-split-quaternion'', $\os{g} g = 1$, as
$$
v \to v' = g \, v \os{g}
$$
preserving its conventional norm
$$
\os{v}' v' = \widetilde{(g \, v \os{g})} (g \, v \os{g}) = g \, \os{v} \os{g} g \, v \os{g} = \os{v} v
$$
If we identify $v=v_R$ as a right chiral vector, we can identify $\os{g} = U_R^\da$ as the Hermitian conjugate of a [[right chiral rotor|chiral Clifford rotation]], in agreement with the vector's [[chiral Clifford rotation]].
A ''biquaternion'', $q = q^a e_a \in \mathbb{C} \otimes \mathbb{H}$, is a [[complex|complex number]] [[quaternion]], with $q^a = q^a_\mathbb{R} + i \, q^a_\mathbb{I}$. This eight dimensional [[division algebra]] is represented by and isomorphic to complex 2x2 matrices, $\mathbb{C}(2)$. There are two conjugations, complex, $q^* = q^{a *} e_a$, and quaternionic, $\tilde{q} = q^{a} \tilde{e}_a$, which combine to give a conjugation equivalent to [[Hermitian]] conjugation of the representative matrix, $q^\da = \tilde{q}^* = q^{a *} \tilde{e}_a = q^{a *} e^\da_a$. Note that although the $\da$ conjugation here does correspond to Hermitian conjugation of a representative matrix, $*$ is not complex conjugation of the matrix but of the complex coefficients in the biquaternion. Quaternionic conjugation can also be done by the [[2D matrix conjugate|determinant]], $\tilde{q} = - \ep \, q^T \, \ep$. The biquaternion basis multiplication table is
| $e_0$ | $ie_0$ | $e_1$ | $ie_1$ | $e_2$ | $ie_2$ | $e_3$ | $ie_3$ |
| $ie_0$ | $-e_0$ | $ie_1$ | $-e_1$ | $ie_2$ | $-e_2$ | $ie_3$ | $-e_3$ |
| $e_1$ | $ie_1$ | $-e_0$ | $-ie_0$ | $e_3$ | $ie_3$ | $-e_2$ | $-ie_2$ |
| $ie_1$ | $-e_1$ | $-ie_0$ | $e_0$ | $ie_3$ | $-e_3$ | $-ie_2$ | $e_2$ |
| $e_2$ | $ie_2$ | $-e_3$ | $-ie_3$ | $-e_0$ | $-ie_0$ | $e_1$ | $ie_1$ |
| $ie_2$ | $-e_2$ | $-ie_3$ | $e_3$ | $-ie_0$ | $e_0$ | $ie_1$ | $-e_1$ |
| $e_3$ | $ie_3$ | $e_2$ | $ie_2$ | $-e_1$ | $-ie_1$ | $-e_0$ | $-ie_0$ |
| $ie_3$ | $-e_3$ | $ie_2$ | $-e_2$ | $-ie_1$ | $e_1$ | $-ie_0$ | $e_0$ |
The quaternions are the real linear subspace of the biquaternions, $q^a \in \mathbb{R}$, so satisfying $q^* = q$, corresponding to a flat Euclidean spacetime, with positive definite norm. Another four dimensional linear subspace, the ''hyperbolic quaternions'', corresponding to positions or vectors in a Minkowski spacetime [[rest frame]], is
$$
v_M = v_M^\mu \si_\mu = v^0_{M\mathbb{R}} e_0 + v^1_{M\mathbb{I}} i e_1 + v^2_{M\mathbb{I}} i e_2 + v^3_{M\mathbb{I}} i e_3 \in M
$$
Note that, via their representative matrices, this is a [[right chiral vector|Dirac matrices]], $v_M=v_R$, in [[Cl(1,3)]]. Its conjugates are
$$
v_M^* = \os{v}_M = v_M^0 \si_0 - v_M^\va \si_\va = v_M^\mu \bar{\si}_\mu = v_L
$$
so $M$ is the ''Hermitian subspace'' of the biquaternions, $v_M^\da = v_M$.
The conventional semi-norm on the biquaternions is the same as for quaternions,
$$
|v|^2 = \os{v} v = v^0 v^0 + v^1 v^1 + v^2 v^2 + v^3 v^3 \; \in \; \mathbb{C}
$$
but is complex, as is the induced [[metric]], $(u,v) = \ha \lp \os{u} v + \os{v} u \rp$. This implies a real semi-norm on the hyperbolic quaternions,
$$
|v_M|^2 = \os{v}_M v_M = v^0_{M\mathbb{R}} v^0_{M\mathbb{R}} - v^1_{M\mathbb{I}} v^1_{M\mathbb{I}} - v^2_{M\mathbb{I}} v^2_{M\mathbb{I}} - v^3_{M\mathbb{I}} v^3_{M\mathbb{I}} = {\rm Det}(v_M) \; \in \; \mathbb{R}
$$
related to the [[determinant]]. An alternative positive definite real norm on the biquaternions is
$$
||v||^2 = v^\da v = v^{0*} v^0 + v^{1*} v^1 + v^{2*} v^2 + v^{3*} v^3 \; \in \; \mathbb{R}_+
$$
This second norm treats the complex and quaternionic parts of the biquaternion equally, but we loose the nice relationship between the norm and the determinant.
A [[Lorentz rotation]], as a [[Clifford rotation]], is carried out by any ''unit biquaternion'', $\os{g} g = 1$, as
$$
v \to v' = \os{g} v g^*
$$
preserving its conventional norm
$$
\os{v}' v' = \widetilde{(\os{g} v g^*)} (\os{g} v g^*) = \os{g}^* \os{v} g \os{g} v g^* = \os{v} v
$$
If we identify $v=v_R$ as a right chiral vector, we can identify $g^* = U_R^\da$ as the Hermitian conjugate of a [[right chiral rotor|chiral Clifford rotation]], and $\os{g} = g^{* \da} = U_R$, in agreement with the vector's [[chiral Clifford rotation]].
The biquaternion multiplication table can't be used to build $Cl(8)$, but a $3 \times 3$ matrix of biquaternions is [[su(6)]].
A [[Dirac spinor]] describes both a fermion and an anti-fermion, via positive and negative energy [[Dirac solutions]]. This suggests a re-arrangement of degrees of freedom, using the [[charge conjugate]], to
$$
\Ps = \lb \ba{c} \Ps_1 \\ \Ps_2 \\ \Ps_3 \\ \Ps_4 \ea \rb \s
\Ps^C = i \ga_2 \Ps^* = \lb \ba{c} -\Ps_4^* \\ \Ps_3^* \\ \Ps_2^* \\ -\Ps_1^* \ea \rb \s
\Ps_Q = \lb \ba{cc} \Ps_1 & -\Ps_4^* \\ \Ps_2 & \Ps_3^* \\ \Ps_3 & \Ps_2^* \\ \Ps_4 & -\Ps_1^* \ea \rb
\sim \lb \ba{c} \ps_{\mathbb{H}L} \\ \ps_{\mathbb{H}R} \ea \rb
$$
in which the Dirac spinor degrees of freedom can inhabit either the left or right chiral Dirac [[biquaternion]]s, $\ps_{\mathbb{H}L}$ or $\ps_{\mathbb{H}R}$. Here we make use of the representation of [[quaternion]]s, $e_\mu \in \mathbb{H}$, using [[Pauli matrices]], $\{e_0 \!\sim\! \si_0, \, e_\pi \!\sim\! - i \si_\pi \}$, and the definition of biquaternions as quaternions with complex coefficients.
$$
\ps_{\mathbb{H}L} = \ps^\mu_{\mathbb{H}L} e_\mu
\sim \ps_{QL} =
\lb \ba{cc}
\Ps_1^\mathbb{R} + i \Ps_1^\mathbb{I} & - \Ps_4^\mathbb{R} + i \Ps_4^\mathbb{I} \\
\Ps_2^\mathbb{R} + i \Ps_2^\mathbb{I} & \Ps_3^\mathbb{R} - i \Ps_3^\mathbb{I}
\ea \rb
= \lb \ps_L \; \bar{\ps}_L \rb
\s
\ps_{\mathbb{H}L} \in \mathbb{C} \otimes \mathbb{H}
\s
\ps_{\mathbb{H}R} \sim \ps_{QR} = i \si_2 \ps_{QL}^* \si_1
$$
$$
\ba{rcl}
\ps_{\mathbb{H}L} &=& \ha \lp \Ps_1^\mathbb{R} + \Ps_3^\mathbb{R} - i \Ps_1^\mathbb{I} + i \Ps_3^\mathbb{I} \rp e_0
+ \ha \lp \Ps_2^\mathbb{I} + \Ps_4^\mathbb{I} + i \Ps_2^\mathbb{R} - i \Ps_4^\mathbb{R} \rp e_1 \\[2pt]
& + & \ha \lp -\Ps_2^\mathbb{R} - \Ps_4^\mathbb{R} + i \Ps_2^\mathbb{I} - i \Ps_4^\mathbb{I} \rp e_2
+ \ha \lp \Ps_1^\mathbb{I} + \Ps_3^\mathbb{I} + i \Ps_1^\mathbb{R} - i \Ps_3^\mathbb{R} \rp e_3
\ea
$$
Describing the biquaternions and their representation requires juggling several conjugations. Since the Pauli matrices satisfy $\si_\mu^* = \si_2 \bar{\si}_\mu \si_2$, we can define a similar conjugation for biquaternions, $\ps_{QL}^* \sim - e_2 \ps_{\mathbb{H}L}^* e_2$; and since the Pauli matrices are Hermitian, we also have $\ps_{QL}^\da \sim \tilde{\ps}{}_{\mathbb{H}L}^*$. The invariant bilinear form on the biquaternions directly relates to the scalar Dirac spinor bilinear product,
$$
\ba{rcl}
( \ps_{\mathbb{H}L},\ps_{\mathbb{H}L} ) &=& \lp \Ps_1^\mathbb{R} \Ps_3^\mathbb{R} + \Ps_2^\mathbb{R} \Ps_4^\mathbb{R} +\Ps_1^\mathbb{I} \Ps_3^\mathbb{I} + \Ps_2^\mathbb{I} \Ps_4^\mathbb{I} \rp
+ i \lp - \Ps_1^\mathbb{R} \Ps_3^\mathbb{I} - \Ps_2^\mathbb{R} \Ps_4^\mathbb{I} +\Ps_1^\mathbb{I} \Ps_3^\mathbb{R} + \Ps_2^\mathbb{I} \Ps_4^\mathbb{R} \rp \\[2pt]
&=& \tilde{\ps}_{\mathbb{H}L} \ps_{\mathbb{H}L} = \det(\ps_{QL}) \\[2pt]
\bar{\Ps} \Ps &=& 2 \lp \Ps_1^\mathbb{R} \Ps_3^\mathbb{R} + \Ps_2^\mathbb{R} \Ps_4^\mathbb{R} +\Ps_1^\mathbb{I} \Ps_3^\mathbb{I} + \Ps_2^\mathbb{I} \Ps_4^\mathbb{I} \rp
= 2 \, \Re(\tilde{\ps}_{\mathbb{H}L} \ps_{\mathbb{H}L})
\ea
$$
The action of [[spin(1,3)]] Lorentz generators (rotations and boosts) on biquaternionic spinors is
$$
J_\pi \ps = \lp - \fr{i}{2} \si_0 \otimes \si_\pi \rp \ps \;\; \sim \;\; J_\pi \ps_\mathbb{H} = \fr{1}{2} e_\pi \ps_\mathbb{H} \s \s
K_\pi \ps = \lp \fr{1}{2} \si_3 \otimes \si_\pi \rp \ps \;\; \sim \;\; K_\pi \ps_\mathbb{H} = \fr{i}{2} e_\pi \ps_\mathbb{H}
$$
showing $Spin^+(1,3) = SL(2,\mathbb{C}) = \mathbb{C} \otimes \mathbb{H}^\mathbb{I}$.
We can also calculate [[CPT symmetry]] conjugations for biquaternionic fermions,
$$
\ba{rclcrclcrclcrcl}
\Ps^C \!&\!=\!&\! i \ga_2 \Ps^* &\s& \ps_{QL}^C \!&\!=\!&\! \ps_{QL} \si_1 &\s& \ps_{\mathbb{H}L}^C \!&\!=\!&\! i \ps_{\mathbb{H}L} e_1 &\s& C \!&\!\sim\!&\! i e_1 \\[2pt]
\Ps^P \!&\!=\!&\! i \ga_0 \Ps &\s& \ps_{QL}^P \!&\!=\!&\! - \si_2 \ps_{QL}^* \si_1 &\s& \ps_{\mathbb{H}L}^P \!&\!=\!&\! - \ps_{\mathbb{H}L}^* e_3 &\s& P \!&\!\sim\!&\! -K e_3 \\[2pt]
\Ps^{T} \!&\!=\!&\! \ga_{13} \Ps^* &\s& \ps_{QL}^{T} \!&\!=\!&\! i \si_2 \ps_{QL}^* &\s& \ps_{\mathbb{H}L}^{T} \!&\!=\!&\! - \ps_{\mathbb{H}L}^* e_2 &\s& T \!&\!\sim\!&\! -K e_2
\ea
$$
in which the antiunitary time conjugation operator, $T$, is introduced, instead of $T_U$, to correctly reproduce the $CPT$ Group action on fermions. Two other relations are $\ps_{\mathbb{H}R} = -i \ps_{\mathbb{H}L}^P$ and $\ps_{\mathbb{H}L}^{CPT'} = -i \ps_{\mathbb{H}L}$.
As a nice warm up, before tackling the standard model, it's instructive to see how the [[SU(3)]] [[Lie group geometry]] might go wobbly and become a [[Cartan geometry]]. It has a U(2) [[subgroup]], constructed from the [[simple]] [[SU(2)]] and U(1) Lie groups. The [[homogeneous space]], [[CP2]], is
$$
CP2 = \fr{SU(3)}{U(2)} = \fr{SU(3)}{SU(2) \times U(1)}
$$
The ''center'', $Z(G) \triangleleft G$, of a [[group]], $G$, is an abelian [[normal subgroup]] consisting of all elements of $G$ that commute with all other elements,
$$
Z(G) = \lc z \in G \; | \; gz = zg \; \forall \, g \in G \rc
$$
A ''central element'', $C$, is a [[Lie algebra]] element that commutes with all others -- it is in the Lie algebra's ''center''. The nonzero Lie brackets will be
$$
\lb T_A, T_B \rb = C_{AB}{}^C T_C + \ep(T_A,T_B) C
$$
with $\ep(T_A,T_B)=-\ep(T_B,T_A)$ a ''2-cocycle'', satisfying the Jacobi-ish identity,
$$
\ep(T_A, [T_B, T_C]) = \ep([T_A, T_B], T_C) - \ep( [T_A, T_C], T_B)
$$
For example, for an [[affine Lie algebra]] the 2-cocycle is $\, \ep(T^m_A, T^n_B) = m \, g_{AB} \de^{m+n,0}$.
The ''centralizer'', $C_G(a) \subset G$, of an element $a$ of a [[group]], $G$, is a [[subgroup]] consisting of all elements, $c \in G$, that commute with $a$,
$$
C_G(a) = \lc c \in G \; | \; ca = ac \rc
$$
The centralizer is the largest subgroup of $G$ having $a$ in its [[center]], $a \in Z(C_G(a))$. The centralizer of a subset, $S$, in $G$ is the subgroup consisting of all elements commuting with the elements of $S$,
$$
C_G(S) = \lc c \in G \; | \; cs = sc \; \forall s \in S \rc
$$
The centralizer of $G$ in $G$ is the center, $C_G(G) = Z(G)$. The centralizer of a subset in a subgroup, $H \subset G$, is
$$
C_H(S) = \lc c \in H \; | \; cs = sc \; \forall s \in S \rc
$$
Refs:
*Jeffrey A. Harvey
**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]
***good brief intro
The ''charge conjugate'' of a [[Dirac spinor]] field, $\Ps$, using the Weyl representation of the [[Dirac matrices]], results from the action of an [[antiunitary]] operator,
$$
\Ps^C = i \ga_2 \Ps^* =
\lb \begin{array}{cc}
0 & -i \si_2 \\
i \si_2 & 0
\end{array} \rb
\lb \begin{array}{c}
\ps^*_L \\
\ps^*_R
\end{array} \rb
=
\lb \begin{array}{c}
-i \si_2 \ps^*_R \\
i \si_2 \ps^*_L
\end{array} \rb
=
\lb \begin{array}{c}
(\ps_R)^C \\
(\ps_L)^C
\end{array} \rb
=
\lb \begin{array}{c}
\bar{\ps}_L \\
\bar{\ps}_R
\end{array} \rb
$$
in which we define the ''[[Weyl spinor]] conjugates'',
$$
\bar{\ps}_R = (\ps_L)^C = i \si_2 \ps_L^* = - \ep \, \ps_L^* \;\;\; \text{ and } \;\;\; \bar{\ps}_L = (\ps_R)^C = - i \si_2 \ps_R^* = \ep \, \ps_R^*
$$
using the spinor [[skew]] and the [[complex conjugate|complex structure]] of [[flipped spin]]ors. Note that charge conjugation flips [[chiral]]ity and squares to the identity, $C^2=1$. The [[Dirac equation]] with a charge conjugate [[gauge field|principal bundle]], $\f{A} = \f{A}^B T_B \to \f{A}^C = \f{A}^B T_B^*$, is satisfied by the charge conjugate spinor,
$$
\begin{array}{rcl}
\lp i \ga^\mu (\pa_\mu + A_\mu^C ) - m \rp \Ps^C
\!\!&\!\!=\!\!&\!\! \lp i \ga^\mu ( \pa_\mu + A_\mu^* ) - m \rp i \ga_2 \Ps^* \\
\!\!&\!\!=\!\!&\!\! i \ga_2 \lp - i \ga^{ \mu *} ( \pa_\mu + A_\mu^* ) - m \rp \Ps^* \\
\!\!&\!\!=\!\!&\!\! i \ga_2 \lp \lp i \ga^\mu (\pa_\mu + A_\mu) - m \rp \Ps \rp^* = 0
\end{array}
$$
using $\ga_\mu \ga_2 = - \ga_2 \ga_\mu^*$. Note that if there is a chiral coupling between gauge field and spinor, such as in the weak interaction, $W_\mu P_L \Ps$, this will not be invariant under charge conjugation since $\ga_2 W^*_\mu P_L \Ps^* = W^*_\mu P_R \ga_2 \Ps^*$, in which $P_{L/R}$ is the [[left/right chirality projector]].
Charge conjugation of a [[quantum Dirac spinor]] results from the action of a corresponding [[unitary]] operator in the [[infinite-dimensional unitary representation]],
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^C = \hat{\cal{C}} \ud{\hat{\Ps}} \hat{\cal{C}}^-
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp ( \ud{\hat{a}}_p^{\wedge/\vee})^C {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + (\ud{\hat{b}}_p^{\wedge/\vee \, \da})^C {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! i \ga_2 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp^* \\
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee \, \da} (i\ga_2 {u}_p^{\wedge/\vee \, *}) e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee} (i\ga_2 {v}_p^{\wedge/\vee \, *}) e^{-i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp
\end{array}
$$
using [[Dirac solution identities]], $i \ga_2 \, u_p^{\wedge/\vee \, *} = v_p^{\wedge/\vee}$ and $i \ga_2 \, v_p^{\wedge/\vee \, *} = u_p^{\wedge/\vee}$. The resulting charge conjugation transformation of the [[creation and annihilation operators]] is thus
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^C = \ud{\hat{b}}_p^{\wedge/\vee} \s\; \lp \ud{\hat{b}}_p^{\wedge/\vee \, \da} \rp^C = \ud{\hat{a}}_p^{\wedge/\vee \, \da}
$$
A fermionic Dirac spinor creation operator of some [[momentum]] and spin is used to describe a one-particle state in [[Fock space]]. So, for example, the charge conjugate of a positron with spin up and momentum $p$,
$$
| \bar{e}^{\wedge}_p \!\!> \; = \, \ud{\hat{b}}_p^{\wedge \da} |0\!\!>
$$
is an electron with spin up and momentum $p$,
$$
| e^{\wedge}_p \!\!> \; = \, \ud{\hat{a}}_p^{\wedge \da} |0\!\!>
$$
For massless particles, represented by a [[massless quantum Dirac spinor]], the charge conjugate of a left-handed fermion is a left-handed anti-fermion field.
For a [[quantum Majorana spinor]], satisfying $\ud{\hat{\Ps}}^C = \ud{\hat{\Ps}}$, we must have
$$
\ud{\hat{a}}_p^{\wedge/\vee} = \lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^C = \ud{\hat{b}}_p^{\wedge/\vee}
$$
A [[Clifford matrix representation]] is ''chiral'' if all [[Clifford basis vectors]] are represented by matrices non-zero only in the second and third quadrant blocks. This is the case iff the first [[Pauli matrix|Pauli matrices]] in the [[Kronecker product]] expression of each vector is either $\si_1$ or $\si_2$,
$$
\ga_\al = \si_{1 \, {\rm or} \, 2} \otimes \dots = \lb \ba{cc} & \ga{}^-_\al \\ \ga{}^+_\al & \ea \rb
$$
A representation may be said to be chiral if this holds for the first Pauli matrices in the product, or it may be chiral for the level where the [[spin connection]] lives -- to be concrete, a rep should be called ''n'th level chiral'' if the n'th Kronecker product matrices are all $\si^P_1$ or $\si^P_2$. In a (1st level) chiral representation, all odd [[Clifford grade]] elements are represented by matrices non-zero only in the second and third quadrants, while all even elements are represented by matrices only non-zero in the first and fourth quadrant,
$$
A =
\lb \begin{array}{cc}
A^{e-} & A^{o-} \\
A^{o+} & A^{e+}
\end {array} \rb
\in Cl
$$
A ''chiral spinor'' is half of a [[spinor]]. Expressed in a chiral rep, it can be ''negative chiral'',
$$
\Psi^- =
\lb \begin{array}{c}
\ps^- \\
0
\end {array} \rb
= \Ps^{-a} Q^-_a = \Ps^{a} Q_a
\;\; \in S^-
$$
or ''positive chiral'',
$$
\Psi^+ =
\lb \begin{array}{c}
0 \\
\ps^+
\end {array} \rb
= \Ps^{+b} Q^+_b = \Ps^{\od{b}} Q_\od{b}
\;\; \in S^+
$$
showing the use of ''dotted indices'' or undotted indices, denoting positive or negative chiral spinor indices, which we mostly avoid in favor of using $+$ or $-$ to denote different chiralities. A full spinor may be built by adding two chiral spinors, $\Psi = \Psi^- + \Psi^+$, such as to make a [[Dirac spinor]]. A spinor may also be broken up into n'th level chiral pieces.
The ''chirality projector'', $P_\mp$, is one of a pair of Clifford elements that operate on the suitable level to project out the desired chiral degrees of freedom,
$$
\begin{array}{cc}
P_- =
\lb \begin{array}{cc}
1 & 0 \\
0 & 0
\end {array} \rb &
P_+ =
\lb \begin{array}{cc}
0 & 0 \\
0 & 1
\end {array} \rb &
\end{array}
$$
It is often built using the [[pseudoscalar]].
As a representation space of a spin algebra, the weight vectors (eigenvectors with respect to a Cartan subalgebra) have an even number of positive weights for a positive chiral spinor and an odd number of positive weights for a negative chiral spinor, or vice versa.
In an abuse of language, a ''chiral'' spinor representation space is sometimes alternatively used to mean a complex [[representation space]], as opposed to a real or quaternionic (pseudo-real) one.
Using the Weyl representation of the [[Cl(1,3)]] [[Dirac matrices]], a [[spacetime rotor|Clifford rotation]],
$$
U = \lb \begin{array}{cc}
U_L & 0 \\
0 & U_R
\end{array} \rb
$$
breaks into ''left and right chiral rotor''s, $U^{L/R}$, which are the product of left and right chiral rotation and boost rotors corresponding to a [[spatial rotation]] and [[Lorentz boost]],
$$
U_{L/R} = U_a^{L/R} U_\nu^{L/R} = (\cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}})(\cosh{\fr{\ze}{2}} \mp n'^\rh \si_\rh \sinh{\fr{\ze}{2}}) = U_{L/R}^\mu \si_\mu
$$
in which $U_{L/R}^\mu \in \mathbb{C}$ are ''chiral rotor coefficients''. Since $U_a^{L/R}$ are identical and [[unitary]], and $U_\nu^{L/R}$ are inverses and [[Hermitian]], the left and right chiral rotor's inverses are each others' Hermitian conjugates,
$$
U_{L/R}^\da = U_\nu^{L/R \, \da} U_a^{L/R \, \da} = U_\nu^{L/R} U_a^{L/R \, -}
= U_\nu^{R/L \, -} U_a^{R/L \, -} = U_{R/L}^-
$$
For a [[Clifford rotation]] of a vector,
$$
v \to v' = U v U^- =
\lb \begin{array}{cc}
U_L & 0 \\
0 & U_R
\end{array} \rb
\lb \begin{array}{cc}
0 & v_L \\
v_R & 0
\end{array} \rb
\lb \begin{array}{cc}
U^-_L & 0 \\
0 & U^-_R
\end{array} \rb
=
\lb \begin{array}{cc}
0 & v'_R \\
v'_L & 0
\end{array} \rb
$$
the [[left and right chiral vector|Dirac matrices]]s rotate as
$$
v_{L/R} \to v'_{L/R} = U_{L/R} v_{L/R} U^\da_{L/R}
$$
and [[Weyl spinor]]s rotate as $\ps_{L/R} \to \ps'_{L/R} = U_{L/R} \ps_{L/R}$.
A [[differential form]] [[field|cotangent bundle]], $\nf{f}(x)$, over a [[manifold]], $M$, is ''closed'' iff its [[exterior derivative]] vanishes,
$$
\f{d} \nf{f} = 0
$$
The [[vector space]] of closed $p$-forms over $M$ is labeled $C^p$.
The ''codifferential'' is the ''adjoint exterior derivative''. Operating on a [[differential form]] of grade $p$, it is defined, using the [[exterior derivative]] and [[Hodge dual]] operators, as
$$
\ve{\de} = (-1)^p *^- \f{d} \, *
$$
and decreases the grade of the form by 1. The codifferential is defined to satisfy
$$
<\!\!<\! \f{d} \nf{a},\nf{b} \!>\!\!> \, = \, <\!\!<\! \nf{a},\ve{\de} \nf{b} \!>\!\!>
$$
for the interior product of the exterior derivative of a p-form, $\nf{a}$, and a (p+1)-form, $\nf{b}$, with the interior product of two p-forms defined as
$$
<\!\!<\! \nf{a},\nf{b} \!>\!\!> \, = \, \int \nf{a} * \nf{b}
$$
On manifolds with vanishing [[torsion]] the codifferential is equivalent (//check this//) to the ''covariant vector derivative'', $\ve{\na} = g^{ij} \ve{\pa_i} \na_j = \ve{e}^\al \na_\al$, when applied to a p-form using the [[tangent bundle covariant derivative|tangent bundle connection]] and [[vector-form algebra]]. Working through examples... for 1-forms,
$$
\ve{\na} \f{A} = \ve{e}^\al \na_\al ( \f{e}^\be A_\be )
= \ve{e}^\al ( \f{e}^\be \pa_\al A_\be + w_{\al\ga}{}^{\be} \f{e}^\ga A_\be )
= \pa^\al A_\al + w_{\al}{}^{\al\be} A_\be
= \mathrm{div}(\f{A})
$$
in which we see the [[divergence]], and for 2-forms,
\begin{eqnarray}
\ve{\na} \ff{F} &=& \ve{e}^\al \na_\al ( \ha \f{e}^\be \f{e}^\ga F_{\be\ga})
= \ve{e}^\al ( \ha \f{e}^\be \f{e}^\ga \pa_\al F_{\be\ga} + \ha w_{\al\de}{}^\be \f{e}^\de \f{e}^\ga F_{\be\ga} + \ha \f{e}^\be w_{\al\de}{}^\ga \f{e}^\de F_{\be\ga} ) \\
&=& \f{e}^\al ( \pa^\be F_{\be\al} + w_{\ga}{}^{\ga\be} F_{\be\al} + w^\be{}_\al{}^\ga F_{\be\ga} )
\end{eqnarray}
Ref:
http://en.wikipedia.org/wiki/Codifferential
also see Nakahara, p253
The $p$-th (de Rham) ''cohomology'' of a [[manifold]], $M$, is the [[vector space]],
$$
H^p(M) = C^p / E^p
$$
equal to the [[coset]] of all [[closed]] $p$-forms over $M$ that are not [[exact]]. An element of the cohomology is specified by a closed coset representative, $[\nf{f^C}] \in H^p(M)$, with $\nf{f^C} \in C^p$.
The ''commutator bracket'' (or simply //''commutator''//) of any two arbitrary [[Lie algebra]] generators, [[Clifford element]]s, or operators is another generator, element, or operator equal to
$$
\lb A, B \rb = A B - B A
$$
employing the appropriate product or operator composition. It relates to the [[antisymmetric bracket]] (and cross product) by a factor of $\ha$,
$$
A \times B = \lb A, B \rb_A = \ha \lb A, B \rb = \ha \lp A B - B A \rp
$$
The commutator (called more precisely the ''graded commutator'') does not commute [[coordinate basis 1-forms]] -- these must be taken out of the bracket first. For example, for the commutator of two grade 1 [[Lieform]]s or [[Clifform]]s,
$$
\lb \f{A},\f{B} \rb = \f{dx^i} \f{dx^j} \lb A_i, B_j \rb
= \f{dx^i} \f{dx^j} \lp A_i B_j - B_j A_i \rp = \f{A} \f{B} + \f{B} \f{A}
$$
So, in general, for two $p$ and $k$ forms,
$$
\lb \nf{A}, \nf{B} \rb = \nf{A} \nf{B} - \lp -1 \rp^{pk} \nf{B} \nf{A}
$$
and [[tangent vector]]s are considered $k$ forms with $k=-1$.
The ''complex numbers'', $\mathbb{C}$, are a two-dimensional [[division algebra]], spanned by two basis elements, $e_0$ and $e_1$. The complex identity element (spanning the ''real part'') is $e_0=1$, and the other, $e_1 = i = \sqrt{-1}$, (spanning the ''imaginary part'') is an ''imaginary'' direction. So a complex number, $z = a + i \, b \in \mathbb{C}$, has a ''real component'', $a \in \mathbb{R}$ (equal to its real part, $\text{Re}(z) = z_{\mathbb{R}} = a$), and an ''imaginary component'', $b = z_{\mathbb{I}} \in \mathbb{R}$ (the imaginary part being $\text{Im}(z) = i \, b$). Complex multiplication is commutative and associative. The multiplication table for the basis elements is
| | $1$ | $i$ |
| $1$ | $1$ | $i$ |
| $i$ | $i$ | $-1$ |
which can be written using a ''complex multiplication coefficient matrix'' as $e_a e_b = M_{ab}{}^c e_c$, so, for example, $e_1 e_1 = -e_0$ and $M_{11}{}^0 = -1$. ''Complex conjugation'' can be written synonymously as
$$
K e_0 = e^*_0 = \os{e}_0 = \bar{e}_0 = e_{\os{0}} = e_0 = 1 \;\;\;\;\; \;\;\;\;\; K e_1 = e^*_1 = \os{e}_1 = \bar{e}_1 = e_{\os{1}} = - e_1 = - i
$$
using the ''conjugation operator'', $K$, and satisfies $(z_1 z_2)^* = z_2^* z_1^* = z_1^* z_2^*$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$.
Multiplying a complex number, $z = a + i \, b$, by its conjugate, $z^* = a - i \, b$, gives its norm,
$$
|z|^2 = z^* z = z^a z^b \os{e}_a e_b = a^2 + b^2 \; \in \; \mathbb{R}
$$
allowing us to calculate the inverse of any nonzero complex number, $z^- = \frac{z^*}{|z|^2}$. The [[metric]] on the complex numbers is then
$$
(e_a, e_b) = \ha ( \os{e}_a e_b + \os{e}_b e_a ) = n_{ab} = \de_{ab}
\s \s
(z_1,z_2) = \ha \lp z^*_1 z_2 + z^*_2 z_1 \rp = a_1 a_2 + b_1 b_2
$$
The complex multiplication table (and metric $n_{dc} = \de_{dc}$) can be used to define
$$
\Ga_{cab} = M_{\os{a} \os{b} c} = M_{\os{a}\os{b}}{}^d n_{dc} \s \text{and so} \s \Ga_{000} = 1 \s \Ga_{011} = \Ga_{110} = \Ga_{101} = -1
$$
Further defining $\overline{\Ga}_{cab} = \Ga_{cba}$ (so $\overline{\Ga}_{c} = \Ga_{c}^T$ in terms of a matrix [[transpose]]), these can be used to construct a $4 \times 4$ real [[chiral]] [[Clifford matrix representation]] of $Cl(2)$, the ''cyclic complex representation'', with the Clifford basis vectors represented as [[Hermitian]] matrices (symmetric matrices since they're real),
$$
\ga'_c =
\lb \begin{array}{cc}
0 & \overline{\Ga}_c \\
\Ga_c & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \overline{\Ga}_c{}^b{}_a \\
\Ga_c{}^b{}_a & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \Ga_{ca}{}^b \\
\Ga_c{}^b{}_a & 0
\end{array} \rb
$$
Explicity, from the complex multiplication coefficient matrix, the chiral Clifford vector and bivector representatives are
\begin{eqnarray}
\Ga_0 &=& \lb \ba{cc} 1 & 0 \\ 0 & -1 \ea \rb = \si_3 = K \\
\Ga_1 &=& \lb \ba{cc} 0 & -1 \\ -1 & 0 \ea \rb = - \si_1 = -JK = - L \\
\Ga_{01} &=& \lb \ba{cc} 0 & -1 \\ 1 & 0 \ea \rb = - i \, \si_2 = \ep = J \\
\end{eqnarray}
which relate directly to the [[Pauli matrices]], the [[skew]], the matrix representation of [[sl(2)]], and the [[realification|realify]] of complex structure. These three operators, $\{J,K,L\}$, conflated with their matrix [[representation]], $\{\pi(J)=-i \si_2$, $\pi(K)=\si_3$, $\pi(L)=\si_1\}$, constitute a basis for the [[sl(2)]] Lie algebra, and generally a [[complex structure]], which acts on the representation space, $V = \pi(z) = \lb \ba{c} a \\ b \ea \rb$. A ''phase rotation'', a $spin(2)$ [[Clifford rotation]], with rotor $U = e^{\ha \ph \, \Ga_{01}}$, similar to a [[spatial rotation]], preserves the complex norm.
The complex numbers also have a different faithful real matrix representation that comes from their multiplication table,
$$
(e_c)^b{}_a = M^\os{b}{}_{ac} \s \s
e_0 = \lb \ba{cc} 1 & \\ & 1 \ea \rb = 1 \s \s
e_1 = \lb \ba{cc} & 1 \\ -1 & \ea \rb = -J \s \s
\ga_c =
\lb \begin{array}{cc}
0 & e_c \\
\os{e}_c & 0
\end{array} \rb
$$
that can also be used to build the ''direct complex representation'' for $Cl(2)$, related to $\ga'_c$ by a [[similarity transformation|Dirac matrices]].
A ''complex structure'' is a linear automorphism, $J : V \mapsto V$, of a [[vector space]] that squares to the negative of the identity, $J J = -1$. This complex structure splits the real but ''J-complex'' vector space, $V = V^J_{\mathbb C} = V^J_{\mathbb R} + V^J_{\mathbb I}$, into two (''J-real'' and ''J-imaginary'') distinct equal halves, $V^J_{\mathbb R}$ and $V^J_{\mathbb I}$, satisfying
$$
J \, V^J_{\mathbb R} = V^J_{\mathbb I} \s J \, V^J_{\mathbb I} = - V^J_{\mathbb R}
$$
These halves are [[eigen]]spaces of a ''complex conjugation'' operator, $K$, such that $K \, V^J_{{\mathbb R}/{\mathbb I}} = \pm V^J_{{\mathbb R}/{\mathbb I}}$. (If $V$ is a [[Lie algebra]] then $K$ is a [[Lie algebra involution]], and $J$ is a complex structure on the Lie algebra if it commutes with the adjoint.) As operators, $K$ and $J$ anticommute, and anticommute with their product, $L = J K$. These three operators, $\{J,K,L\}$, constitute a basis for the [[sl(2)]] Lie algebra, which has a [[representation]] with basis $\{J=-i \si_2$, $K=\si_3$, $L=\si_1\}$ using [[Pauli matrices]] acting on $V= \left[\begin{array}{c} V^J_{\mathbb R} \\ V^J_{\mathbb I} \end{array}\right]$. If a vector $v \in V$ is written as $v = u + J w$, with $u, w \in V^J_{\mathbb R}$, then the usual notation for the complex conjugate is $v^* = Kv = u - J w$, and $J$ can be written as $J=i=\sqrt{-1}$, with $v$ a [[complex number]]; however, it is often useful to keep track of $J$ and $i$ separately as commuting complex structures. For example, acting on a J-complex vector space, the $J$ operator splits the space into positive and negative ''i-complex'' eigenspaces,
$$
J \, V_\pm = \pm i \, V_\pm \s
V_\pm = V_{\mathbb R} \mp i V'_{\mathbb R} = V^J_{\mathbb R} \mp i V^J_{\mathbb I} \s
V^J_{\mathbb R} = \ha \lp V_+ + V_- \rp \s
V^J_{\mathbb I} = \fr{i}{2} \lp V_+ - V_- \rp \s
$$
with the i-complex structure matching the J-complex structure of the space (i.e. $V_{\mathbb R} = V^J_{\mathbb R}$ and $V'_{\mathbb R} = V^J_{\mathbb I}$). These conjugate as $K \, V_\pm = V_\mp$.
Similarly, a ''real structure'' is a linear automorphism of a vector space that squares to the identity, $J J = 1$.
The crux of [[triality]] is an isomorphism between [[Clifford vectors|Clifford basis vectors]], positive [[chiral]] [[spinor]]s, and negative chiral spinors, in eight dimensions -- but this also exists in two dimensions as ''complex triality''. This relates to an automorphism of a set of three [[complex number]]s, and to an inner automorphism of [[su(3)]].
The relevant triality is a non-degenerate trilinear function of three [[complex number]]s (or [[split-complex number]]s),
\begin{eqnarray}
T(v,\ps,\ch) &=& v^c \ps^a \ch^b M_{ab\os{c}} = v^0 \ps^0 \ch^0 - v^0 \ps^1 \ch^1 - v^1 \ps^1 \ch^0 - v^1 \ps^0 \ch^1 \\
T'(v,\ps,\ch) &=& v^c \ps^a \ch^b M'_{ab\os{c}} = v^0 \ps^0 \ch^0 + v^0 \ps^1 \ch^1 + v^1 \ps^1 \ch^0 + v^1 \ps^0 \ch^1 \\
\end{eqnarray}
or of the equivalent vector and spinors, using [[division algebra confusion]] with $Cl(2)$ (or $Cl(1,1)$).
A triality automorphism, permuting the three elements in the triplet, corresponds to an inner [[triality automorphism of su(3)]].
A non-null spacetime [[momentum]], $p = p^\mu \ga_\mu$, with mass $m^2 = p^2 = 2 \, r \cdot q$, may be constructed from two non-colinear null momenta,
$$
p = r + q
$$
A corresponding [[Dirac spinor]] is made from two Weyl spinors. We use [[helicity notation]], and build $r$ from a right-handed [[Weyl spinor]], $r ]$, and $q$ from a left-handed Weyl spinor, $q \g$. Positive energy [[Dirac solution|Dirac solutions]] basis are $u_+ = r ] + \al \, q \g$ and $u_- = r \g + \be \, q ]$, which when put in the [[Dirac equation]], using $p_R = r ] \l r + r ] \l r$ and $p_L = r \g [ r + q \g [ q$,
$$
0 = \lb \ba{cc} - m & p_L \\ p_R & - m \ea \rb \lb \ba{c} \al \, q \g \\ r ] \ea \rb
= \lb \ba{c}
- m \, \al \, q \g + q \g [ q \, r ] \\
- m \, r ] + r ] \l r \, \al \, q \g]
\ea \rb
$$
$$
0 = \lb \ba{cc} - m & p_L \\ p_R & - m \ea \rb \lb \ba{c} r \g \\ \be \, q ] \ea \rb
= \lb \ba{c}
- m \, r \g + r \g [ r \, \be \, q ] \\
- m \, \be \, q ] + q ] \l q \, r \g]
\ea \rb
$$
requires $\al = \fr{m}{\l r \, q \g} = \fr{ [ q \, r ]}{m} = - \fr{1}{\be}$, which is sensible since $m^2 = \l r \, q \g [ q \, r ]$.
A ''compound triality decomposition'' of a [[Lie algebra]] is a [[triality decomposition]] in which each triplet element is the outer product of two [[division algebra]] elements, $\mathbb{D}_1 \otimes \mathbb{D}_2$, and the triality subalgebra, $\mathfrak{h} = \mathfrak{h}_1 + \mathfrak{h}_2 = {\rm Tri}(\mathbb{D}_1) + {\rm Tri}(\mathbb{D}_2)$, is the union of their subalgebras, so
$$
\mathfrak{g} = \mathfrak{h}_1 + \mathfrak{h}_2 + v_1 \otimes v_2 + m_1 \otimes m_2 + p_1 \otimes p_2
$$
This produces the [[exceptional magic square]] of Lie algebras. If the structure constants and [[Killing form]] from the triality decomposition of a Lie algebra from one division algebra are
$$
C_{\mathfrak{h}\mathfrak{h}}{}^\mathfrak{h} \;\; C_{\mathfrak{h}v}{}^v \;\; C_{\mathfrak{h}m}{}^m \;\; C_{\mathfrak{h}p}{}^p \;\; C_{vv}{}^\mathfrak{h} \;\; C_{mm}{}^\mathfrak{h} \;\; C_{pp}{}^\mathfrak{h} \;\; C_{vm}{}^p \;\; C_{mp}{}^v \;\; C_{pv}{}^m \;\; g_{\mathfrak{h}\mathfrak{h}} \;\; g_{vv} \;\; g_{mm} \;\; g_{pp}
$$
then the structure constants for the ''compound triality composition'' of two such Lie algebras are
$$ \ba{rclcrclcrcl}
& & &\!\!\!\! \!\!\!\! C^1_{\mathfrak{h}_1\mathfrak{h}_1}{}^{\mathfrak{h}_1} \!\!\!\! \!\!\!\!& &
& &\!\!\!\! \!\!\!\! C^2_{\mathfrak{h}_2\mathfrak{h}_2}{}^{\mathfrak{h}_2}
\!\!\!\! \!\!\!\!& & & \\
C_{\mathfrak{h}_1 v}{}^v \!\!&\!\!=\!\!&\!\! C^1_{\mathfrak{h}_1 v_1}{}^{v_1} \de_{v_2}^{v_2} & &
C_{\mathfrak{h}_1 m}{}^m \!\!&\!\!=\!\!&\!\! C^1_{\mathfrak{h}_1 m_1}{}^{m_1} \de_{m_2}^{m_2} & &
C_{\mathfrak{h}_1 p}{}^p \!\!&\!\!=\!\!&\!\! C^1_{\mathfrak{h}_1 p_1}{}^{p_1} \de_{p_2}^{p_2} \\
C_{\mathfrak{h}_2 v}{}^v \!\!&\!\!=\!\!&\!\! C^2_{\mathfrak{h}_2 v_2}{}^{v_2} \de_{v_1}^{v_1} & &
C_{\mathfrak{h}_2 m}{}^m \!\!&\!\!=\!\!&\!\! C^2_{\mathfrak{h}_2 m_2}{}^{m_2} \de_{m_1}^{m_1} & &
C_{\mathfrak{h}_2 p}{}^p \!\!&\!\!=\!\!&\!\! C^2_{\mathfrak{h}_2 p_2}{}^{p_2} \de_{p_1}^{p_1} \\
C_{vv}{}^{\mathfrak{h}_1} \!\!&\!\!=\!\!&\!\! C^1_{v_1 v_1}{}^{\mathfrak{h}_1} g^2_{v_2 v_2} & &
C_{mm}{}^{\mathfrak{h}_1} \!\!&\!\!=\!\!&\!\! C^1_{m_1 m_1}{}^{\mathfrak{h}_1} g^2_{m_2 m_2} & &
C_{pp}{}^{\mathfrak{h}_1} \!\!&\!\!=\!\!&\!\! C^1_{p_1 p_1}{}^{\mathfrak{h}_1} g^2_{p_2 p_2} \\
C_{vv}{}^{\mathfrak{h}_2} \!\!&\!\!=\!\!&\!\! C^2_{v_2 v_2}{}^{\mathfrak{h}_2} g^1_{v_1 v_1} & &
C_{mm}{}^{\mathfrak{h}_2} \!\!&\!\!=\!\!&\!\! C^2_{m_2 m_2}{}^{\mathfrak{h}_2} g^1_{m_1 m_1} & &
C_{pp}{}^{\mathfrak{h}_2} \!\!&\!\!=\!\!&\!\! C^2_{p_2 p_2}{}^{\mathfrak{h}_2} g^1_{p_1 p_1} \\
C_{vm}{}^p \!\!&\!\!=\!\!&\!\! C^1_{v_1 m_1}{}^{p_1} C^2_{v_2 m_2}{}^{p_2} & &
C_{mp}{}^v \!\!&\!\!=\!\!&\!\! C^1_{m_1 p_1}{}^{v_1} C^2_{m_2 p_2}{}^{v_2} & &
C_{pv}{}^m \!\!&\!\!=\!\!&\!\! C^1_{p_1 v_1}{}^{m_1} C^2_{p_2 v_2}{}^{m_2}
\ea
$$
in which it may be necessary to rescale $C^1$, $C^2$, $g^1$, and $g^2$ to satisfy the [[Jacobi identitiy|Lie algebra]].
Some examples of compound triality decompositions are
\begin{eqnarray}
su(6) &=& su(2) + su(2) + su(2) + u(1) + u(1) + 4^v \otimes 2 + 4^- \otimes 2 + 4^+ \otimes 2 \\
e6 &=& spin(8) + u(1) + u(1) + 8^v \otimes 2 + 8^- \otimes 2 + 8^+ \otimes 2 \\
e7 &=& spin(8) + su(2) + su(2) + su(2) + 8^v \otimes 4^v + 8^- \otimes 4^- + 8^+ \otimes 4^+ \\
e8 &=& spin(8) + spin(8) + 8^v \otimes 8^v + 8^- \otimes 8^- + 8^+ \otimes 8^+ \\
\end{eqnarray}
This relates directly to [[Clifford compound division algebra representation]]s
The ''conformal algebra'', a [[Lie algebra]] isomorphic to [[spin(2,4)]], derives from the Lie brackets of the [[conformal generators|conformal transformation]],
\begin{eqnarray}
\lb L_{\mu\nu}, L_{\rh\si} \rb &=& \et_{\nu\rh} L_{\mu\si} + \et_{\mu\si} L_{\nu\rh} - \et_{\mu\rh} L_{\nu\si} - \et_{\nu\si} L_{\mu\rh} \\
\lb L_{\mu\nu}, P_\rh \rb &=& \et_{\nu\rh} P_\mu - \et_{\mu\rh} P_\nu \\
\lb L_{\mu\nu}, K_\rh \rb &=& \et_{\nu\rh} K_\mu - \et_{\mu\rh} K_\nu \\
\lb P_\mu, K_\nu \rb &=& 2 \lp L_{\mu\nu} - \et_{\mu\nu} D \rp \\
\lb D, P_\mu \rb &=& - P_\mu \\
\lb D, K_\mu \rb &=& K_\mu \\
\end{eqnarray}
The conformal generators for rotations, translations, special conformal transformations, and dilations relate to $spin(2,4)$ bivectors and $spin(2,4)_- \sim Cl(1,3)$ elements as
\begin{eqnarray}
L_{\mu\nu} &\sim& \ga'_{\mu\nu} \sim \ga_{\mu\nu} \\
P_\mu &\sim& -\ga'_{\mu 5} - \ga'_{\mu 6} \\
K_\mu &\sim& -\ga'_{\mu 5} + \ga'_{\mu 6} \\
D &\sim& \ha \ga'_{5 6} \\
\end{eqnarray}
which satisfy the Lie brackets above.
Elements of the [[conformal group]] act as auto[[diffeomorphism]]s of [[Minkowski spacetime|rest frame]], $M$. But, as a [[Lie group]], the conformal group is $Spin(2,4)$ and acts on vectors, $v' \in Cl(2,4)_1$, on ''conformal spacetime'', $M'$, of signature $(2,4)$. The ''conformal embedding'' of Minkowski spacetime into conformal spacetime is via a function, $X : M \to M'$, which induces a map on tangent vectors, $v' : TM \to TM'$, with the corresponding map on Clifford vectors, $v' : Cl(1,3)_1 \to Cl(2,4)_1$. We will establish that the [[Cl(2,4)]] chiral vector from this mapping of a [[Cl(1,3)]] vector is
$$
v'_+(v) = v + \ha (v^2+1) \ga + \ha i (v^2 -1) =
\lb \ba{cc}
-i & v_L \\
v_R & i \, v_R v_L
\ea \rb
$$
This chiral vector transforms under [[conformal group]] translations by $p$ to the translated vector,
$$
U^P_+ v'_+ U^{P-}_- =
\lb \ba{cc}
1 & 0 \\
i \, p_R & 1
\ea \rb
\lb \ba{cc}
-i & v_L \\
v_R & i \, v_R v_L
\ea \rb
\lb \ba{cc}
1 & i \, p_L \\
0 & 1
\ea \rb
=
\lb \ba{cc}
-i & v_L + p_L \\
v_R + p_R & i \, (v_R + p_R) (v_L + p_L)
\ea \rb
= v'_+(v+p)
$$
Every mapped $Cl(1,3)$ vector sits on a null cone in conformal spacetime, $0 = v'_- v'_+$, and is the [[Clifford reflection]] of a null vector, $n' = \ha (v^2+1)(\ga'_6 - \ga'_5)$, through a vector, $( v - \ga'_5)$, with $v = v^\mu \ga'_\mu$, so every mapped vector in $Cl(2,4)_1$,
$$
v'(v) = - \frac{(v - \ga'_5) \, n' (v - \ga'_5)}{(v^2+1)} = v - \ha (v^2+1) \ga'_5 + \ha (v^2-1) \ga'_6
= v + v^2 \ha (\ga'_6 - \ga'_5) - \ha (\ga'_5 + \ga'_6)
$$
is on a paraboloid over $Cl(1,3)_1$ along $ \ha (\ga'_6 - \ga'_5)$ lowered by $\ha (\ga'_5 + \ga'_6)$ -- the ''conformal embedding'' of spacetime. A conformal embedding can also be in the vector space of [[Cl(4,4)]], via [[realification|realify]] of $Cl(2,4)$.
Hmm, if the conformal embedding of Minkowski spacetime is a parabolic section of the null cone, then the conformal embedding of [[de Sitter spacetime]] is a hyperbolic section?
Ref:
*[[Doran, Lasenby - Geometric Algebra for Physicists (2003)|papers/Doran, Lasenby - Geometric Algebra for Physicists (2003).pdf]]
Elements of the ''conformal group'', $Spin(2,4)$, are exponentiated elements of the conformal Lie algebra, [[spin(2,4)]] -- [[Clifford rotation]] rotors, $U' = e^{\ha B'} \in Cl(2,4)$. Via the [[chiral]] representatives of $spin(2,4)$, these then act on spacetime [[Cl(1,3)]] Clifford vectors, multivectors, and spinors, via a ''conformal rotor'', $U_\pm = e^{\ha B_\pm} \in Cl(2,4)_\pm$, with
$$
B_\pm = B + P_\pm + K_\pm + dD_\pm
$$
and ''conformal transformations'',
$$
v_\pm' \to U_\pm v'_\pm U^-_\mp
\s
\Ps_\pm \to U_\pm \Ps_\pm
$$
in which $v'_\pm = v^\mu \ga_\mu - v^5 \ga \pm i \, v^6$ are chiral [[Cl(2,4)]] vectors.
The spacetime bivector, $B$, generates spacetime [[Clifford rotation]]s via a spacetime rotor, $U^B_\pm = e^{\ha B}$.
The translation generators, $P_\pm = \pm 2 i p P_{L/R}$, square to zero, so the ''translation rotor'' is
$$
U^P_\pm = e^{\ha P_\pm} = 1 + \ha P_\pm = 1 \pm i p P_{L/R}
$$
Using the chiral [[Dirac matrices]], these are
$$
U^P_+ = \lb \begin{array}{cc} 1 & 0 \\ i \, p_R & 1 \end{array} \rb
\s
U^P_- = \lb \begin{array}{cc} 1 & - i \, p_L \\ 0 & 1 \end{array} \rb
$$
in which the [[momentum]] is $p_{L/R} = p^0 \si_0 \mp p^\va \si_\va$. Working within $Spin(2,4)_+$, operating on a chiral spinor, or [[twistor]], translation is
$$
U^P_+ \Ps_+ = \lb \ba{cc} 1 & 0 \\ i \, p_R & 1 \ea \rb \lb \ba{c} \ps_L \\ \ps_R \ea \rb
= \lb \begin{array}{c} \ps_L \\ \ps_R + i \, p_R \ps_L \end{array} \rb
$$
which relates to the incidence relation, $\ps_R = i \, p_R \ps_L$.
The special conformal generator, $K_\pm = \mp 2 i k P_{R/L}$, also squares to zero, so the ''special conformal rotor'' is
$$
U^K_\pm = e^{\ha K_\pm} = 1 + \ha K_\pm = 1 \mp i k P_{R/L}
$$
which are
$$
U^K_+ = \lb \begin{array}{cc} 1 & - i \, k_L \\ 0 & 1 \end{array} \rb
\s
U^K_- = \lb \begin{array}{cc} 1 & 0 \\ i \, k_R & 1 \end{array} \rb
$$
in which the ''special conformal vector'' is $k_{L/R} = k^0 \si_0 \mp k^\va \si_\va$.
The dilation generator, $dD_\pm = \pm \ha i d \ga$, squares to $\fr{d^2}{4}$, so generates spacetime ''dilations'' via a ''dilation rotor'',
$$
U^D_\pm = e^{\ha dD_\mp} = \cosh{\fr{\ze}{2}} \pm i \ga \sinh{\fr{\ze}{2}}
$$
in which $\tanh{\ze}=\fr{d}{2}$.
A [[conformal transformation]], or ''global conformal transformation'', is a [[local conformal transformation]] of flat [[Minkowski spacetime|rest frame]] -- a restricted auto[[diffeomorphism]], $x^\mu \to x'^\mu = \ph^\mu(x)$, that leaves the [[metric]] unchanged up to a scaling,
$$
\et_{\mu\nu} \to \et'_{\mu\nu}(x') = \pa_\mu \ph^\rh(x) \, \pa_\nu \ph^\si(x) \, \et_{\rh\si} = \La(x)^2 \et_{\mu\nu}
$$
resulting in a ''conformally flat'' metric. (A [[Lorentz transformation]] is a conformal transformation with unit scaling, $\La=1$.) Equivalently, a conformal transformation is an autodiffeomorphism that corresponds to a Clifford rotation and scaling of the [[frame]],
$$
\f{e} \to \f{e}'(x') = \f{\ve{L}} \f{e} = \La(x) \, U \f{e} U^-
$$
for some [[rotor|Clifford rotation]], $U(x)$, with the [[vector valued form]] ''conformal transformation operator'' defined as
$$
\f{\ve{L}}(x) = \f{dx^i} L_i{}^j \ve{\pa'_j} \s\s L_i{}^j(x) = \pa_i \ph^j(x)
$$
If we consider conformal transformations near the identity, $\ph^\mu(x) \simeq x^\mu + \ep^\mu(x)$ for small $\ep$, then our defining equation to first order is
$$
\pa_\mu \ep_\nu + \pa_\nu \ep_\mu = f(x) \et_{\mu\nu}
$$
with $f(x) = \ha \pa^\mu \ep_\mu$ (from tracing) and $\La^2(x) \simeq 1 + f(x)$. Messing around with this equation, we get
$$
0 = 3 \pa_\nu \pa^\nu \pa^\mu \ep_\mu
\s
2 \pa_\mu \pa_\nu \ep_\rh = \ha (-\et_{\mu\nu} \pa_\rh + \et_{\rh\mu} \pa_\nu + \et_{\nu \rh} \pa_\mu) \pa^\si \ep_\si
$$
which is solved by $\ep_\mu = a_\mu + b_{\mu\nu} x^\nu + c_{\mu\nu\rh} x^\nu x^\rh$. Using this ansatz, and converting these infinitesimal coefficients, restricted by our PDE's, into generators, we get ''conformal generators'' for ''translations'', ''rotations'', ''dilations'', and ''special conformal transformations'',
\begin{eqnarray}
L_{\mu\nu} &=& x_\mu \pa_\nu - x_\nu \pa_\mu \\
P_\mu &=& \pa_\mu \\
K_\mu &=& x^\rh x_\nu \pa_\nu - 2 x_\mu x^\nu \pa_\nu \\
D &=& x^\mu \pa_\mu \\
\end{eqnarray}
The Lie brackets between these generators give the [[conformal algebra]].
A [[spinor]], $\Ps$, undergoes a [[Clifford rotation]] by a rotor, $U = e^{\ha B}$, that is the [[exponentiation]] of a $Cl(p,q)$ [[Clifford algebra]] bivector, $B = \ha B^{\al \be} \ga_{\al \be}$. For some [[Clifford matrix representation]], the ''conjugate spinor'' is
$$
\bar{\Ps} = \Ps^\da \, n
$$
which is the [[Hermitian]] conjugate of the spinor times the ''spinor metric'',
$$
n = \ga_1 \ga_2 ... \ga_p
$$
composed from multiplying the $p$ positive signature [[Clifford basis vectors]]. The spinor metric squares to $n^2 = -(-1)^p$ and commutes or anticommutes with the basis vectors based on their signature. Since the positive signature basis vectors are represented by Hermitian matrices, and the negative signature basis vectors by anti-Hermitian matrices, we have $\ga_\al^\da = n \ga_\al n$. Basis bivectors then satisfy
$$
\ga_{\al \be}^\da = \ga_\be^\da \ga_\al^\da = - (-1)^p n \ga_\be \ga_\al n = (-1)^p n \ga_{\al \be} n
$$
so a bivector, bivector squared, to the third, and to the fourth, are
$$
B^\da = (-1)^p n B n \s ( B B )^\da = - (-1)^p n B B n \s ( B B B )^\da = (-1)^p n B B B n \s ( B B B B )^\da = - (-1)^p n B B B B n
$$
The Hermitian conjugate of a rotor,
$$
U = e^{\ha B} = 1 + \ha B + \ha (\ha B)^2 + \fr{1}{3!} (\ha B)^3 + \fr{1}{4!} (\ha B)^4 + ...
$$
is
\begin{eqnarray}
U^\da &=& - (-1)^p n n + (-1)^p \ha n B n - (-1)^p \ha n (\ha B)^2 n + (-1)^p \fr{1}{3!} n (\ha B)^3 n - (-1)^p \fr{1}{4!} n (\ha B)^4 n \, + ... \\
&=& - (-1)^p n e^{-\ha B} n = - (-1)^p n U^- n
\end{eqnarray}
and we have $U^\da n = n U^-$. This justifies the definition of the conjugate spinor and spinor metric, which are used to define a [[Hermitian form]] for spinors,
$$
\bar{\Ps} \Ps = \Ps^\da n \Ps \;\;\; \to \;\;\; \Ps^\da U^\da n U \Ps = \Ps^\da n U^- U \Ps = \bar{\Ps} \Ps
$$
invariant unter $Spin(p,q)$ Clifford rotations.
The [[charge conjugate]], $\Ps^C = i \ga_2 \Ps^*$, of a [[massless quantum Dirac spinor]] is
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^C &=& i \ga_2 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{L/R} u_p^{L/R} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{R/L \, \da} v_p^{L/R} e^{+i p_\mu x^\mu} \rp^* \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{L/R \, \da} v_p^{R/L} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{R/L} u_p^{R/L} e^{-i p_\mu x^\mu} \rp
\end{array}
$$
using the [[massless Dirac solution identities]], $i \ga_2 u_p^{L/R \, *} = v_p^{R/L}$ and $i \ga_2 v_p^{L/R \, *} = u_p^{R/L}$. The equivalent charge conjugation transformations of the [[creation and annihilation operators of a massless quantum Dirac spinor]] are thus
$$
\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^C = \ud{\hat{b}}_{\, p}^{L/R}
\s \s
\lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^C = \ud{\hat{a}}_{\, p}^{R/L \, \da}
$$
The [[parity conjugate]], $\Ps^P = i \ga_0 \Ps(t,-x)$, of a massless quantum Dirac spinor is
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^P &=& i \ga_0 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, -p}^{L/R} u_{-p}^{L/R} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, -p}^{R/L \, \da} v_{-p}^{L/R} e^{+i p_\mu x^\mu} \rp \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \pm \ud{\hat{a}}_{\, -p}^{L/R} u_p^{R/L} e^{-i p_\mu x^\mu} \mp \ud{\hat{b}}_{\, -p}^{R/L \, \da} v_p^{R/L} e^{+i p_\mu x^\mu} \rp
\end{array}
$$
using $i \ga_0 \, u_{-p}^{L/R} = \pm u_{p}^{R/L}$ and $i \ga_0 \, v_{-p}^{L/R} = \mp v_{p}^{R/L}$. The equivalent parity conjugation transformation of the creation and annihilation operators are thus
$$
\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^P = \mp \ud{\hat{a}}_{\, -p}^{R/L}
\s \s
\lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^P = \pm \ud{\hat{b}}_{\, -p}^{L/R \, \da}
$$
The [[unitary]] [[time conjugate]], $\Ps^{T_U} = i \ga_0 \ga \, \Ps(-t,x)$, of a massless quantum Dirac spinor is
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^{T_U} &=& i \ga_0 \ga \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, -p}^{L/R} u_{-p}^{L/R} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{\, -p}^{R/L \, \da} v_{-p}^{L/R} e^{-i p_\mu x^\mu} \rp \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp i \ud{\hat{a}}_{\, -p}^{L/R} v_p^{R/L} e^{+i p_\mu x^\mu} - i \ud{\hat{b}}_{\, -p}^{R/L \, \da} u_p^{R/L} e^{-i p_\mu x^\mu} \rp
\end{array}
$$
using $i \ga_0 \ga \, u_{-p}^{L/R} = +i \, v_{p}^{R/L}$ and $i \ga_0 \ga \, v_{-p}^{L/R} = -i \, u_{p}^{R/L}$. The equivalent unitary time conjugation transformation of the creation and annihilation operators are thus
$$
\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^{T_U} = -i \, \ud{\hat{b}}_{\, -p}^{L/R \, \da}
\s \s
\lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^{T_U} = +i \, \ud{\hat{a}}_{\, -p}^{R/L}
$$
The time conjugate, $\Ps^T = \ga_{13} \Ps(-t,x)$, of a massless quantum Dirac spinor corresponds to an [[antiunitary]] operator on [[Fock space]],
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^T &=& \hat{T}' \ud{\hat{\Ps}} \hat{T}'^-
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \lp \ud{\hat{a}}_{\, p}^{L/R} \rp^T u_p^{L/R \, *} e^{+i p_\mu x^\mu} + \lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^T v_p^{L/R \, *} e^{-i p_\mu x^\mu} \rp \\
&=& \ga_{13} \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, -p}^{L/R} u_{-p}^{L/R} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{\, -p}^{R/L \, \da} v_{-p}^{L/R} e^{-i p_\mu x^\mu} \rp \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp -i \ud{\hat{a}}_{\, -p}^{L/R} u_p^{L/R \, *} e^{+i p_\mu x^\mu} +i \ud{\hat{b}}_{\, -p}^{R/L \, \da} v_p^{L/R \, *} e^{-i p_\mu x^\mu} \rp \\
\end{array}
$$
using $\ga_{13} \, u_{-p}^{L/R} = -i u_p^{L/R \, *}$ and $\ga_{13} \, v_{-p}^{L/R} = +i v_p^{L/R \, *}$. The time conjugation transformation of the creation and annihilation operators for particle and antiparticles is thus
$$
\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^T = -i \, \ud{\hat{a}}_{\, -p}^{L/R}
\s \s
\lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^T = +i \, \ud{\hat{b}}_{\, -p}^{R/L \, \da}
$$
Applying time conjugation twice, we get
$$
\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^{T^2} = +i \lp \ud{\hat{a}}_{\, -p}^{L/R} \rp^T = - \ud{\hat{a}}_{\, p}^{L/R}
\s \s
\lp \ud{\hat{b}}_{\, p}^{R/L \, \da} \rp^{T^2} = -i \lp \ud{\hat{b}}_{\, -p}^{R/L \, \da} \rp^T = - \ud{\hat{b}}_{\, p}^{R/L \, \da}
$$
and so $\lp \ud{\hat{a}}_{\, -p}^{L/R} \rp^T = +i \, \ud{\hat{a}}_{\, p}^{L/R}$ and $\lp \ud{\hat{b}}_{\, -p}^{R/L \, \da} \rp^T = -i \, \ud{\hat{b}}_{\, p}^{R/L \, \da}$, which looks very strange when we have $\lp \ud{\hat{a}}_{\, p}^{L/R} \rp^T = -i \, \ud{\hat{a}}_{\, -p}^{L/R}$. Somehow the $-p$ case is giving a different phase than for $p$. What this implies is that one simply should not use the $L/R$ basis solutions when dealing with time conjugation. Instead, one should use the $\wedge/\vee$ up/down spin basis massive solutions in the massless limit. Nevertheless, it is sometimes useful to overlook this fact.
We can use these conjugations to convert between any two operators of equal or opposite [[momentum]]. For example, a complete set is
$$
\begin{array}{rclcrcl}
\ud{\hat{a}}_{\, p}^L \!\!&\!\!=\!\!&\!\! \ud{\hat{a}}_{\, p}^L
&\;\;\;\;&
\ud{\hat{b}}_{\, p}^L \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^C
\\
\ud{\hat{a}}_{\, p}^R \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^{P T}
&\;\;\;\;&
\ud{\hat{b}}_{\, p}^R \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^{C P T}
\\
\ud{\hat{a}}_{\, -p}^L \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^T
&\;\;\;\;&
\ud{\hat{b}}_{\, -p}^L \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^{C T}
\\
\ud{\hat{a}}_{\, -p}^R \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^P
&\;\;\;\;&
\ud{\hat{b}}_{\, -p}^R \!\!&\!\!\sim\!\!&\!\! (\ud{\hat{a}}_{\, p}^L)^{C P}
\\
\end{array}
$$
with phases left as an exercise for the reader. These can also be related to creation operators via [[creation conjugation|creation conjugate]].
Note that since the right-handed partner of a left-handed [[quantum Weyl spinor]] may not exist, its charge conjugate and parity conjugate may not be well defined, but its charge-parity conjugate and time conjugates are.
A ''connection'' (or ''gauge field'') completely encodes the local geometry of a [[fiber bundle]]. Specifically, it describes how the local trivializations change as one moves around on the base manifold. The group of these changes is the same as the structure group, $G$, of the fiber bundle. From any point, the infinitesimal change of a local trivialization when moving in any direction is described by the operation of a [[Lie algebra]] element. These changes may be described via a [[Lie algebra]] valued [[1-form]] over the base, the connection,
$$
\f{A} = \f{dx^i} A_i{}^B(x) T_B
$$
with the appropriate action on the fiber elements. Using this connection, the [[covariant derivative]] of a section, $\si(x)$, (valued in the fiber) is
$$
\f{\na} \si = \f{d} \si + \f{A} \si = \f{dx^i} \lp \pa_i \si + A_i{}^B T_B \si \rp
$$
in which the Lie algebra basis elements, $T_B$, act on the fiber. The connection changes under a [[gauge transformation]] so as to keep this derivative covariant.
The ''coordinate basis 1-forms'', $\f{dx^i}$, are [[1-form]]s dual to the [[coordinate basis vectors]],
\[ \f{dx^{i}}(\ve{\partial_j}) = \ve{\partial_j} \f{dx^{i}} = \delta_{j}^{i} \]
The ''coordinate basis forms'' (//''coordinate basis p-forms''//) are constructed by taking the [[wedge product]] of (p) [[coordinate basis 1-forms]],
$$
\nf{dx^{i \dots j}} = \f{dx^i} \dots \f{dx^j}
$$
For example, the ''coordinate basis 2-forms'' are
$$
\ff{dx^{ij}} = \f{dx^i} \f{dx^j}
$$
The wedge product between basis 1-forms is implied but never written, as the antisymmetric nature of the form product is assumed in the [[vector-form algebra]]. The coordinate basis forms are [[antisymmetric|index bracket]], changing sign under the interchange of any two adjacent indices, $\ff{dx^{ij}}=-\ff{dx^{ji}}$. On an $n$ dimensional manifold, the highest grade coordinate basis form is the ''coordinate basis $n$-form'',
\[ \nf{d^n x} = \f{dx^0} \dots \f{dx^{n-1}} \]
Technically, there is also a coordinate basis $0$-form,
$$
1
$$
For any grade, $p$, there are $\frac{n!}{\left(n-p\right)!p!}$ distinct coordinate basis $p$-forms. Adding these up over the $n+1$ possible grades, including the basis 0-form, there are $2^n$ distinct coordinate basis forms.
The coordinate basis vectors,
$$
\ve{\pa_i} = \ve{\frac{\partial}{\partial x^i}} \in T_p M
$$
with coordinate [[index|indices]], $i$, span the space, $T_p M$, of [[tangent vector]]s to possible curves passing through each point, $p$, of a [[manifold]], $M$. The basis vectors may not be colinear, but are not otherwise inherently related — unlike the [[Clifford basis vectors]], they are not necessarily orthogonal or of unit length.
The coordinate basis vectors may be visualized as little arrows pointing along each coordinate curve. Each ''coordinate basis vector'', $\ve{\pa_i}$, is the [[tangent vector]] to the curve formed by varying the $x^i$ coordinate while holding the others fixed.
The coordinates, $x^i$, used to describe points in a [[manifold]] patch may always be abandoned in favor of a new set of coordinates, $y^i$. Since coordinates in the old and new set describe the same manifold points, the new coordinates may be written as functions of the old, $y^i(x)$, and the old as functions of the new, $x^i(y)$. Similarly, two such sets of coordinates must be used in coordinate patch overlaps on the manifold.
The [[coordinate basis vectors]] are different in the two sets of coordinates, and are related by the partial derivatives of the old and new coordinates as functions of each other:
$$
\ve{\pa^y_i} = \ve{\fr{\pa}{\pa y^i}} = \fr{\pa x^j}{\pa y^i} \ve{\fr{\pa}{\pa x^j}} = \fr{\pa x^j}{\pa y^i} \ve{\pa^x_j}
\quad \quad \quad
\ve{\pa^x_i} = \ve{\fr{\pa}{\pa x^i}} = \fr{\pa y^j}{\pa x^i} \ve{\fr{\pa}{\pa y^j}} = \fr{\pa y^j}{\pa x^i} \ve{\pa^y_j}
$$
With the partial derivative matrices satisfying
$$
\fr{\pa x^j}{\pa y^i} \fr{\pa y^i}{\pa x^k} = \de_k^j
$$
Since $x$ and $y$ are coordinates for the same point, the partial derivative matrices may equivalently be considered functions of $x$ or $y$ as necessary. Similarly, the [[coordinate basis 1-forms]] are related by
$$
\f{dy^i} = \fr{\pa y^i}{\pa x^j} \f{dx^j}
\quad \quad \quad
\f{dx^i} = \fr{\pa x^i}{\pa y^j} \f{dy^j}
$$
and the [[partial derivative]]s, $\pa_i$, of a function (or field components), $f(x)$, in different coordinate systems are related by
$$
\pa^x_i f(x) = \fr{\pa}{\pa x^i} f(x) = \fr{\pa y^j}{\pa x^i} \fr{\pa}{\pa y^j} f(x(y)) = \fr{\pa y^j}{\pa x^i} \pa^y_j f(y)
$$
A [[natural]] geometric object is invariant under coordinate change. For example, [[tangent vector]]s and [[1-form]]s may be expressed in terms of either set of coordinate basis vectors and forms,
\begin{eqnarray}
\ve{v} &=& v^i \ve{\fr{\pa}{\pa x^i}} = v^i \fr{\pa y^j}{\pa x^i} \ve{\fr{\pa}{\pa y^j}} = v'^j \ve{\fr{\pa}{\pa y^j}} = \ve{v'}\\
\f{f} &=& f_i \f{dx^i} = f_i \fr{\pa x^i}{\pa y^j} \f{dy^j} = f'_j \f{dy^j} = \f{f'}
\end{eqnarray}
In old terminology, tangent vectors are described by "contravariant" (upper) indexed components transforming as $v'^j = v^i \fr{\pa y^j}{\pa x^i}$ and forms are described by "covariant" (lower) indexed components transforming as $f'_j = f_i \fr{\pa x^i}{\pa y^j}$ under coordinate change. Any indexed object transforming this way under coordinate change is called a ''tensor''.
Another way of looking at coordinate change is as the identity map from the manifold to itself -- technically a [[diffeomorphism]]. However, a coordinate change is a ''passive diffeomorphism'' as it does not move the manifold points, but only mixes (re-labels) their coordinates.
A collection of elements called a ''coset'', $G/H$, can be formed by modding a [[group]], $G$, by a [[subgroup]], H. Specifically, a ''left coset'' element, $[g] \in G/H$, consists of all elements of $G$ related by the right action of elements of $H$,
$$
[g] = gH = \left\{ gh : \forall \; h \in H \right\}
$$
A coset is not a group unless $H$ is a [[normal subgroup]], in which case $G/H$ is called the ''quotient group''.
An example, let $G$ be the set of integers,
$$
G = \left\{ \dots, -2, -1, 0, 1, 2, \dots \right\}
$$
with addition, $+$, as the group product. Choose the subgroup, $H$, to be all elements of $G$ that are multiples of $4$,
$$
H = \left\{ \dots, -8, -4, 0, 4, 8, \dots \right\}
$$
The left coset consists of four ( $=$ the ''index'' of $H$ in $G$) elements,
\begin{eqnarray}
G/H &=& \left\{
\lb \dots, -3, 1, 5, \dots \rb,
\lb \dots, -2, 2, 6, \dots \rb,
\lb \dots, -1, 3, 7, \dots \rb,
\lb \dots, 0, 4, 8, \dots \rb \right\} \\
&=& \left\{ [1], [2], [3], [0] \right\}
\end{eqnarray}
The notation "$[g]$" means that $g \in G$ is a ''coset representitive'' -- any other representative related by $h \in H$ is equivalent, $[g]=[gh]$. Every element of $G$ is in exactly one of the coset elements, and each coset element is isomorphic to $H$ -- in fact, one of the coset elements, $[0]$, is $H$. A set of representatives of all the cosets is called a ''transversal'' (or ''coset representative section''), and is a map, $\si : G/H \mapsto G$. There is a product between coset elements determined by the product of their representatives and representative equivalence. For example, $[1]+[3] = [4] = [0]$. And, in this example, the coset does form a group, since $H$ is normal in $G$,
$$
ghg^- = g + h - g = h \in H
$$
A ''right coset'' element, $[g] \in H \backslash G$, consists of all elements of $G$ related by the left action of elements of $H$,
$$
[g] = Hg = \left\{ hg : \forall \; h \in H \right\}
$$
ref:
[[Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)|http://arxiv.org/abs/1205.3365]]
The ''cotangent bundle'' (//''1-form bundle''//), $T^* M = \Om^1 M$, with $n$ dimensional base [[manifold]], $M$, is a [[vector bundle]] with $n$ fiber basis elements identified as the [[coordinate basis 1-forms]], $\f{dx^i}$, for the manifold. It is the dual bundle to the [[tangent bundle]]. The fiber at a base manifold point, $p$, is the $n$ dimensional cotangent space, $T_p^* M$, spanned by the basis 1-forms, and the cotangent bundle is the union of all cotangent spaces, $T^* M = \bigcup_{p \in M} T_p^* M$. The transition functions for the basis elements, $\f{dx^i_2} = \lp t^{21} \rp_j{}^i \f{dx^j_1}$, over overlapping patches, $U_1$ and $U_2$, are given by the ''Jacobian matrix'',
$$\lp t^{21} \rp_j{}^i = \fr{\pa x_2^i}{\pa x_1^j}$$
The structure group is thus the group of general linear transformations, $G = GL(n,\Re)$. A ''covector field'' (//''1-form field''//), $\f{f} = \f{f}(x) = f_i(x) \f{dx^i}$, over the manifold is a section of the cotangent bundle, and consists of a [[1-form]] at each manifold point.
When a [[metric]] exists for the tangent bundle the [[frame]] basis forms, $\f{e^\al} = \f{dx^i} \lp e_i \rp^\al$, may alternatively be used as local fiber basis elements for the cotangent bundle. The transition functions are then [[Lorentz transformations|Lorentz rotation]], $\f{e^\al_2} = \lp L^{21} \rp_\be{}^\al \f{e^\be_1}$. This [[reduction of the structure group]] is the same as for the tangent bundle. Similarly, through equating the Lorentz transition functions and using the [[frame]], $\ve{e} \f{e^\al} = \ga^\al$, the cotangent bundle may be [[associated]] to the [[Clifford vector bundle]].
Since the cotangent bundle is dual to the tangent bundle, its geometric elements -- including the [[cotangent bundle connection]], holonomy, curvature, etc. -- are the dual constructions to those for the tangent bundle, and provide no new geometric insight.
Grade $p$ [[differential form]] fields are sections of the ''p-form bundle'', $\Omega^p M$, which has the $\frac{n!}{\left(n-p\right)!p!}$ [[coordinate basis p-forms|coordinate basis forms]], $\nf{dx^{i \dots j}}=\f{dx^i} \dots \f{dx^j}$, as basis. The combined collection of these p-form product bundles is the ''differential form bundle'', $\Omega M = \bigoplus_{p=0}^{n} \Omega^{p} M$, having dimension $2^{n}$.
The [[vector bundle connection]] for the [[cotangent bundle]] is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on [[coordinate basis 1-forms]], giving the ''cotangent bundle covariant derivative'',
\begin{eqnarray}
\na_i \f{dx^j} &=& -\Ga^j{}_{ik} \f{dx^k} \\
\f{\na} \f{dx^j} &=& -\f{\Ga}^j{}_k \f{dx^k}
\end{eqnarray}
The coefficients, $\Ga^j{}_{ik}$, of the ''cotangent bundle connection'', $\f{\Ga}^j{}_k = \f{dx^i} \Ga^j{}_{ik}$, are referred to as the [[Christoffel symbols]], and the index positions are arranged to agree with convention rather than having the usual order for connection coefficients. These are the same Christoffel symbols that arise in the [[tangent bundle connection]] since the basis elements are in [[dual space]]s,
$$
0 = \na_i \de_j^k = \na_i \lp \ve{\pa_j} \f{dx^k} \rp
= \lp \Ga^m{}_{ij} \ve{\pa_m} \rp \f{dx^k} - \ve{\pa_j} \lp \Ga^k{}_{im} \f{dx^m} \rp
= \Ga^k{}_{ij} - \Ga^k{}_{ij}
$$
An alternative expression for the cotangent bundle connection may be found by calculating the covariant derivative of the [[vielbein 1-forms|frame]],
\begin{eqnarray}
\na_i \f{e^\al} &=& \lp \pa_i \lp e_j \rp^\al - \lp e_k \rp^\al \Ga^k{}_{ij} \rp \f{dx^j} = w_{i\be}{}^\al \f{e^\be} \\
\f{\na} \f{e^\al} &=& \f{d} \f{e^\al} - \f{dx^i} \f{dx^j} \lp e_k \rp^\al \Ga^k{}_{ij} = \f{d} \f{e^\al} - \ff{T^\al} = \f{w}{}_\be{}^\al \f{e^\be}
\end{eqnarray}
in which the [[tangent bundle spin connection|tangent bundle connection]] coefficients, $ w_{i\be}{}^\al$, appear. This last equation, involving the [[torsion]] coefficients, $T^\al{}_{ij} = 2 \Ga^\al{}_{\lb ij \rb}$, may be solved for the spin connection coefficients by solving [[Cartan's equation]]. Note that the cotangent bundle covariant derivative is equivalent to the [[exterior derivative]] if and only if the torsion vanishes.
The cotangent bundle covariant derivative extends via the [[distributive rule|derivation]] to act on [[differential form]]s of higher order,
\begin{eqnarray}
\na_i \f{dx^j} \f{dx^k} \dots \f{dx^l} &=& \lp \na_i \f{dx^j} \rp \f{dx^k} \dots \f{dx^l} + \f{dx^j} \lp \na_i \f{dx^k} \rp \dots \f{dx^l} + \f{dx^j} \f{dx^k} \dots \lp \na_i \f{dx^l} \rp \\
&=& - \Ga^j{}_{im} \f{dx^m} \f{dx^k} \dots \f{dx^l} - \f{dx^j} \Ga^k{}_{im} \f{dx^m} \dots \f{dx^l} - \f{dx^j} \f{dx^k} \dots \Ga^l{}_{im} \f{dx^m}
\end{eqnarray}
The ''covariant derivative'' operator is a grade $1$ [[derivative|derivation]] of a [[fiber bundle]] section (field) that accounts for the local trivialization (change of basis) via the appropriate [[connection]]. It may be written as a [[1-form]] operator, or as a derivative with respect to a specific coordinate direction,
$$
\f{\na} = \f{dx^i} \na_i
$$
It is defined to have the following properties,
$$
\f{\na} \lp \nf{B} + f \nf{C} \rp = \f{\na} \,\nf{B} + f \f{\na} \, \nf{C} + \lp \f{d} f \rp \nf{C}
$$
where $f$ is any scalar [[function]] over the base manifold, $\f{d}$ is the [[exterior derivative]], and $\nf{B}$ and $\nf{C}$ are any tangent vector, differential form, Clifford element, or generally any fiber bundle section or direct product of sections. Using the [[partial derivative]] and connection, the covariant derivative of a section is
\begin{eqnarray}
\na_i B &=& \pa_i B + A_i B \\
\f{\na} B &=& \f{\pa} B + \f{A} B
\end{eqnarray}
A section is ''horizontal'' at a point iff its covariant derivative vanishes, $\f{\na} B = 0$.
In general, the covariant derivative may be defined for any fiber valued form that varies under a [[gauge transformation]] as $\nf{B'} = g \, \nf{B}$. Using the [[exterior derivative]],
$$
\f{\na} \nf{B} = \f{d} \nf{B} + \f{A} \nf{B}
$$
This operator generalizes further to a covariant derivative operating as forms that transform arbitrarily under the gauge group and aren't necessarily sections. For example, operating on the the [[curvature]], which transforms as $\ff{F'} = g \, \ff{F} \, g^-$, the covariant derivative is
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \ff{F} - \ff{F} \f{A}
$$
"Covariance" refers to the property that the covariant derivative of any field transforms under the same group action as the field. The covariant derivative thus plays an essential role in constructing gauge invariant objects, and this restriction provides the rule for the behavior of the connection under gauge transformation. When it is not obvious, the covariant derivative should be labeled with the symbol(s) of the connection(s) for the bundle for which it is covariant,
$$
\f{\na}{}^A C = \f{dx^i} \na^A_i C = \f{dx^i} \lp \pa_i C + A_i C \rp = \f{d} C + \f{A} C
$$
A ''creation operator'', $\hat{a}^\da$, and corresponding ([[Hermitian]] conjugate) ''annihilation operator'', $\hat{a}$, acting on a [[Hilbert space]] satisfy the ''canonical commutation relation'',
$$
\lb \hat{a}, \hat{a}^\da \rb = \hat{1}
$$
Together, the above three basis elements span the ''Heisenberg Lie algebra''. Considering their universal enveloping algebra, one can form other operators, such as the ''number operator'',
$$
\hat{N} = \hat{a}^\da \hat{a}
$$
satisfying
$$
\lb \hat{N}, \hat{a}^\da \rb = \hat{a}^\da \s\;\;\; \lb \hat{N}, \hat{a} \rb = - \hat{a}
$$
This Hilbert space is spanned by ''harmonic energy states'', $|n\rangle$, with $n \ge 0$, and the action of these operators on these basis states is
$$
\hat{N} |n\rangle = n |n\rangle \s \hat{a}^\da |n\rangle = \sqrt{n+1} |n+1\rangle \s \hat{a} |n\rangle = \sqrt{n} |n-1\rangle
$$
Thus the creation and annihilation operators are ''ladder operators'', with the ''ground state'' satisfying $\hat{a} | 0 \rangle = 0$ and the level $n$ state equal to
$$
|n\rangle = \fr{1}{\sqrt{n!}} \lp \hat{a}^\da \rp^n |0\rangle
$$
These operators describe the kinematics of a [[quantum harmonic oscillator]].
A [[massless quantum Dirac spinor]],
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{L/R} u_p^{L/R} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{R/L \, \da} v_p^{L/R} e^{+i p_\mu x^\mu} \rp
$$
includes [[creation and annihilation operators]] for antiparticles and particles of left and right chirality. Using chiral basis state orthonormality, such as $u^{L \, \da} u^L = 1$, and the [[Dirac delta function]], these creation and annihilation operators can be written as
$$
\begin{array}{rcl}
\ud{\hat{b}}_{\, p}^{R/L \, \da} &=& \int{d^3x} \, e^{-i p_\mu x^\mu} v_p^{L/R \, \da} \ud{\hat{\Ps}} \\
\ud{\hat{a}}_{\, p}^{L/R} &=& \int{d^3x} \, e^{+i p_\mu x^\mu} u_p^{L/R \, \da} \ud{\hat{\Ps}}
\end{array}
$$
These operators interact with other operators on an [[infinite-dimensional unitary representation]] space, such as with the [[Hermitian]] [[helicity]] operator,
$$
\lb \hat{h}, \ud{\hat{\Ps}} \rb = (\hat{p} \cdot \vec{S}) \, \ud{\hat{\Ps}}
$$
which implies
$$
\begin{array}{rcl}
\lb \hat{h}, \ud{\hat{a}}_{\, p}^{L/R} \rb u_p^{L/R} \!\!&\!\!=\!\!&\!\! \ud{\hat{a}}_{\, p}^{L/R} \, (\hat{p} \cdot \vec{S}) \, u_p^{L/R} \\
\lb \hat{h}, \ud{\hat{b}}_{\, p}^{R/L \, \da} \rb v_p^{L/R} \!\!&\!\!=\!\!&\!\! \ud{\hat{b}}_{\, p}^{R/L \, \da} \, (\hat{p} \cdot \vec{S}) \, v_p^{L/R} \\
\end{array}
$$
and so, since $\lb \hat{A}, \hat{B} \rb^\da = - \lb \hat{A}^\da, \hat{B}^\da \rb$ and $\hat{h}$ is Hermitian, we have the interaction between helicity and annihilation or creation operators,
$$
\begin{array}{rclcrcl}
\lb \hat{h}, \ud{\hat{a}}_{\, p}^{L/R} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{a}}_{\, p}^{L/R} & \s &
\lb \hat{h}, \ud{\hat{b}}_{\, p}^{R/L} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{b}}_{\, p}^{R/L} \\
\lb \hat{h}, \ud{\hat{a}}_{\, p}^{L/R \, \da} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{a}}_{\, p}^{L/R \, \da} & \s &
\lb \hat{h}, \ud{\hat{b}}_{\, p}^{R/L \, \da} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{b}}_{\, p}^{R/L \, \da} \\
\\
\end{array}
$$
This means the helicity of annihilating a left-handed particle or left-handed antiparticle is $-\ha$.
The creation and annihilation operators, as operator-valued functions of position, have canonical brackets, while their [[Fourier transform]]ed versions have scaled brackets,
$$
\lb \ud{\hat{a}}_{x}^{a}, \ud{\hat{a}}_{x'}^{b \, \da} \rb = \hat{1} \de^{ab} \de^3( x-x') \s \lb \ud{\hat{a}}_{p}^{a}, \ud{\hat{a}}_{p'}^{b \, \da} \rb = \hat{1} \de^{ab} (2 E) (2\pi)^3 \de^3( p-p')
$$
We can construct a ''creation conjugation operator'', $\hat{K}$, that is [[antiunitary]] like the [[time conjugation operator|time conjugate]] and corresponds to [[complex conjugation|complex structure]], similar to the [[charge conjugation operator|charge conjugate]],
$$
\Ps(t,x)^K = \Ps(t,x)^*
$$
Since the [[Dirac Lagrangian]] is real, it is invariant under this operation. Acting on a [[quantum Dirac spinor]], ''creation conjugation'' is
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^K &=& \hat{K} \ud{\hat{\Ps}} \hat{K}^-
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \lp \ud{\hat{a}}_{p}^{\wedge/\vee} \rp^K {u}_{p}^{\wedge/\vee \, *} e^{+i p_\mu x^\mu} + \lp \ud{\hat{b}}_{p}^{\wedge/\vee \, \da} \rp^K {v}_{p}^{\wedge/\vee \, *} e^{- i p_\mu x^\mu} \rp \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp^* \\
&=& \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee \, \da} {u}_p^{\wedge/\vee \, *} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee} {v}_p^{\wedge/\vee \, *} e^{-i p_\mu x^\mu} \rp \\
\end{array}
$$
The creation conjugation transformation of the [[creation and annihilation operators]] is thus
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^K = \ud{\hat{a}}_{p}^{\wedge/\vee \, \da} \s\;
\lp \ud{\hat{b}}_p^{\wedge/\vee} \rp^K = \ud{\hat{b}}_{p}^{\wedge/\vee \, \da}
$$
and is similar for any other particle field, such as a [[massless quantum Dirac spinor]]. Creation conjugation changes particle annihilation into particle creation. Because creation conjugation maps positive energy states to nonphysical negative energy states, it is not considered a symmetry of nature -- but can be part of other symmetries.
The curvature is perhaps the most important object characterizing the local geometry of a [[fiber bundle]] and [[connection]]. Its expression and action depends on the action of the structure group. Taking, for example, the structure group to act from the left, the ''curvature'' is then a local, [[Lie algebra]] (of the structure group) valued [[2-form|differential form]] defined as
\begin{eqnarray}
\ff{F}(x) &=& \ha \f{dx^i} \f{dx^j} F_{ij}{}^B T_B \\
&=& \f{d} \f{A} + \f{A} \f{A} \\
&=& \f{d} \f{A} + \f{A} \times \f{A} \\
&=& \f{d} \f{A} + \ha \lb \f{A}, \f{A} \rb
\end{eqnarray}
The curvature coefficients are
$$
F_{ij}{}^C = \pa_i A_j{}^C - \pa_j A_i{}^C + A_i{}^A A_j{}^B C_{AB}{}^C
$$
in which $C_{AB}{}^C$ are the [[structure constants|Lie algebra]].
The curvature is most intuitively derived in terms of the [[holonomy]] of infinitesimal loops. It may also be derived by applying the [[covariant derivative]] twice to any fiber bundle section,
$$
\f{\na} \f{\na} C = \lp \f{d} + \f{A} \rp \lp \f{d} + \f{A} \rp C =
\f{d} \f{d} C + \f{d} \f{A} C + \f{A} \f{d} C + \f{A} \f{A} C
= \lp \f{d} \f{A} + \f{A} \f{A} \rp C = \ff{F} C
$$
The curvature changes under [[gauge transformation]]s, $C \mapsto C'=gC$, as $\ff{F} \mapsto \ff{F'}=g \ff{F} g^-$. Note again that the expression of the curvature, and its action, depends on the form of the group action.
In [[spacetime]], for each direction in the Lie algebra, the curvature, $\ff{F} = \ha F_{\mu \nu} \f{e}^\mu \f{e}^\nu = \f{E} \f{e}^0 + \ff{B}$, decomposes into a spatial ''electric part'' 1-form and ''magnetic part'' 2-form, $\f{E} = \f{e}^\va E_\va = \f{dx}^e E_e$ and
$$
\ff{B} = *_3 \f{B} = B_1 \f{e}^2 \f{e}^3 - B_2 \f{e}^1 \f{e}^3 + B_3 \f{e}^1 \f{e}^2
$$
(using the [[Hodge dual]] of the magnetic part 1-form, $\f{B} = \f{e}^\va B_\va$, in three dimensional space), with components
$$
F_{\mu \nu} =
\lb
\begin{array}{cccc}
0 & -E_1 & -E_2 & -E_3 \\
E_1 & 0 & B_3 & -B_2 \\
E_2 & -B_3 & 0 & B_1 \\
E_3 & B_2 & -B_1 & 0
\end{array}
\rb
$$
in analogy with the [[electromagnetic field]].
[[Contracting|vector-form algebra]] the [[coordinate basis vectors]] with the [[Ricci curvature]] and the [[tangent bundle]] [[metric]] gives the ''curvature scalar'' (//''Ricci scalar''//),
$$
R = g^{im} \ve{\pa_i} \f{R}{}_m = R^i{}_i
$$
This equals a full contraction of the [[Riemann curvature]] tensor,
$$
R = g^{mj} R_{ij}{}^i{}_m = 2 g^{mj} \lp \pa_{\lb i \rd} \Ga^i{}_{\ld j \rb m} + \Ga^i{}_{\lb i \rd l} \Ga^l{}_{\ld j \rb m} \rp
$$
In terms of the [[tangent bundle spin connection|tangent bundle connection]] and [[frame]], the curvature scalar is
$$
R = \ve{e^\al} \f{R}{}_\al = \ve{e^\al} \ve{e_\be} \ff{R}^\be{}_\al
= 2 \lp e_\al \rp^j \lp e_\be \rp^i \lp \pa_{\lb i \rd} w_{\ld j \rb}{}^\be{}_\al + w_{\lb i \rd}{}^\be{}_\ga w_{\ld j \rb}{}^\ga{}_\al \rp
$$
If the spin connection is torsionless, the curvature scalar may also be written as
\begin{eqnarray}
R = R_\al{}^\al &=& 2 \pa_\be w_\al{}^{\be \al} + w_{\be \al}{}^\ep w_\ep{}^{\be \al} - w_\be{}^{\ep \be} w_{\al \ep}{}^\al
\end{eqnarray}
''De Sitter spacetime'' is the unique geometry of a [[spacetime]], $M$, satisfying [[Einstein's equation]] with no matter and a positive cosmological constant, $\La$. It may be embedded in the five dimensional, flat, Lorentzian $(\et_{00}=1)$ spacetime, $\mathbb{R}(1,4)$ -- in which it is a [[hyperboloid|http://en.wikipedia.org/wiki/Hyperboloid]] of one sheet:
$$
x^0 x^0 - x^w x^w = - \al^2
$$
The spaces corresponding to each time, $x^0=t$, are [[3-sphere]]s -- growing larger for $t>0$ and for $t<0$. De Sitter spacetime may also be described as a [[homogeneous space]], $M = SO(1,4)/SO(1,3)$.
The geometry is succinctly expressed by the [[frame]],
$$
\f{e} = \f{dt} \ga_0 + e^{\fr{t}{\al}} \f{d x}^\pi \ga_\pi
$$
with flat spatial sections at each $x^0 = t$, and the $\pi$ [[index|indices]] ranging over $\{1,2,3\}$. The coframe is
$$
\ve{e} = \ga^0 \ve{\pa_t} + e^{-\fr{t}{\al}} \ga^\pi \ve{\pa_\pi}
$$
The [[torsion]]less [[spin connection]] for the [[Clifford vector bundle]], found by solving [[Cartan's equation]], $0=\f{d} \f{e} + \f{\om} \times \f{e}$, is
\begin{eqnarray}
\f{\om} &=& - \ve{e} \times \f{d} \f{e} + \fr{1}{4} \lp \ve{e} \times \ve{e} \rp \lp \f{e} \cdot \f{d} \f{e} \rp \\
&=& -\fr{1}{\al} e^{\fr{t}{\al}} \f{d x}^\pi \ga_{0 \pi}
\end{eqnarray}
The [[Clifford-Riemann curvature]] is
\begin{eqnarray}
\ff{R} &=& \f{d} \f{\om} + \ha \f{\om} \, \f{\om} \\
&=& - \fr{1}{\al^2} e^{\fr{t}{\al}} \f{d t} \f{d x}^\pi \ga_{0 \pi}
- \fr{1}{2 \al^2} e^{\fr{2 t}{\al}} \f{d x}^\rh \f{d x}^\pi \ga_{\rh \pi} \\
&=& - \fr{1}{2 \al^2} \f{e} \f{e}
\end{eqnarray}
The [[Clifford-Ricci curvature]] is
$$
\f{R} = \ve{e} \times \ff{R} = - \fr{1}{2 \al^2} \ve{e} \times \f{e} \f{e} = - \fr{3}{\al^2} \f{e} = - \La \f{e}
$$
showing that the de Sitter spacetime satisfies the vacuum [[Einstein's equation]] with positive cosmological constant, $\La = \fr{3}{\al^2}$.
From the frame, the non-zero [[metric]] components are $\lc g_{tt}=1, \, g_{ss} = - e^{\fr{2 t}{\al}} \rc$ and the non-zero [[Christoffel symbols]] are $\lc \Ga^t{}_{ss} = \fr{1}{\al} e^{\fr{2 t}{\al}}, \, \Ga^s{}_{ts} = \Ga^s{}_{st} = \fr{1}{\al} \rc$.
Since our universe appears to have a positive cosmological constant, at large $t$ this dominates the matter content and our universe is well approximated by a de Sitter spacetime. In such a universe, the distant galaxies will accelerate away from us until, at the ''de Sitter horizon'', $r=\al$, they are receding from us faster than the speed of light -- so their light cannot reach us. Our galaxy appears to have a lonely future.
Ref:
*http://en.wikipedia.org/wiki/De_Sitter_space
*[[The Case for a Gravitational de Sitter Gauge Theory|papers/9610068.pdf]]
**Overview of how a Poincare gauge theory description of gravity, which has no Lagrangian formulation, needs to have terms added in order to be renormalizeable. These terms turn out to produce de Sitter gauge theory, with Lagrangian.
*[[Some Implications of the Cosmological Constant to Fundamental Physics|papers/0702065.pdf]]
Any graded operator, $\nf{D}$, that acts distributively over [[products|vector-form algebra]] of [[differential form]]s of grades $f$ and $g$ according to the ''graded Liebniz rule'',
$$
\nf{D} \nf{F} \nf{G} = \lp \nf{D} \nf{F} \rp \nf{G} + \lp -1 \rp^{df} \nf{F} \nf{D} \nf{G}
$$
is a ''graded derivation'' (or simply //''derivation''//) of grade $d$.
The most general grade $d$ derivation operator may be written as
$$
\nf{D} = {\cal L}_{\nf{\ve{K}}} + \nf{\ve{L}}
$$
in which $\nf{\ve{K}}$ is a vector valued $d$-form (a [[vector valued form]] of total grade $(d-1)$) and $\nf{\ve{L}}$ is a vector valued $(d+1)$-form (a vector valued form of grade $d$) and ${\cal L}$ is the [[FuN derivative]].
The ''determinant'' of a real or complex $n\times n$ [[matrix|linear operator]], such as the [[frame]] matrix, $A_i{}^\al$, is a real or complex number equal to
\begin{eqnarray}
\ll A \rl &=& \det ( A_i{}^\al ) = \va^{ij\dots k} A_i{}^0 A_j{}^1 \dots A_k{}^{n-1} \\
&=& \va^{ij\dots k} A_i{}^\al A_j{}^\be \dots A_k{}^\ga \fr{1}{n!} \ep_{\al \be \dots \ga} \\
&=& A_0{}^\al A_1{}^\be \dots A_{n-1}{}^\ga \ep_{\al \be \dots \ga}
\end{eqnarray}
using [[permutation symbol]]s. The determinant satisfies many identities
$$
\ll A B \rl = \ll A \rl \ll B \rl
\s
\s
\ll A^- \rl = \ll A \rl^-
\s
\s
\ll A^T \rl = \ll A \rl
\s
\s
\ll c \, A \rl = c^n \ll A \rl
$$
using the [[transpose]]. Also, the determinant of an [[exponentiated|exponentiation]] matrix is the exponential of the [[trace]] of the matrix,
$$
\ll e^A \rl = e^{\li A \ri}
$$
If a matrix is in blocks,
$$
\det \lb \ba{cc} A & B \\ C & D \ea \rb = \ll A \rl \ll D - C A^- B \rl
$$
Even weirder, if $A$ and $B$ are $m \times n$ and $n \times m$,
$$
\ll 1 + A \, B \rl = \ll 1 + B \, A \rl
$$
For a $2 \times 2$ matrix, the determinant also satisfies
$$
\ll A + B \rl = \ll A \rl + \ll B \rl + \li A \ri \li B \ri - \li A B \ri
$$
and can be computed using the [[skew]] as
$$
\ll A \rl = \bar{A} A = - \ep \, A^T \ep \, A = \li \bar{A} A \ri
$$
with the ''2D matrix conjugate'' defined as $\bar{A} = - \ep \, A^T \ep$.
*<<slider chkSlidernatF natF 'nat >' 'natural operators, vectors, forms'>>
*<<slider chkSliderfbF fbF 'fb >' 'fiber bundles'>>
*<<slider chkSliderpbF pbF 'pb >' 'principal bundles'>>
*<<slider chkSliderhamF hamF 'ham >' 'Hamiltonian dynamics, symplectic geometry'>>
*<<slider chkSliderssF ssF 'ss >' 'homogeneous spaces'>>
*<<slider chkSlidercartanF cartanF 'cartan >' 'Cartan geometry'>>
<<ListTagged dg>>
A ''diffeomorphism'' is a smooth invertible map from one [[manifold]] to another, or to itself. A diffeomorphism from a manifold to itself is also an [[automorphism]], so is called an ''autodiffeomorphism'' -- but it's usually just called a diffeomorphism of a manifold.
An ''autodiffeomorphism'', $\ph : x \mapsto x'$, typically maps points, $x$, of a manifold, $M$, to different points, $x'$, of $M$. Typically each of these points is identified by coordinates, $x^i$ and $x'^i$, on manifold patches, and a diffeomorphism is written as $x'^i(x)$ (or as $x'^i = y^i(x)$ or $x'^i = \phi^i(x)$). If these points are all the same, $x'(x) = x \, \forall \, x$, it is a passive diffeomorphism -- a [[coordinate change]].
Differential forms [[pull back|pullback]] and tangent vectors push forward under a diffeomorphism; both pull back and push forward under an autodiffeomorphism.
An autodiffeomorphism is a transformation of the position and shape of fields on a manifold. This is an active transformation of the [[Dirac spinor]], [[frame]], [[spin connection]], and [[connection]] fields:
$$
\Ps(x) \to \Ps'(x) = \Ps(x'(x)) \s \f{e} \to \f{e}' = \f{\ve{L}} \f{e}(x'(x)) \s \f{\om} \to \f{\om}' = \f{\ve{L}} \f{\om}(x'(x)) \s \f{A} \to \f{A}' = \f{\ve{L}} \f{A}(x'(x)) \s \ve{e} \to \ve{e}\,' = \ve{e}(x'(x)) \f{\ve{L}}^-
$$
using [[vector-form algebra]] and the [[vector valued form]] for the autodiffeomorphism,
$$
\f{\ve{L}} = \f{\pa} \ve{\ph} = \f{dx^i} \lp \pa_i \ph^j(x) \rp \ve{\pa'_j} = \f{dx^i} L_i{}^j \ve{\pa'_j}
$$
and its inverse, $\f{\ve{L}}^- = \f{dx'^i} L^-_i{}^j \ve{\pa_j}$.The 1-form fields thus [[pull back|pullback]] along the autodiffeomorphism, such as $\f{\phi^*A} = \f{\ve{L}} \f{A}$, as does the [[partial derivative]] of the transformed Dirac spinor,
$$
\f{d} \Ps (x) \to \f{d} \Ps'(x) = \f{d} \Ps(x'(x)) = \f{dx^i} \lp \pa_i x'^j(x) \rp \pa'_j \Ps(x'(x)) = \f{\ve{L}} \f{d}' \Ps (x'(x)) = \f{\ve{L}} \f{d}' \Ps'
$$
It is sometimes interesting to consider the [[eigen]]vector decompositions of a [[representation space]] under the action of two different [[Cartan subalgebra|Lie algebra structure]]s. Consider [[sl(2)]] acting on a $2$. One could choose the "compact" Cartan subalgebra spanned by $T_2$, producing imaginary weights and complex eigenvectors, or a "non-compact" Cartan subalgebra spanned by $T_1$, producing real weights and eigenvectors. Here we consider something weird, akin to a [[Wick rotation]] -- we choose
$$
T_\al = \lp \fr{(1+\al)}{2} \, T_1 + \fr{(1-\al)}{2} \, T_2 \rp =
\lb
\begin{array}{cc}
0 & \al \\
1 & 0 \\
\end{array}
\rb
$$
as our Cartan subalgebra generator, for parameter $-1 \leq \al \leq 1$. The resulting eigenvalues of $T_\al$ are $\pm \sqrt{\al}$, so $C^+ \to \pm 1$ and $C^- \to \pm i$, with eigenvectors
$$
V^\al_\pm =
\lb
\begin{array}{c}
\pm \sqrt{\al} \\
1
\end{array}
\rb
\s\;\;\;\;\;
\textrm{so}
\;\;\;\;\;
V^+_\pm =
\lb
\begin{array}{c}
\pm 1 \\
1
\end{array}
\rb
\;\;\;\;\;
\textrm{and}
\;\;\;\;\;
V^-_\pm =
\lb
\begin{array}{c}
\pm i \\
1
\end{array}
\rb
$$
At $\al = 0$ the matrix is singular but this still works. The eigenvectors of the compact Cartan, $V^-_\pm$, define a [[complex structure]], $J$, compatible with $C^-$, acting on the $2$,
$$
J = \ha
\lb
\begin{array}{cc}
+i & -i \\
1 & 1 \\
\end{array}
\rb
\lb
\begin{array}{cc}
+i & 0 \\
0 & -i \\
\end{array}
\rb
\lb
\begin{array}{cc}
+i & 1 \\
-i & 1 \\
\end{array}
\rb
=
\lb
\begin{array}{cc}
0 & -1 \\
+1 & 0 \\
\end{array}
\rb
$$
which splits the space into J-real and J-imaginary halves, spanned by
$$
\lb \begin{array}{c}
0 \\ 1
\end{array} \rb
\in V_{\mathbb R}
\s\;\; \textrm{and} \s\;\;
\lb \begin{array}{c}
1 \\ 0
\end{array} \rb
\in V_{\mathbb I}
$$
It is often the case that two Cartans differ by two different pairs of basis generators, instead of differing by one pair as above. In this case, such as in [[sp(4,R)]], there will typically be a linear complex map between clusters of four eigenvectors for each Cartan.
A ''differential form'', or //''p-form''//, or //''grade p form''// is a geometric object acting antisymmetrically on $p$ [[tangent vector]]s at a point to give a real number. It generalizes [[1-form]]s, and may be visualized as a $p$ dimensional volume element sitting at a [[manifold]] point. A ''2-form'' may be written in terms of real coefficients times the [[coordinate basis forms]] as
\[ \ff{a} = \ha a_{ij} \f{dx^i}\f{dx^j} \]
Such a 2-form may be visualized as an infinitesimal area element. The coefficients are [[antisymmetric|index bracket]] in the indices,
\[ a_{ij} = a_{\lb ij \rb} = \ha \lp a_{ij} - a_{ji} \rp \]
A general p-form may be written as
\[ \nf{b} = \fr{1}{p!} b_{i \dots k} \f{dx^i} \dots \f{dx^k} \]
The ''vector-form decoration'' convention requires that [[tangent vector]]s have an over-arrow, while grade $p$ forms have $p$ under-arrows or an under-bar if $p$ is unspecified or greater than 2. Multi-tangent vectors of grade $p$, which arise in [[vector-form algebra]], may be referred to as ''(-p)-form''s and are decorated with $p$ over-arrows or an over-bar. Unlike the case for [[Clifford element]]s, no use is made of differential forms of mixed grade. In an $n$ dimensional manifold, the highest grade form is an $n$-form,
\[ \nf{z} = z_v \f{dx^0} \dots \f{dx^{n-1}} = z_v \nf{d^n x} \]
in which $\nf{d^n x}$ is the coordinate basis n-form. Also, technically, a real number at a manifold point is a 0-form.
Any differential form may also be written in terms of the [[frame]] basis forms as
\[ \nf{b} = \fr{1}{p!} b_{\al \dots \be} \f{e^\al} \dots \f{e^\be} \]
Consider an $n$ dimensional [[manifold]] and its [[tangent bundle]]. At each manifold point, $x$, the tangent bundle fiber, $V_x = T_x M$, is a vector space spanned by the $n$ [[coordinate basis vectors]], $\ve{\pa_i} \in V_x$. An $m$ dimensional subspace, $V^s_x \subset V_x$, can be spanned at each manifold point by a set of $m$ linearly independent basis vectors, $\ve{s_a}$. A collection of such subspaces, one defined at each manifold point, is a ''distrubution'',
$$
\ve{\De} = \left\{ \ve{s_1}, \ve{s_2}, \dots, \ve{s_m} \right\}
$$
specified by $m$ linearly independent vector fields, $\ve{s_a}(x)$. A distribution is a subbundle of the tangent bundle.
A distribution is ''involutive'' (//in ''involution''//) iff the basis vector fields of the distribution close under the [[Lie bracket|Lie derivative]],
$$
\lb \ve{s_a}, \ve{s_b} \rb_L \in V^s
$$
This is sometimes written as $\lb \ve{\De}, \ve{\De} \rb_L = \ve{\De}$. An involutive distribution may be integrated to give the ''foliation'' of the manifold by the collection of $m$ dimensional [[submanifold]]s which have $V^s_x$ as their tangent vector space at each point.
The ''divergence'' of a [[vector field|tangent bundle]], $\ve{v}$, is a function describing how much the vector field is spreading away from (or converging towards) each point. The divergence operator takes a vector field as argument and returns a real valued field, and is defined implicitly by the [[Lie derivative]] of the [[volume form]] along the vector field,
$$
{\cal L}_{\ve{v}} \nf{e} = \f{d} \lp \ve{v} \nf{e} \rp = \nf{e} \, \mathrm{div}(\ve{v})
$$
In terms of the [[tangent bundle covariant derivative|tangent bundle connection]] or [[Christoffel symbols]] it is
$$
\mathrm{div}(\ve{v}) = D_i v^i = \pa_i v^i + v^j \Ga^i{}_{ij} = D_\al v^\al = \pa_\al v^\al + v^\al \om_\be{}^\be{}_\al
$$
This involves the [[trace]] of the [[Christoffel symbols]] or [[spin connection]], which can be used to produce a formula relating the divergence to the [[partial derivative]] of the [[frame determinant|volume form]],
$$
\pa_i \ll e \rl v^i = \ll e \rl D_i v^i
$$
It also gives the divergence of [[frame]] vectors, $\mathrm{div}(\ve{e}_\al) = w_\be{}^\be{}_\al$, and, by [[duality|dual space]], the divergence of 1-forms, $\mathrm{div}(\f{e}^\al) = \ve{\na} \f{e}^\al = w_{\be}{}^{\be\al} = \ve{\de} \f{e}^\al$, equivalent to the operation of the [[codifferential]].
A ''division algebra'', $\mathbb{D}$, over a field is an algebra (having addition and multiplication) that also has division, so that for any $a$ and non-zero $b$, there is a unique $x$ such that $a = b \, x$, and a unique $y$ such that $a = y \, b$. The possible normed division algebras are the reals, $\mathbb{R}$, [[complex number]]s, $\mathbb{C}$, [[quaternion]]s, $\mathbb{H}$, and [[octonion]]s, $\mathbb{O}$. A related set of algebras, called ''composition algebras'' have a non-positive definite norm, implying nonzero null elements do not have inverses (a failing for which they might earn the name ''derision algebra''). These include the [[split-complex number]]s, $\mathbb{C}'$, [[split-quaternion]]s, $\mathbb{H}'$, and [[split-octonion]]s, $\mathbb{O}'$. These ''split-division algebra''s (''derision algebras'') are still loosely referred to as division algebras.
There is a natural ''confusion'' between $n$ dimensional [[division algebra]] elements and [[Clifford algebra]] [[vectors|Clifford basis vectors]], $v=v^c \ga_c \, \in \, n_v$, negative real [[chiral]] [[spinor]]s, $\ps=\ps^a Q^-_a \, \in \, n_-$, and positive real chiral spinors, $\ch=\ch^b Q^+_b \, \in \, n_+$. If we have a [[chiral]] [[Clifford matrix representation]] with gamma matrix coefficients, $\Ga_{c}{}^b{}_a$, we can relate a triplet of vector and opposite-chiral spinor components by
$$
\ch^b = v^c \ps^a \Ga_c{}^b{}_a
$$
Similarly, if we have a triplet of $n$ dimensional, signature-compatible division algebra elements, $v = v^c e_c$, $\ps = \ps^a e_a$, and $\ch = \ch^b e_b$ in $\mathbb{D}$, we can relate them by
$$
\os{\ch} = v \, \ps = v^c \ps^a M_{ca}{}^\os{b} e_\os{b} = \ch^b e_\os{b}
$$
using division algebra multiplication coefficients, $M_{ca}{}^b$, and conjugation. From the above similar relations, and $\Ga_c{}^b{}_a = M_{ca}{}^\os{b}$, we have a useful confusion of division algebra elements with vectors and spinors and their structure,
\begin{eqnarray}
v = v^c e_c & \;\; \leftrightarrow \;\; & v = v^c \ga_c \\
\ps = \ps^a e_a & \;\; \leftrightarrow \;\; & \ps=\ps^a Q^-_a \\
\ch = \ch^b e_b & \;\; \leftrightarrow \;\; & \ch=\ch^b Q^+_b \\
\tilde{\ch} = v \, \ps & \;\; \leftrightarrow \;\; & \ch = v \, \ps
\end{eqnarray}
This also allows us to confuse [[triality]],
$$
\ch^b v^c \ps^a \Ga_{cba} = \bar{\ch} \, v \, \ps = T(v, \ps, \ch) = \lp \os{\ch}, v \, \ps \rp = \ha \lp \ch (v \ps) + \widetilde{\ch (v \ps)} \rp = \ch^b v^c \ps^a M_{ca\os{b}}
$$
If we wish to use a different Clifford matrix representation we can use a [[similarity transformation|Dirac matrices]]. This confusion allows us to compute triality automorphisms in [[Lie algebra]]s and other structures related to Clifford algebras that are related to division algebras.
The idea that Lorentz transformations should be modified to preserve a constant minimum length as well as the speed of light.
This makes sense, since it's basically a modification of special relativity to take place in a [[de Sitter spacetime]] instead of Minkowski space -- which is appropriate in a universe with a positive cosmological constant. But I don't see how the effect is going to be measurable in QFT, since this [[spacetime]] curvature is usually so small. Proponents of "DSR" claim the effect is increased locally by large local energy density. But it seems to me like a hack -- and what we really want to do is QFT in an arbitrarily curved spacetime.
Good introductory paper:
http://arxiv.org/abs/gr-qc/0207085
Speculation:
A momentum cutoff... maybe the momentum is a closed manifold rather than a plane, similar to how the tangent space in [[Cartan geometry]] is a curved surface rather than a plane. Ah, this is supported by these papers:
http://arxiv.org/abs/hep-th/0207279
http://arxiv.org/abs/gr-qc/0612093
and mentioned here:
http://math.ucr.edu/home/baez/week232.html
Hey, does Derek's paper on [[Cartan geometry]] mention that?
ah, here, relation to [[de Sitter gravity]]:
*[[de Sitter special relativity|paper/0606122.pdf]]
**wow, the connection between this and a minimal length is rather tenuous -- it relies on the assumption that a high energy process will change the local value of $\La$. Why would that happen? Should consider non-constant $\La$...
**basically, approximate the whole universe by de Sitter spacetime instead of by a flat Minkowski spacetime, and do particle physics in this universe the way it's normally done in Minkowski. This is less general than our reality, which is a bumpy spacetime.
Hmm, Lorentz transformations shouldn't need to be modified to preserve a minimal finite [[proper time]].
The space of real linear operators on a [[vector space]], $V$, is called the ''dual space'', $V^*$. For every $f \in V^*$ and $v \in V$ we have $f(v) \in {\mathbb R}$. For every basis element, $e_i \in V$, there is a unique ''dual basis element'', $e^i \in V^*$, such that $e^i(e_j) = \de^i_j$. If there is a nondegenerate [[metric]] on the vector space, $(e_i,e_j)=g_{i j}$, then this induces the ''inverse metric'' on the dual space, $(e^i,e^j)=g^{i j}$, with $g^{i j} g_{j k} = \de^i_k$. This metric also allows us to define the ''metric-dual basis element''s, $e_i \in V^*$, such that $e_i(e_j) = g_{ij}$. If $v = v^i e_i$ is a vector, then $v^* = v_i e^i$ is its ''dual vector'', in which $v_i = g_{ij} v^j$. If $v^i$ is thought of as a column of vector components, $u_i$ is then a row of dual vector components, and $u^* \, v = u_i v^i = u^j g_{ji} v^i = (u,v)$.
The rank $6$ exceptional [[Lie group]], [[E6]], is described by its $78$ dimensional [[Lie algebra]], ''e6''. This Lie algebra may be decomposed as a $45$ dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special orthogonal group]] Lie algebra, $so(10)$, acting on the $32$ dimensional space of, complex, positive, $Cl(10)$ [[spinor]]s, $16_-$, and a $u(1)$,
$$
e6 = so(10) + u(1) + 16_- + \bar{16}_-
$$
Also, using [[f4]] and its fundamental representation,
$$
e6 = f4 + 26 = so(8) + u(1) + u(1) + 2 \times 8_v + 2 \times 8_- + 2 \times 8_+
$$
The fundamental representation of $e6$ is $27$.
The rank $8$ exceptional [[Lie group]], [[E8]], is described by its $248$ dimensional [[Lie algebra]], ''e8''. This Lie algebra may be decomposed as a $120$ dimensional subalgebra, the [[spin Lie algebra]], $spin(16)$, acting on the $128$ dimensional representation space of, real, positive [[chiral]], [[Cl(16)]] [[spinor]]s, $S^{\lp16\rp+}$. In this way, any $e_8$ element may be written in terms of basis generators as:
\begin{eqnarray}
E &=& B + \Ps = \ha b^{\al\be} \ga^{(16)+}_{\al\be} + \ps^a Q^+_a \\
&& \in spin(16) + 128^+ = e_8
\end{eqnarray}
Explicitly, a $spin(16)$ element is expressed above as the first (upper left) quadrant of a Cl(16) bivector, $B = \ha b^{\al \be} \ga^{\lp16\rp+}_{\al \be}$, in a real, [[chiral]] [[Clifford matrix representation]], with $1 \le \al,\be \le 16$. This is a $128\times128$ real, antisymmetric matrix that is part of a $256\times256$ dimensional matrix of the Cl(16) rep. In terms of matrix components, with matrix indices $1 \le a,b \le 128$, this positive chiral part of the bivector is
$$
\lp B \rp^a{}_b = \ha b^{\al \be} \lp \ga^{\lp16\rp+}_{\al \be} \rp^a{}_b
$$
The $120$ unique, positive chiral, basis bivectors, $\ga^{\lp16\rp+}_{\al \be} \sim T_A$, are Lie algebra generators of $spin(16)$ and of $e8$, represented as $128\times128$ matrices. A positive chiral, real spinor, $\Ps = \ps^a Q^+_a$, is a column of $128$ real numbers on which these bivectors act. In terms of matrix components, these generators are $\lp Q^+_a \rp^b = \de_a^b$ and the spinor components are $\lp \Ps \rp^b = \ps^b$. The action of a bivector on a spinor gives a spinor, which can be written three different ways as:
\begin{eqnarray}
\ps'^{a} &=& \lp B \rp^a{}_b \psi^b \\
\ps'^{d} \lp Q^+_d \rp^a &=& \ha b^{\al \be} \lp \ga^{\lp16\rp+}_{\al \be} \rp^a{}_b \psi^{c} \lp Q^+_c \rp^b \\
\Ps' = \ps'^c Q^+_c &=& B \Ps = \ha b^{\al \be} \psi^{c} \ga^{\lp16\rp+}_{\al \be} Q^+_c
\end{eqnarray}
The $248$ dimensional Lie algebra, e8, is spanned by these two sets of generators. The Lie brackets between bivectors, and between bivector generators, are determined by a [[Clifford basis identity|Clifford basis identities]],
\begin{eqnarray}
\lb B_1, B_2 \rb &=& B_1 B_2 - B_2 B_1 \\
\lb \ga^{\lp16\rp+}_{\al \be}, \ga^{\lp16\rp+}_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga^{\lp16\rp+}_{\be \de} + \et_{\al \de} \ga^{\lp16\rp+}_{\be \ga} + \et_{\be \ga} \ga^{\lp16\rp+}_{\al \de} - \et_{\be \de} \ga^{\lp16\rp+}_{\al \ga} \right\}
\end{eqnarray}
giving the same structure constants as for the spin Lie algebra,
$$
C_{\lb\al\be\rb\lb\ga\de\rb}{}^{\lb\ep\up\rb} = 2 \left\{ - \et_{\al \ga} \de^{\lb\ep \up\rb}_{\be \de} + \et_{\al \de} \de^{\lb\ep \up\rb}_{\be \ga} + \et_{\be \ga} \de^{\lb\ep \up\rb}_{\al \de} - \et_{\be \de} \de^{\lb\ep \up\rb}_{\al \ga} \right\}
$$
in which the appropriate Clifford algebra metric for Cl(16,0) is $\et_{\al \be} = \de_{\al \be}$. The Lie brackets between bivector and spinor, and between their generators, are
\begin{eqnarray}
\lb B, \Ps \rb &=& B \Ps \\
\lb \ga^{\lp16\rp+}_{\al \be}, Q^+_a \rb &=& \ga^{\lp16\rp+}_{\al \be} Q^+_a = \lp \ga^{\lp16\rp+}_{\al \be} \rp^b{}_c \lp Q^+_a \rp^c Q^+_b
\end{eqnarray}
giving structure constants:
$$
C_{\lb\al\be\rb a}{}^{b} = \lp \ga^{\lp16\rp+}_{\al \be} \rp^b{}_a = - \lp \ga^{\lp16\rp+}_{\al \be} \rp_a{}^b
$$
Finally, the e8 Lie algebra description is completed by letting the structure constants be completely antisymmetric -- the [[Killing form]] identity,
$$
C_{ab}{}^{\lb\al\be\rb} = C^{\lb\al\be\rb}{}_{a b} = \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ba} = - \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab}
$$
giving the Lie brackets between spinor generators,
\begin{eqnarray}
\lb Q^+_a, Q^+_b \rb &=& - \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab} \ga^{\lp16\rp+}_{\al \be} \\
\lb \Ps_1, \Ps_2 \rb &=& - \ps_1^a \ps_2^b \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{ab} \ga^{\lp16\rp+}_{\al \be}
\end{eqnarray}
To summarize the above expressions, if elements of $e8$ are expressed as a combinations of $16\times16$ antisymmetric matrices of coefficients, $b = - b^T$, and $128$ elements columns, $\ps$, then the $e8$ Lie brackets can be defined heuristically in terms of matrix operations between these elements as:
\begin{eqnarray}
\lb b_1, b_2 \rb_{e8} &=& 2 \lp b_1 \et b_2 - b_2 \et b_1 \rp \\
\lb b, \ps \rb_{e8} &=& \big< \ha b \ga^{(16)+} \big> \ps \\
\lb \ps_1, \ps_2 \rb_{e8} &=& - 2 \big( \ps_1^T \ga^{(16)+} \ps_2 \big)_B
\end{eqnarray}
The Killing form for e8 is
\begin{eqnarray}
g_{\lb \al \be \rb \lb \ga \de \rb} &=&
C_{\lb \al \be \rb \lb \ep \up \rb}{}^{\lb \ze \et \rb} C_{\lb \ga \de \rb \lb \ze \et \rb}{}^{\lb \ep \up \rb}
+ C_{\lb \al \be \rb a}{}^b C_{\lb \ga \de \rb b}{}^a
= 240 \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp \\
g_{ab} &=& 2 C_{a \lb \al \be \rb}{}^c C_{b c}{}^{\lb \al \be \rb}
= 2 \lp \ga^{\lp16\rp+}_{\al \be} \rp_a{}^c \lp {\ga^{\lp16\rp+}}^{\al \be} \rp_{cb} = 2 \lp \de_\al^\be \de_\be^\al - \de_\al^\al \de_\be^\be \rp \de_{ab} = - 480 \, \de_{ab}
\end{eqnarray}
Since we use a real representation for Cl(16,0), the above describes the compact real form of E8. If we use a complex rep for Cl(16,0), we get a complex form of e8. And if we use a Clifford algebra of different signature, like [[Cl(4,12)]], we sometimes get a (real or complex) non-compact E8. For all of these choices, the above structure constants remain symbolically the same, with the appropriate choice of $\et_{\al \be}$. Note though that it is always necessary to choose a chiral rep for Cl.
The $e8$ Lie algebra has many subalgebras other than the $spin(16)$ discussed above. Two maximal subgroups of $E8$ are $(E7 \times SU(2))/(\mathbb{ Z}/2 \mathbb{ Z})$ and $(E6 \times SU(3))/(\mathbb{ Z}/3 \mathbb{ Z})$ -- involving the other exceptional Lie algebras, [[e7]] and [[e6]], and the special unitary Lie algebras, [[su(3)]] and [[su(2)]]. The two corresponding decompositions of $e8$ are:
\begin{eqnarray}
e8 &=& e7 + su(2) + 56 \times 2 \\
&=& \lp su(2) + su(2) + su(2) + so(8) + (2,2,1) \times 8^v + (2,1,2) \times 8^- + (1,2,2) \times 8^+ \rp \\
& & + \, su(2) + \lp (1,1,2) \times 8^v + (1,2,1) \times 8^- + (2,1,1) \times 8^+ \rp \times 2
\end{eqnarray}
and
\begin{eqnarray}
e8 &=& e6 + su(3) + 27 \times 3 + \bar{27} \times \bar{3} \\
&=& so(10) + u(1) + 32 + su(3) + 27 \times 3 + \bar{27} \times \bar{3} \\
&=& so(4) + su(2) + su(2) + u(1) + 4 \times 8 + u(1) + 32 + su(3) + 27 \times 3 + \bar{27} \times \bar{3}
\end{eqnarray}
Yet another way $e8$ can be broken up is via the [[e8 triality decomposition]]:
\begin{eqnarray}
e8 &=& spin(8) + spin(8) + 8^v \! \times \! 8^v + 8^- \! \times \! 8^- + 8^+ \! \times \! 8^+ \\
&=& so(4) + so(4) + 4 \times 4 + so(6) + so(2) + 6 \times 2 + 3 \times 8 \times 8 \\
&=& so(4) + su(2) + su(2) + su(2) + 4 \times 4 + su(4) + u(1) + 6 \times 2 + 3 \times 8 \times 8
\end{eqnarray}
Ref:
*[[http://en.wikipedia.org/wiki/E8_(mathematics)|http://en.wikipedia.org/wiki/E8_(mathematics)]]
*G,S,&W, [[Superstring Theory|http://www.amazon.com/Superstring-Cambridge-Monographs-Mathematical-Physics/dp/0521357527/ref=pd_bbs_sr_3/104-9709999-3726336?ie=UTF8&s=books&qid=1179001057&sr=8-3]]
*J. F. Adams, [[Lectures on Exceptional Lie groups|http://www.amazon.com/gp/reader/0226005267/ref=sib_dp_pt/104-6593454-7361512#reader-link]]
*S. Adler
**[[Should E8 SUSY Yang-Mills be Reconsidered as a Family Unification Model?|http://arxiv.org/abs/hep-ph/0201009]]
*P. Ramond
**[[Exceptional Groups and Physics|http://arxiv.org/abs/hep-th/0301050]]
The [[root system]] of the [[e8]] Lie algebra is quite pretty, as can be seen when it is projected into 2D:
@@display:block;text-align:center;[img[images/png/e8 root system.png]]@@
The [[Cartan matrix|root system]] for e8 is
$$
\left[\begin{array}{cccccccc}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & -1\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 2\\
\end{array}\right]
$$
A set of [[simple roots|root system]], $\{ \al_{1}, \al_{2}, \al_{3}, \al_{4}, \al_{5}, \al_{6}, \al_{7}, \al_{8} \}$, in 8D are
$$
\left[\begin{array}{cccccccc}
1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0\\
-\ha & -\ha & -\ha & -\ha & -\ha & -\ha & -\ha & - \ha\\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0\\
\end{array}\right]
$$
The resulting roots are all 112 + 128 = 240 permutations of
$$
\begin{eqnarray}
& ( \pm 1, \pm 1, 0, 0, 0, 0, 0, 0 ) \\
& ( \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha, \pm \ha ) \textrm{ with even number plus}
\end{eqnarray}
$$
and all have length $2$.
The ''E8 lattice'', Q8, is the integral span of the simple roots -- the even self-dual lattice in 8D.
The [[e8]] [[Lie algebra]], $e_8={\rm Lie}(E_8)$, corresponding to the [[Lie group]], [[E8]], breaks into a $120$ dimensional $spin(16)$ [[spin Lie algebra]] acting on a $128$ dimensional [[chiral]] [[Cl(16)]] [[spinor]], $S^{\lp16\rp+}$. However, this $spin(16)$ can be decomposed into two $28$ dimensional $spin(8)$'s and a $64$ dimensional piece, related to two pieces of the $128=64+64$ spinor through [[triality]] -- a decomposition described by [[John Baez]] in [[TWF90|http://math.ucr.edu/home/baez/week90.html]]:
<<<
Emboldened with our success, we now look at the vector space
so(8) + so(8) + end(S+) + end(S-) + end(V)
Here end(S+) is the space of all linear transformations of the vector space S+, so if you like, it's just the space of 8x8 matrices. Similarly for end(S-) and end(V). Now the dimension of this space is
28 + 28 + 64 + 64 + 64 = 248
Hey! This is just the dimension of E8! Maybe this space is E8!
Yes indeed. Again, you can cook up a bracket operation on this space using all the stuff we've got. Here's the basic idea. end(S+), end(S-), and end(V) are already Lie algebras, where the bracket of two guys x and y is just the commutator [x,y] = xy - yx, where we multiply using matrix multiplication. Since so(8) has a representation as linear transformations of V, it has two representations on end(V), corresponding to left and right matrix multiplication; glomming these two together we get a representation of so(8) + so(8) on end(V). Similarly we have representations of so(8) + so(8) on end(S+) and end(S-). Putting all this stuff together we get a Lie algebra, if we do it right - and it's E8. At least that's what Kostant said; I haven't checked it.
<<<
We can build this by breaking up the $spin(16)^+ + S^{\lp16\rp+}$ generators and structure constants into the new ones. Letting the indices run $1 \le \al,\be \le 8$ and $1 \le a,b \le 8$, and using the [[chiral Cl(16) bivector|Cl(16)]] decomposition into [[Cl(8)]] elements using the [[Kronecker product]], we define the new set of e8 generators in terms of the old:
$$
\begin{array}{rclcccl}
H_{\al\be} &=& \ga^{\lp16\rp+}_{\al\be} &=& \Ga^+_{\al\be} \otimes 1 &\in& spin(8)^+ \otimes 1 \\
G_{\al\be} &=& \ga^{\lp16\rp+}_{\lp\al+8\rp\lp\be+8\rp} &=& P^{\lp8\rp}_+ \otimes \Ga_{\al\be} &\in& 1 \otimes spin(8) \\
\Ps^I_{\al\be} &=& \ga^{\lp16\rp+}_{\al\lp\be+8\rp} &=& -\Ga^+_\al \otimes \Ga_\be &\in& V^{\lp8\rp+} \otimes V^{\lp8\rp}\\
\Ps^{II}_{ab} &=& Q^+_{16\lp a-1\rp+b} &=& q^+_a \otimes q^+_b &\in& S^{\lp8\rp+} \otimes S^{\lp8\rp+}\\
\Ps^{III}_{ab} &=& Q^+_{16\lp a-1\rp+b+8} &=& q^+_a \otimes q^-_b &\in& S^{\lp8\rp+} \otimes S^{\lp8\rp-}
\end{array}
$$
in which $P^{\lp8\rp}_+ = \ha \lp 1 + \Ga \rp$ is the positive chirality projector for Cl(8), giving $\Ga^+_{\al\be} = P^{\lp8\rp}_+ \Ga_{\al\be}$ and $\Ga^+_{\al} = P^{\lp8\rp}_+ \Ga_{\al}$, and $q^\pm_a$ are positive and negative chiral Cl(8) spinors. Since this is just a re-labeling, the new Lie brackets (and structure constants) come from the old structure constants.
Since the triality decomposition of $e8$,
$$
e8 = spin(8) + spin(8) + 8^v \! \times \! 8^v + 8^- \! \times \! 8^- + 8^+ \! \times \! 8^+
$$
closely matches the triality decomposition of [[f4]],
$$
f4 = spin(8) + 8^v + 8^- + 8^+
$$
an easy way to obtain the $e8$ structure constants is by [[compound triality decomposition]] with respect to two $f4$'s. Explicitly,
\begin{eqnarray}
\lb \ga_{\al \be}, \ga_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} \\
\lb \ga_{\al' \be'}, \ga_{\ga' \de'} \rb &=& 2 \left\{ - \et_{\al' \ga'} \ga_{\be' \de'} + \et_{\al' \de'} \ga_{\be' \ga'} + \et_{\be' \ga'} \ga_{\al' \de'} - \et_{\be' \de'} \ga_{\al' \ga'} \right\} \\
\lb \ga_{\al \be}, \ga_{\ga \de'} \rb &=& 2 \left\{ \et_{\be \ga} \ga_{\al \de'} - \et_{\al \ga} \ga_{\be \de'} \right\} \\
\lb \ga_{\al' \be'}, \ga_{\ga \de'} \rb &=& 2 \left\{ \et_{\be' \de'} \ga_{\ga \al'} - \et_{\al' \de'} \ga_{\ga \be'} \right\} \\
\lb \ga_{\al \be}, Q^+_{ab'} \rb &=& ( \ga^+_{\al \be} )^b{}_a Q^+_{bb'} \\
\lb \ga_{\al' \be'}, Q^+_{ba'} \rb &=& ( \ga^+_{\al' \be'} )^{b'}{}_{a'} Q^+_{bb'} \\
\lb \ga_{\al \be}, Q^-_{ab'} \rb &=& ( \ga^-_{\al \be} )^b{}_a Q^-_{bb'} \\
\lb \ga_{\al' \be'}, Q^-_{b'a} \rb &=& ( \ga^-_{\al' \be'} )^{b'}{}_{a'} Q^-_{bb'} \\
\lb \ga_{\al \al'}, \ga_{\be \be'} \rb &=& -2 \ga_{\al \be} \et_{\al' \be'} - 2 \et_{\al \be} \ga_{\al' \be'} \\
\lb Q^+_{aa'}, Q^+_{bb'} \rb &=& \ha ( \ga^+{}^{\al \be} )_{ab} \ga_{\al \be} g_{a'b'} + \ha g_{ab} ( \ga^+{}^{\al' \be'} )_{a'b'} \ga_{\al' \be'} \\
\lb Q^-_{aa'}, Q^-_{bb'} \rb &=& \ha ( \ga^-{}^{\al \be} )_{ab} \ga_{\al \be} g_{a'b'} + \ha g_{ab} ( \ga^-{}^{\al' \be'} )_{a'b'} \ga_{\al' \be'} \\
\lb \ga_{\al\al'}, Q^+_{aa'} \rb &=& ( \bar{\ga}_\al )^b{}_a ( \bar{\ga}_{\al'} )^{b'}{}_{a'} Q^-_{bb'} \\
\lb \ga_{\al\al'}, Q^-_{aa'} \rb &=& - ( \ga_\al )^b{}_a ( \ga_{\al'} )^{b'}{}_{a'} Q^+_{bb'} \\
\lb Q^+_{aa'}, Q^-_{bb'} \rb &=& - ( \ga^\al )_{ab} ( \ga^{\al'} )_{a'b'} \ga_{\al\al'}
\end{eqnarray}
in which the greek indices are raised and lowered by the Clifford signature metrics, $\eta_{\al \be}$ and $\eta_{\al' \be'}$, and the latin spinor indices are raised and lowered by the appropriate spinor metric, $g_{ab}$ or $g_{a'b'}$. Note that this description holds for the compact, split, or quaternionic real forms of $e8$, via appropriate choice of $spin(4,4)$ or $spin(8)$.
Ref:
*[[John Baez]]
**http://math.ucr.edu/home/baez/week90.html
**[[Octonions|papers/oct.pdf]]
*Barton and Sudbery
**[[Magic Squares and Matrix Models of Lie algebras|papers/0203010v2.pdf]]
Any (real or complex) $n \times n$ [[matrix|linear operator]], $A^i{}_j$, has a corresponding set of $n$ complex ''eigenvalues'', $\la_\al$, ''right eigenvectors'', $R_\al$, and ''left eigenvectors'', $L^\al$, satisfying the ''eigenequations'',
$$
\ba{lcr}
A \, R_\al \!\!&\!\!=\!\!&\!\! \la_\al R_\al \\
L^\al A \!\!&\!\!=\!\!&\!\! L^\al \la^\al
\ea
\s \s
\ba{lcr}
A^i{}_j R^j{}_\be \!\!&\!\!=\!\!&\!\! R^i{}_\al \La^\al_\be \\
L^\al{}_i A^i{}_j \!\!&\!\!=\!\!&\!\! \La^\al_\be L^\be{}_j
\ea
$$
in which $\La^\al_\be$ is the diagonal matrix with the eigenvalues, $\la_\al$, on the diagonal,
$$
\La = \lb
\begin{array}{cccc}
\la_1 & 0 & \dots & 0\\
0 & \la_2 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \la_n\\
\end{array}
\rb
$$
The eigenvectors, $R_\al = e_i R^i{}_\al$, are elements of some [[vector space]], and $L^\al$ elements of its [[dual space]]. From the the above equations, the eigenvalues are the roots of the ''characteristic polynomial'',
$$
0 = p_A(\la) = \ll \la - A \rl = \det \lp \la \de_j^i - A^i{}_j \rp
$$
using the [[determinant]]. The matrix is ''singular'' iff at least one of the eigenvalues is $0$ and ''degenerate'' iff two or more of the eigenvalues are the same, with each eigenvalue having a ''multiplicity'', $m$. The matrix ''rank'' is the number of nonzero eigenvalues. If $A$ is real and symmetric, $A^T = A$, ($A_j{}^i = A^i{}_j$), or complex and [[Hermitian]], $A^\da = A$, ($A^*_j{}^i = A^i{}_j$), the eigenvalues are real and the left and right eigenvectors are dual and orthonormal. If $A$ is real but not symmetric, the eigenvalues and eigenvectors are complex. The eigenvectors are only determined up to a scaling factor by the eigenequations. And if $A$ is degenerate, the $m$ eigenvectors corresponding to an eigenvalue need only span that $m$ dimensional ''eigenspace''.
A matrix is ''defective'' iff the (left or right) eigenvectors don't span the $n$ dimensional vector space. If a matrix is not defective the left and right eigenvectors can be scaled to satisfy ''normality conditions'',
$$
L^\al{}_i R^i{}_\be = \de^\al_\be
$$
and the matrix may be written as
$$
A^i{}_j = R^i{}_\be \La^\be_\al L^\al{}_j
$$
the ''spectral decomposition''. A symmetric or Hermitian matrix is never defective, and has $R = L^- = L^\da$. The determinant of a matrix equals the product of its eigenvalues, and its [[trace]] is their sum,
$$
\ll A \rl = \ll \La \rl = \prod_\al \la_\al \s\;\;\; \li A \ri = \li \La \ri = \sum_\al \la_\al
$$
If we are dealing with a [[vector valued 1-form|vector valued form]], $\f{\ve{A}} = \f{dx^i} A_i{}^j \ve{\pa_j}$, then the [[vector-form algebra]] gives the eigenequations for the ''eigenforms'' and //''eigenvectors''//,
\begin{eqnarray}
\f{\ve{A}} \f{R^\be} &=& \f{R^\al} \La_\al{}^\be\\
\ve{L_\al} \f{\ve{A}} &=& \La_\al{}^\be \ve{L_\be}
\end{eqnarray}
and, if the eigenvectors and eigenforms can be scaled to satisfy $\ve{L_\al} \f{R^\be} = \de_\al^\be$, the spectral decomposition is
$$
\f{\ve{A}} = \f{R^\be} \La_\be^\al \ve{L_\al}
$$
In four dimensional [[spacetime]], the ''electromagnetic field'', $\f{A} = \ph \f{e}^0 + \f{a}$, is a $U(1)$ [[connection]] [[1-form]] field, with $\ph = A_0$ the ''electric potential'' and $\f{a}=A_\va \f{e}^\va$ the spatial ''magnetic potential''. The [[curvature]],
$$
\ff{F} = \f{d} \f{A} = \ha F_{\mu \nu} \f{e}^\mu \f{e}^\nu = \f{E} \f{e}^0 + \ff{B}
$$
decomposes into spatial ''electric field'' 1-form and ''magnetic field'' 2-form parts, $\f{E} = \f{e}^\va E_\va = \f{dx}^e E_e$ and
$$
\ff{B} = *_3 \f{B} = B_1 \f{e}^2 \f{e}^3 - B_2 \f{e}^1 \f{e}^3 + B_3 \f{e}^1 \f{e}^2
$$
(using the [[Hodge dual]] of the magnetic field 1-form, $\f{B} = \f{e}^\va B_\va$, in three dimensional space), with components
$$
F_{\mu \nu} =
\lb
\begin{array}{cccc}
0 & -E_1 & -E_2 & -E_3 \\
E_1 & 0 & B_3 & -B_2 \\
E_2 & -B_3 & 0 & B_1 \\
E_3 & B_2 & -B_1 & 0
\end{array}
\rb
$$
If we like, we can use a [[frame]] and [[vector-form algebra]] to convert (''Cliffordize'') these 1-form and 2-form fields to the corresponding [[Cl(1,3)]] [[Clifford algebra]] fields,
$$
A = \ve{e} \f{A} = \ga^\mu A_\mu \s \s F = \ve{e} \ve{e} \ff{F} = \ha \ga^{\mu\nu} F_{\mu \nu}
$$
with the above formulas looking mostly the same, but Cliffordized.
In flat spacetime, $\f{d} \f{e}^\mu = 0$, with negative spatial [[Minkowski metric]], we have
$$
\ff{F} = \f{E} \f{e}^0 + \ff{B} = \f{d} \f{A} = \lp \f{e}^0 \pa_t + \f{d}{}_3 \rp \lp \ph \f{e}^0 + \f{a} \rp
$$
and the electric and magnetic fields are thus $\f{E} = \f{d}{}_3 \ph - \pa_t \f{a}$ and $\ff{B} = \f{d}{}_3 \f{a}$.
The dynamics of the electromagnetic field are dictated by the [[Yang-Mills Lagrangian]] $U(1)$, which gives [[Maxwell's equations]] and, in flat spacetime with no current, [[Maxwell solutions]].
Note that a [[gauge transformation]] of the electromagnetic field, $\f{A}' = \f{A} - \f{d} \Ph$, results in the same curvature, $\ff{F}' = \ff{F}$, and therefore the same electric and magnetic fields.
A [[differential form]] [[field|cotangent bundle]], $\nf{f}(x)$, over a [[manifold]], $M$, is ''exact'' iff it is the [[exterior derivative]] of some other differential form field,
$$
\nf{f} = \f{d} \nf{g}
$$
The [[vector space]] of exact $p$-forms over $M$ is labeled $E^p$. Since the exterior derivative is nilpotent, all exact forms are [[closed]], $E^p \subset C^p$.
The algebraic structure of real, [[complex|complex number]], [[quaternion]], and [[octonion]] [[division algebra]]s, and their split versions, allows the existance of exceptional [[Lie algebra]]s,
| | $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{O}$ |
| $\mathbb{R}$ | [[so(2)]] $\subset$ [[su(2)]] | [[so(3)]] $\subset$ [[su(3)]] | [[so(5)]] $\subset$ [[sp(3)]] | [[so(9)]] $\subset$ [[f4]] |
| $\mathbb{C}$ | [[so(3)]] $\subset$ [[su(3)]] | [[so(4)]] $\subset$ 2[[su(3)]] | [[so(6)]] $\subset$ [[su(6)]] | [[so(10)]] $\subset$ [[e6]] |
| $\mathbb{H}$ | [[so(5)]] $\subset$ [[sp(3)]] | [[so(6)]] $\subset$ [[su(6)]] | [[so(8)]] $\subset$ [[so(12)|special orthogonal group]] | [[so(12)]] $\subset$ [[e7]] |
| $\mathbb{O}$ | [[so(9)]] $\subset$ [[f4]] | [[so(10)]] $\subset$ [[e6]] | [[so(12)]] $\subset$ [[e7]] | [[so(16)]] $\subset$ [[e8]] |
| | $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{O}$ |
| $\mathbb{R}'$ | [[so(2)]] $\subset$ [[su(2)]] | [[so(3)]] $\subset$ [[su(3)]] | [[so(5)]] $\subset$ [[sp(3)]] | [[so(9)]] $\subset$ [[f4]] |
| $\mathbb{C}'$ | [[so(2,1)]] $\subset$ [[sl(3)]] | [[so(3,1)]] $\subset$ [[sl(3,C)]] | [[so(5,1)]] $\subset$ [[sl(3,H)]] | [[so(9,1)]] $\subset$ [[e6(26)]] |
| $\mathbb{H}'$ | [[so(3,2)]] $\subset$ [[sp(6,R)]] | [[so(4,2)]] $\subset$ [[su(3,3)]] | [[so(6,2)]] $\subset$ [[sp(6,H)]] | [[so(10,2)]] $\subset$ [[e7(25)]] |
| $\mathbb{O}'$ | [[so(5,4)]] $\subset$ [[f4(4)]] | [[so(6,4)]] $\subset$ [[e6(2)]] | [[so(8,4)]] $\subset$ [[e7(5)]] | [[so(12,4)]] $\subset$ [[e8(24)]] |
| | $\mathbb{R}'$ | $\mathbb{C}'$ | $\mathbb{H}'$ | $\mathbb{O}'$ |
| $\mathbb{R}'$ | [[su(2)]] | [[sl(3)]] | [[sp(6,R)]] | [[f4(4)]] |
| $\mathbb{C}'$ | [[sl(3)]] | 2[[sl(3)]] | [[sl(6,R)]] | [[e6(6)]] |
| $\mathbb{H}'$ | [[sp(6,R)]] | [[sl(6,R)]] | [[so(6,6)]] | [[e7(7)]] |
| $\mathbb{O}'$ | [[f4(4)]] | [[e6(6)]] | [[e7(7)]] | [[e8(8)]] |
and relies on [[compound triality decomposition]]s, and [[Clifford compound division algebra representation]]s, with [[reductive]] subalgebras shown.
Further details are available in:
*Evans, "Trialities and Exceptional Lie Algebras: Deconstructing the Magic Square", https://arxiv.org/abs/0910.1828
*Barton and Sudbury, "Magic squares and matrix models of Lie algebras", http://arxiv.org/abs/math/0203010
Any [[linear operator]] may be ''exponentiated'',
\[ e^A = 1 + A + \ha A A + \fr{1}{3!} A A A + \dots \]
Or, equivalently, ''exponentiation'' may be defined as
\[ e^A = \lim_{N \to \infty} \lp 1 + \fr{1}{N} A \rp^N \]
The derivative of a parameterized exponential is
\[ \fr{d}{dt} e^{tA} = A + t A^2 + \fr{1}{2!} t^2 A^3 + \dots = A e^{tA} = e^{tA} A \]
More rigorously, the solution of any set of first order ODE's,
$$
\fr{d}{dt} E = A E
$$
is used to define the exponentiation of that operator,
$$
E(t) = e^{t A} E(0)
$$
If $A$ may be written in terms of an [[adjoint|Clifford adjoint]] operator and a diagonal matrix of [[eigen]]values, $A = U \La U^-$, then
\[ e^A = U e^\La U^- \]
is easily computed by exponentiating the eigenvalues.
The ''exterior derivative'' operator is the [[partial derivative]] operator, $\f{d}=\f{\pa}$, applied to [[differential form]]s,
$$
\f{d} \nf{A} = \f{\pa} \nf{A} = \lp \f{dx^i} \pa_i \rp \lp \f{dx^j} \dots \f{dx^k} \fr{1}{p!} A_{j \dots k} \rp = \f{dx^i} \f{dx^j} \dots \f{dx^k} \fr{1}{p!} \pa_i A_{j \dots k}
$$
This operation is conventionally written as $dA$ but is written in these notes using the under-arrow, since it has a form grade of $1$. Even though it is defined using the un-natural partial derivative it is a [[natural]] (coordinate independent) operator since the non-tensor terms arising from [[coordinate change]], $x \mapsto x(y)$, vanish by the symmetry of partial derivation and antisymmetry of collections of forms,
$$
\f{d} \f{f} = \f{dx^i} \f{dx^j} \pa^x_i f_j = \f{dy^k} \f{dy^m} \fr{\pa x^i}{\pa y^k} \fr{\pa x^j}{\pa y^m} \fr{\pa y^n}{\pa x^i} \pa^y_n f_j = \f{dy^k} \f{dy^m} \lp \pa^y_k \fr{\pa x^j}{\pa y^m} f_j - \fr{\pa^2 x^j}{\pa y^k \pa y^m} f_j \rp = \f{dy^k} \f{dy^m} \pa^y_k f'_m = \f{d'} \f{f'}
$$
This doesn't work for the partial derivative applied to [[tangent vector]]s, so there is no such thing as the exterior derivative of a [[vector valued form]]. The exterior derivative of a scalar function, $\f{d} f = \f{dx^i} \pa_i f$, is called the ''gradient'' in old fashioned vector calculus, while the exterior derivative of a 1-form, $\f{d} \f{f}$, is associated to the ''curl'' in three dimensional space. The exterior derivative is equivalent to the [[cotangent bundle covariant derivative|cotangent bundle connection]], $\f{d}=\f{\na}$, if and only if the [[torsion]] vanishes.
The operator is nilpotent,
$$
\f{d} \f{d} = \f{dx^i} \pa_i \f{dx^j} \pa_j = \f{dx^i} \f{dx^j} \pa_i \pa_j = 0
$$
and, as a grade $1$ [[derivation]], distributes over the [[product|vector-form algebra]] of a $f$-form, $\nf{F}$, and $g$-form, $\nf{G}$, via the graded Liebniz rule,
$$
\f{d} \lp \nf{F} \nf{G} \rp = \lp \f{d} \nf{F} \rp \nf{G} + \lp -1 \rp^f \nf{F} \lp \f{d} \nf{G} \rp
$$
But it does not distribute over the product of vectors and forms, since instead
$$
\f{d} \lp \ve{v} \nf{G} \rp = \f{\pa} \lp \ve{v} \nf{G} \rp = \lp \f{\pa} \ve{v} \rp \nf{G} - \ve{v} \lp \f{d} \nf{G} \rp + \lp \ve{v} \f{\pa} \rp \nf{G} = {\cal L}_{\ve{v}} \nf{G} - \ve{v} \lp \f{d} \nf{G} \rp
$$
in which the pair of unnatural terms reassemble into the natural [[Lie derivative]].
The rank $4$ exceptional [[Lie group]], [[F4]], is described by its $52$ dimensional [[Lie algebra]], ''f4''. The compact [[real form]], $f_{4(-52)}$, of this Lie algebra may be decomposed as a $36$ dimensional subalgebra, the [[spin Lie algebra]], $spin(9)$, acting on the $16$ dimensional space of real Cl(9) [[spinor]]s, $16^s$,
$$
f4 = spin(9) + 16^s = spin(8) + 8^v + 8^- + 8^+
$$
which breaks up further into $28$ dimensional $spin(8)$ and three $8$ dimensional elements: the vector, $8^v$, positive [[chiral]] [[spinor]], $8^+$, and negative chiral spinor, $8^-$ -- all related through [[triality]]. Starting with an [[octonionic representation of Cl(8)]] as $16 \times 16$ real, [[chiral]] matrices, these [[Hermitian]] $Cl(8)$ basis vectors (with $1 \le \al \le 8$) can be supplemented by a ninth real, non-chiral basis vector, which also relates to an anti-Hermitian basis of $Cl(0,8)$,
$$
\ga'_\al =
\lb \begin{array}{cc}
0 & \overline{\Ga}_\al \\
\Ga_\al & 0
\end{array} \rb
\s
\ga'_0 = \ga =
\lb \begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array} \rb
\s
\ga_\al = \ga'_0 \ga'_\al =
\lb \begin{array}{cc}
0 & -\overline{\Ga}_\al \\
\Ga_\al & 0
\end{array} \rb
$$
The resulting bivector basis generators span a $spin(9)$ representation acting on a $S^{(16)}$ spinor space,
$$
\ga_{\al \be} =
\lb \begin{array}{cc}
- \overline{\Ga}_\al \Ga_\be & 0 \\
0 & - \Ga_\al \overline{\Ga}_\be
\end{array} \rb
\s \s
\Ps =
\lb \begin{array}{c}
\ps \\
\ch
\end{array} \rb
= \ps^a Q^-_a + \ch^b Q^+_b
$$
The $f_4$ Lie brackets can be written explicitly from these (with $\et_{\al \be} = -\de_{\al \be}$) as
\begin{eqnarray}
\lb \ga_{\al \be}, \ga_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} \\
\lb \ga_{\al \be}, \ga_{\ga} \rb &=& 2 \left\{ \et_{\be \ga} \ga_{\al} - \et_{\al \ga} \ga_{\be} \right\} \\
\lb \ga_{ \al}, \ga_{ \be} \rb &=& 2 \, \ga_{\al \be} \\
\lb \ga_{\al \be}, Q^-_a \rb &=& Q^-_b ( - \overline{\Ga}_\al \Ga_\be )^b{}_a \\
\lb \ga_{\al \be}, Q^+_a \rb &=& Q^+_b ( - \Ga_\al \overline{\Ga}_\be )^b{}_a\\
\lb \ga_{ \al}, Q^-_a \rb &=& Q^+_b ( \Ga_\al )^b{}_a \\
\lb \ga_{ \al}, Q^+_a \rb &=& Q^-_b ( - \bar{\Ga}_\al )^b{}_a \\
\lb Q^-_a, Q^-_b \rb &=& \ga_{\al \be} ( - \overline{\Ga}{}^\al \Ga^\be )_{ab} \\
\lb Q^+_a, Q^+_b \rb &=& \ga_{\al \be} ( - \Ga^\al \overline{\Ga}{}^\be )_{ab} \\
\lb Q^-_a, Q^+_b \rb &=& \ga_{ \al} ( - \overline{\Ga}{}^\al )_{ab}
\end{eqnarray}
in which some of the spinor indices are lowered by $g_{ab} = -\de_{ab}$ for the compact real form. These can be written as a [[matrix of f4 structure constants]].
The triality construction of f4 via [[octonion]]s or [[split-octonion]]s produces the three [[real form]]s of f4, labeled by the trace of their [[Killing form]]s as the compact, $f_{4(-52)}$, split, [[$f_{4(4)}$|f4(4)]], and other, [[$f_{4(-20)}$|f4(-20)]], real forms. These have subalgebras $spin(9)$, $spin(5,4)$, and $spin(1,8)$. There is a [[triality automorphism of f4]].
The smallest irreducible [[representation]] of f4 is 26 dimensional.
The [[Lie algebra structure]] of [[f4]] is described by choosing a Cartan subalgebra,
$$
C = c^1 \ga_{12} + c^2 \ga_{34} + c^3 \ga_{56} + c^4 \ga_{78}
$$
and computing the [[eigen]]vector decomposition of the rest of the Lie algebra under the adjoint action of $C$. For the compact [[real form]], $f_{4(-52)}$, all eigenvalues (roots) are pure imaginary. The decomposition
$$
f4 = spin(8) + 8^v + 8^- + 8^+
$$
corresponds to the [[root system]],
$$
\begin{array}{|r|cccc|l|}
\hline
& c^1 & c^2 & c^3 & c^4 & \mathrm{dimension} \\
\hline
spin(8) & ( \pm 1 & \pm 1 & 0 & 0 ) & 24 \\
8^v & ( \pm 1 & 0 & 0 & 0) & 8 \\
8^- & ( \pm \ha & \pm \ha & \pm \ha & \pm \ha ) & 8 (\mathrm{odd\#}+) \\
8^+ & ( \pm \ha & \pm \ha & \pm \ha & \pm \ha ) & 8 (\mathrm{even\#}+) \\
\hline
\end{array}
$$
with these coefficients assumed to be multiplying $\mathbb{i}$.
A [[triality automorphism of f4]] corresponds to a transformation of these roots (and corresponding root vectors and the Cartan subalgebra generators) by a [[triality matrix]], such as
$$
T =
\lb \begin{array}{cccc}
- \ha & -\ha & - \ha & -\ha \\
\ha & \ha & - \ha & - \ha \\
\ha & - \ha & \ha & - \ha \\
\ha & - \ha & - \ha & \ha \\
\end{array} \rb
$$
that maps $8^v \mapsto 8^- \mapsto 8^+ \mapsto 8^v$ and mixes roots within $spin(8)$.
This "other" [[real form]] of the [[f4]] Lie algebra, with [[Killing form]] signature $-20$, may be decomposed as a $36$ dimensional subalgebra, the [[spin Lie algebra]], $spin(1,8)$, acting on the $16$ dimensional space of real Cl(1,8) [[spinor]]s, $16^s$,
$$
f_{4(-20)} = spin(1,8) + 16^s = spin(8) + 8^v + 8^- + 8^+
$$
which breaks up further into $28$ dimensional $spin(8)$ and three $8$-dimensional elements: the vector, $8^v$, positive [[chiral]] [[spinor]], $8^+$, and negative chiral spinor, $8^-$, only two of which have positive Killing form signature. Starting with a cyclic [[Clifford division algebra representation]] of $Cl(0,8)$ as $16 \times 16$ real, [[chiral]] matrices, the basis vectors (with $1 \le \al \le 8$) can be supplemented by a ninth real, non-chiral basis vector,
$$
\ga_\al =
\lb \begin{array}{cc}
0 & -\overline{\Ga}_\al \\
\Ga_\al & 0
\end{array} \rb
\s
\ga_0 =
\lb \begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array} \rb
$$
The resulting bivector basis generators span a $spin(1,8)$ representation acting on a $16^s$ spinor space,
$$
\ga_{\al \be} =
\lb \begin{array}{cc}
- \overline{\Ga}_\al \Ga_\be & 0 \\
0 & - \Ga_\al \overline{\Ga}_\be
\end{array} \rb
\s
\ga_{0 \al} =
\lb \begin{array}{cc}
0 & \overline{\Ga}_\al \\
\Ga_\al & 0
\end{array} \rb
\s
\Ps =
\lb \begin{array}{c}
\ps \\
\ch
\end{array} \rb
= \ps^a Q^-_a + \ch^b Q^+_b
$$
The $f_{4(-20)}$ Lie brackets can be written explicitly from these (with $\et_{\al \be} = - \de_{\al \be}$) as
\begin{eqnarray}
\lb \ga_{\al \be}, \ga_{\ga \de} \rb &=& 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} \\
\lb \ga_{\al \be}, \ga_{0 \ga} \rb &=& 2 \left\{ \et_{\be \ga} \ga_{0 \al} - \et_{\al \ga} \ga_{0 \be} \right\} \\
\lb \ga_{0 \al}, \ga_{0 \be} \rb &=& -2 \, \ga_{\al \be} \\
\lb \ga_{\al \be}, Q^-_a \rb &=& Q^-_b ( - \overline{\Ga}_\al \Ga_\be )^b{}_a \\
\lb \ga_{\al \be}, Q^+_a \rb &=& Q^+_b ( - \Ga_\al \overline{\Ga}_\be )^b{}_a\\
\lb \ga_{0 \al}, Q^-_a \rb &=& Q^+_b ( \Ga_\al )^b{}_a \\
\lb \ga_{0 \al}, Q^+_a \rb &=& Q^-_b ( \bar{\Ga}_\al )^b{}_a \\
\lb Q^-_a, Q^-_b \rb &=& \ga_{\al \be} n^-_{ac} ( - \overline{\Ga}{}^\al \Ga^\be )^c{}_b \\
\lb Q^+_a, Q^+_b \rb &=& \ga_{\al \be} n^+_{ac} ( - \Ga^\al \overline{\Ga}{}^\be )^c{}_b \\
\lb Q^-_a, Q^+_b \rb &=& \ga_{0 \al} n^-_{ac} ( - \overline{\Ga}{}^\al )^c{}_b
\end{eqnarray}
in which the positive and negative spinor metrics are $n^\pm_{ab} = \pm \de_{ab}$. These metrics for the vector and spinor components of $f_{4(-20)}$ come from its [[Killing form]], which is diagonal, $-1$ for $\ga_{\al \be}$ and $Q^-_b$ generators, and $+1$ for $\ga_{0 \al}$ and $Q^+_b$ generators, this giving the signature $-28+8-8+8 = -20$. Alternatively, we could construct $f_{4(-20)}$ with a $spin(9)$ subalgebra, with positive Killing signature for the $8^-$ and $8^+$. Is there a triality automorphism of f4(-20)? There sort of is, but not the usual one since two of the 8's need to be multiplied by $i$.
The split [[real form]] of the [[f4]] Lie algebra, with [[Killing form]] signature $4$, may be decomposed as a $36$ dimensional subalgebra, the [[spin Lie algebra]], $spin(5,4)$, acting on the $16$ dimensional space of real Cl(5,4) [[spinor]]s,
$$
f_{4(4)} = spin(5,4) + 16^s = spin(4,4) + 8^v + 8^- + 8^+
$$
which breaks up further into $28$ dimensional $spin(4,4)$ and three $8$ dimensional elements: the vector, $8^v$, negative [[chiral]] [[spinor]], $8^-$, and positive chiral spinor, $8^+$ -- all related through [[split-octonion]]ic [[triality]]. Using a [[split-octonionic representation of Cl(4,4)]] as $16 \times 16$ real, [[chiral]] matrices, these [[Cl(4,4)]] basis vectors and [[spin(4,4)]] bivectors are
$$
\ga'_\al =
\lb \begin{array}{cc}
0 & \overline{\Ga}{}'_\al \\
\Ga'_\al & 0
\end{array} \rb
\s
\ga'_{\al \be} =
\lb \begin{array}{cc}
\overline{\Ga}{}'_\al \Ga{}'_\be & 0 \\
0 & \Ga{}'_\al \overline{\Ga}{}'_\be
\end{array} \rb
$$
The $f_{4(4)}$ Lie brackets can be written explicitly using the corresponding [[Cl(4,4)]] products of [[spin(4,4)]] bivectors, vectors, negative spinors, and positive spinors, using the [[grade|Clifford grade]],
$$
\ba{rclvrcl}
\lb \ga'_{\al \be}, \ga'_{\ga \de} \rb \!\!&\!\!=\!\!&\!\! 2 \left\{ - \et'_{\al \ga} \ga'_{\be \de} + \et'_{\al \de} \ga'_{\be \ga} + \et'_{\be \ga} \ga'_{\al \de} - \et'_{\be \de} \ga'_{\al \ga} \right\} & \;\;\; &
\lb B_1, B_2 \rb \!\!&\!\!=\!\!&\!\! 2 \, B_1 \times B_2 = 2 \, B_3 \\
\lb \ga'_{\al \be}, \ga'_{\ga} \rb \!\!&\!\!=\!\!&\!\! 2 \left\{ \et'_{\be \ga} \ga'_{ \al} - \et'_{\al \ga} \ga'_{\be} \right\} & &
\lb B, v_1 \rb \!\!&\!\!=\!\!&\!\! 2 \, B \times v_1 = 2 \, v_2 \\
\lb \ga'_{ \al}, \ga'_{\be} \rb \!\!&\!\!=\!\!&\!\! - 2 \, \ga'_{\al \be} & &
\lb v_1, v_2 \rb \!\!&\!\!=\!\!&\!\! -2 \, v_1 \times v_2 = -2 \, B \\
\lb \ga'_{\al \be}, Q'^-_a \rb \!\!&\!\!=\!\!&\!\! Q'^-_b ( \overline{\Ga}{}'_\al \Ga'_\be )^b{}_a & &
\lb B, \ps^-_1 \rb \!\!&\!\!=\!\!&\!\! B_- \ps^-_1 = \ps^-_2 \\
\lb \ga'_{\al \be}, Q'^+_a \rb \!\!&\!\!=\!\!&\!\! Q'^+_b ( \Ga'_\al \overline{\Ga}{}'_\be )^b{}_a & &
\lb B, \ps^+_1 \rb \!\!&\!\!=\!\!&\!\! B_+ \ps^+_1 = \ps^+_2 \\
\lb \ga'_{ \al}, Q'^-_a \rb \!\!&\!\!=\!\!&\!\! \mp \, Q'^+_b ( {\Ga}{}'_\al )^b{}_a & &
\lb v , \ps^- \rb \!\!&\!\!=\!\!&\!\! \mp v_+ \ps^- = \mp \ps^+ \\
\lb \ga'_{ \al}, Q'^+_a \rb \!\!&\!\!=\!\!&\!\! \pm Q'^-_b ( \overline{\Ga}{}'_\al )^b{}_a & &
\lb v , \ps^+ \rb \!\!&\!\!=\!\!&\!\! \pm v_- \ps^+ = \pm \ps^- \\
\lb Q'^-_a, Q'^-_b \rb \!\!&\!\!=\!\!&\!\! \ga'_{\al \be} n'^-_{ac} ( \overline{\Ga}{}'^\al \Ga'^\be )^c{}_b & &
\lb \ps^-_1 , \ps^-_2 \rb \!\!&\!\!=\!\!&\!\! \ha \li \bar{\ps}{}^-_2 \overline{\Ga}{}'^\al \Ga'^\be \ps^-_1 \ri \ga'_{\al \be} = \li \ps^-_1 \bar{\ps}{}^-_2 \ri_2 = B \\
\lb Q'^+_a, Q'^+_b \rb \!\!&\!\!=\!\!&\!\! \ga'_{\al \be} n'^+_{ac} ( \Ga'^\al \overline{\Ga}{}'^\be )^c{}_b & &
\lb \ps^+_1 , \ps^+_2 \rb \!\!&\!\!=\!\!&\!\! \ha \li \bar{\ps}{}^+_2 \Ga'^\al \overline{\Ga}{}'^\be \ps^+_1 \ri \ga'_{\al \be} = \li \ps^+_1 \bar{\ps}{}^+_2 \ri_2 = B \\
\lb Q'^-_a, Q'^+_b \rb \!\!&\!\!=\!\!&\!\! \pm \ga'_{ \al} n'^-_{ac} ( \overline{\Ga}{}'^\al )^c{}_b & &
\lb \ps^- , \ps^+ \rb \!\!&\!\!=\!\!&\!\! \pm \li \bar{\ps}{}^- \overline{\Ga}{}'^\al \ps^+ \ri \ga'_{\al} = \pm \li \ps^+ \bar{\ps}{}^- \ri_1 = \pm \li v_+ \ri_1 = \pm v \\
\ea
$$
in which the positive and negative spinor metric is $n'^\pm_{ab} = (\ga'_1 \ga'_2 \ga'_3 \ga'_4)^\pm_{ab}$ if we're using split-octonion ordering, or $n'^\pm_{ab} = (\ga'_4 \ga'_6 \ga'_7 \ga'_8)^\pm_{ab}$ if we're using conformal [[Cl(4,4)]] ordering, which matches ($\pm$) the spinor parts of the $f_{4(4)}$ [[Killing form]]. Some of the signs may seem funny above, such as the negative sign in the commutator of two vectors; this is because it is actually a bivector, $\ga_\al \to \ga_{0\al}$. We needed to introduce a new, non-chiral vector and its bivectors,
$$
\ga'_0 = \pm \ga' =
\lb \begin{array}{cc}
\mp 1 & 0 \\
0 & \pm 1
\end{array} \rb
\s \s
\ga'_{0 \al} =
\lb \begin{array}{cc}
0 & \pm \overline{\Ga}{}'_\al \\
\mp \Ga'_\al & 0
\end{array} \rb
$$
completing a $spin(5,4)$ subalgebra of $f_{4(4)}$. The choice of sign in $\ga_0$ results in the two corresponding equivalent descriptions of $f_{4(4)}$, with the different sign choices above. This description of $f_{4(4)}$ should work for any Clifford matrix representation of $Cl(4,4)$. The split-octonionic structure allows a [[triality automorphism of f4(4)]].
There is also a decomposition with respect to [[sp(3)]],
$$
f_{4(4)} = sp(3) + su(2) + (8+2+2+2) \times 2
$$
in which the $(8+2+2+2) \times 2$ generators have negative signature.
A ''fiber bundle'' is a [[manifold]], $E$, the ''total space'' (''//entire space//''), along with a ''defining map'', $\pi$, to a separate ''base manifold'', $M$. Locally, a patch of the total space, $E_{U_a} \sim U_a \otimes F$, is the product of a base patch, $U_a$, and a ''typical fiber'', $F$ -- and $\pi : E_{U_a} \rightarrow U_a$ is a projection, $\pi : z \mapsto x$. A fiber bundle may be visualized as the base manifold with a copy of the fiber attached at each base manifold point -- the fiber is said to be "over" the base manifold. There are thus two equivalent ways of describing a fiber bundle: as things happening in fibers over the base space, $M$, or as things in the total space, $E$. Each fiber is a [[submanifold]] of $E$, but there is not necessarily any submanifold of $E$ associated with $M$.
The best we can usually do is specify the explicit maps from all the $E_{U_a}$ to $U_a \otimes F$ via a ''local trivialization'', $\phi_a : z \mapsto (x,f)$. By inverting these maps, each typical fiber element, $f \in F$, gives a local section, $\ph_a^-(x,f)$, in $E_{U_a}$ over each patch. A ''local section'', $\si$, is a map from base manifold patches to the total space patches, $\si:U_a \rightarrow E_{U_a}$, which projects trivially, $\pi(\si(x))=x \; \forall \; x \in U_a$. The local trivializations over the patches are glued together such that the ''transition functions'',
$$
t_{ab}(z) = \ph^-_a \circ \ph_b(z)
$$
at each overlap point, $x=\pi(z)$, are [[autodiffeomorphism|diffeomorphism]]s of the fiber there, $\ld t_{ab} \rl_x : F \to F$, corresponding to the action of an element of the ''structure group'', $G$, of the fiber bundle on the fibers, $G:F \to F$. Another way of writing this, letting $z=\ph^-_b(x,f)$, is
$$
t_{ab} \circ \ph^-_b(x,f) = \ph^-_a(x,f) = \ph^-_b(x,g_{ab}(x) f)
$$
in which $g_{ab}(x) \in G$, acts on each fiber element at each overlap point via the [[left action|group]] of the [[Lie group]], $G$. When describing a fiber bundle it is necessary to specify the base, the fiber, the structure group, and the group action on the fiber.
With a local trivialization in hand, a local section, $\si(x) = \ph_a(x,f_\si(x))$, can be specified by choosing a typical fiber element, $f_\si(x)$, at each $x \in U_a$. The collection of fiber valued functions, $f_\si(x) \in F$ (which is sometimes just written as $\si(x) \in F$), is refered to by physicists as a ''field'' over the base manifold. A complete collection of local sections can be glued together to give a ''global section'' iff
$$
\ph^-_a(x,f^a_\si(x)) = \ph^-_b(x,g_{ab}(x) f^a_\si(x))
$$
and hence iff
$$
f^b_\si(x) = g_{ab}(x) f^a_\si(x)
$$
In this way, a ''global section'' (//''section''//) associates $M$ with a particular submanifold, $\si$, of $E$. This change of how a section is represented when the local trivialization is changed, $\si'(x) = g(x) \si(x)$, is the most basic type of [[gauge transformation]].
The [[partial derivative]] is zero when acting on a ''constant'' section, $f_\si(x) = \si(x) = \si$, that is specified locally by a constant field, $\pa_i \si=0$. This derivative doesn't properly keep track of the local trivialization or gluing between patches. To remedy this, a [[covariant derivative]] is introduced which, via a [[connection]], keeps track of how the local trivialization changes over the base when taking the derivative of a section -- it co-varies with a gauge transformation. Using the covariant derivative, any fiber element may be [[parallel transport]]ed along any path on the base to obtain a new fiber element at any point along the path. For a closed path, the parallel transport of a fiber element is represented by a [[holonomy]] -- an element of the structure group which acts on the fiber element. For a small closed path, or loop, the holonomy is given approximately by the [[curvature]] -- an important geometric descriptor of the fiber bundle and connection.
The above description, employing local trivializations, treats fiber bundle geometry as something happening over a base space. Fiber bundle geometry may be described more naturally over the total space by employing an [[Ehresmann connection]]. The defining map, $\pi$, of a fiber bundle gives an involutive [[distribution]], $\ve{\De_\pi}$, and foliation of the total space, $E$, by fibers. This distribution is the kernel of the [[pushforward|pullback]] of the map, $\pi_* \ve{\De_\pi}=0$, and is tangent to the foliating fibers of $E$.
Refs:
*A more thorough description is available at http://en.wikipedia.org/wiki/Fiber_bundle
*[[A Route Towards Gauge Theory|papers/A Route Towards Gauge Theory.pdf]]
*[[Geometrical aspects of local gauge symmetry|papers/Geometrical aspects of local gauge symmetry.pdf]]
*[[Preparation for Gauge Theory|papers/9902027.pdf]]
**excellent mathematical review of the basics
The [[skew]], $\ep$, is an antisymmetric [[metric]] for [[Weyl spinor]]s. If $\ps_{1L}$ and $\ps_{2L}$ are left chiral Weyl spinors, with components $\ps_{1L}^{\wedge/\vee}$ and $\ps_{2L}^{\wedge/\vee}$, then the scalar,
$$
s = \ps_{1L}^T \ep \, \ps_{2L} = - \ps_{1L}^{\wedge} \ps_{2L}^{\vee} + \ps_{1L}^{\vee} \ps_{2L}^{\wedge} = - \mbox{det}
\lb \ba{cc}
\ps_{1L}^{\wedge} & \ps_{2L}^{\wedge} \\
\ps_{1L}^{\vee} & \ps_{2L}^{\vee}
\ea \rb
$$
using the [[transpose]], or the [[determinant]], is invariant under [[Lorentz transformation]]s. In analogy to contravariant and covariant vectors, if $\ps_L$ is a left chiral [[Weyl spinor]], then $\ep \ps_L$ is the ''spin flipped Weyl spinor''. The [[Pauli basis spinors|Weyl spinor]] and ''flipped basis spinors'' are
$$
\ch^\wedge = \lb \ba{c}1 \\ 0 \ea \rb
\s
\ch^\vee = \lb \ba{c} 0 \\ 1 \ea \rb
\s
\xi^\wedge = \ep \ch^\wedge = \ch^\vee = \lb \ba{c} 0 \\ 1 \ea \rb
\s
\xi^\vee = \ep \ch^\vee = -\ch^\wedge = \lb \ba{c} -1 \\ 0 \ea \rb
$$
For a left or right chiral Weyl spinor, $\ps_{L/R}$, its [[charge conjugate]] is the spin flipped [[complex conjugate|complex structure]], $\bar{\ps}_{R/L} = (\ps_{L/R})^C = \mp \ep \ps_{L/R}^*$. Since $\ep^2 = -1$, flipping a spinor twice gives negative the spinor.
Consider a [[vector field|tangent bundle]], $\ve{v}(x)$ over a manifold. There are unique [[path]]s, $x(t)$, called ''integral curves'' of the vector field, such that the [[tangent vector]] at each point along the path is equal to the vector of the vector field at that point,
$$
\fr{d x^i(t)}{d t} = v^i(x(t))
$$
This relation may be thought of, and solved, as a set of ODE's. Consider the integral curves, with $\ve{v}$ as tangent vectors, starting from each manifold point, $x$, with these paths parameterized such that $t=0$ at these points. From these initial conditions, the point of each path, $y(t,x)$, is determined as a function of parameter and starting point, $x$. This is a ''flow'' — a parameterized [[autodiffeomorphism|diffeomorphism]], $\ph_t(x)=y(t,x)$, satisfying the "time symmetry" rule:
$$
\ph_t(\ph_{t'}(x)) = y(t,y(t',x)) = y(t+t',x) = \ph_{t+t'}(x)
$$
A flow may be visualized as a movement of the manifold points beneath any overlying geometric elements. Any chosen initial point, $x$, is carried along by the flow along the path, $\ph_t(x)$, defined by the ''flow equation'',
$$
\lb \fr{\pa}{\pa t} \ph_t^i(x) \rb \ve{\pa_i} = \ve{v}(x)
$$
for all $t$. The flow is completely determined by the vector field, $\ve{v}(x)$ — the ''vector field generator'' of the flow. An observer attached to such a point carried by the flow may either consider herself to be moving through a field of geometric elements or, alternatively, to be having the geometric elements change over her. For short times, the flow is (in terms of coordinates)
$$
\ph_t^i(x) \simeq x^i + t v^i(x)
$$
The solution to the flow equation may be written heuristically as the [[exponentiation]] of the flow,
$$
\ph_t(x) = e^{t\ve{v}\f{d}} x = e^{t {\cal L}_{\ve{v}}} x
$$
and any geometric object may be [[pushed forward|pullback]] along the flow by exponentiating the [[Lie derivative]], with
$$
\ph^*_t X = e^{t {\cal L}_{\ve{v}}} X
$$
It is possible to have a parameterized autodiffeomorphism, $\ph_t(x)$, that satisfies $\ph_0(x)=x$ but does not satisfy the time symmetry rule. This is a ''time dependent flow'', and produces two distinct, time dependent velocity fields. The first, the ''Lagrangian flow field'', is the velocity of each initial manifold point, $x_0$, wherever it might be carried on the manifold:
$$
\ve{v_t}(x_0) = \lb \fr{\pa}{\pa t} \ph_t^i(x_0) \rl_t \ve{\pa_i}
$$
The second, the ''Euler flow field'', is the velocity at each manifold point at time $t$:
$$
\ld \ve{v_t}(x) \rl_{x=\ph_t(x_0)} = \lb \fr{\pa}{\pa t} \ph_t^i(x_0) \rl_t \ve{\pa_i}
$$
If the Euler flow field is constant in time, it is the vector field generator for the corresponding flow.
The ''equivalence principle'' of General Relativity holds that physics in a sufficiently small region near each point in an $n$ dimensional curved [[spacetime]] is locally indistinguishable from physics in a flat spacetime [[rest frame]] at that point. The mathematical implication is that at each point there is a set of $n$ ''orthonormal basis vectors'' (a.k.a. //''frame vectors''//), which can be written in terms of the [[coordinate basis vectors]] as
\[ \ve{e_\al} = \lp e_\al \rp^i \ve{\pa_i} \]
They are orthonormal, $\lp \ve{e_\al},\ve{e_\be} \rp = \et_{\al \be}$, under use of a [[metric]], in which $\et_{\al \be}$ is a [[Minkowski metric]]. The set of their [[1-form]] duals constitute the ''coframe 1-forms'' (a.k.a. //''coframe''//, //''vielbein''//, //''tetrad''//, //''frame 1-forms''//, or sometimes also just called the //''frame''//),
\begin{eqnarray}
\f{e^\al} &=& \f{dx^i} \lp e_i \rp^\al\\
\ve{e_\al} \f{e^\be} &=& \lp e_\al \rp^i \lp e_j \rp^\be \ve{\pa_i} \f{dx^i} = \lp e_\al \rp^i \lp e_i \rp^\be = \de_\al^\be
\end{eqnarray}
As the set of "rulers" on the manifold, the coframe matrix components have [[units]] of time, $T$. The ''orthonormal basis vector matrix'', or //''frame matrix''//, $\lp e_\al \rp^i$, is the [[inverse]] of the ''coframe matrix'', $\lp e_i \rp^\al$ — they satisfy $\lp e_\al \rp^i \lp e_i \rp^\be = \de_\al^\be$ and $\lp e_i \rp^\al \lp e_\al \rp^j = \de_i^j$. (The ${}^-$ in $\lp e^-_\al \rp^i$ is not written but is implied from the position of the [[indices]].) The frame or coframe matrices multiply indexed objects and change their indices between coordinate indices and orthonormal basis (rest frame) labels, $T_{\al i} \lp e_j \rp^\al \lp e_\ga \rp^i = T_{j \ga}$.
The frame (or coframe), along with the Minkowski metric encoding the signature, completely determines a metric on the manifold. But the converse is not true — the metric only determines a frame up to a [[Lorentz rotation]]. And a frame is necessary to define [[spinor]]s in curved spacetime. So the frame (or coframe) should be considered the fundamental object encoding the geometry of the manifold. It is possible to make the correspondence between frame and metric one-to-one by imposing a coordinate dependent restriction on the form of the frame matrix — such as restricting to the use of a [[UT frame]].
It is very useful to employ [[Clifford basis vectors]], $\ga_\al \leftrightarrow \ve{e_\al}(x)$, as the fundamental geometric basis vector elements of each rest frame. We can define the ''coframe'' as
$$
\f{e} = \f{e^\al} \ga_\al = \f{dx^i} \lp e_i \rp^\al \ga_\al
$$
a Clifford vector valued 1-form field (a [[Clifform]]). This can be used to [[Cliffordize|Cliffordization]] tangent vectors into Clifford vectors, $v = \ve{v} \f{e}$, using [[vector-form algebra]]. As we wanted, the Cliffordized orthonormal basis vectors are then the Clifford basis vectors,
$$
\ve{e_\al} \; \leftrightarrow \; \ve{e_\al} \f{e} = \ga_\al
$$
We can similarly define the ''frame'', a Clifford vector valued vector field, as
$$
\ve{e}=\ga^\al \ve{e_\al}=\ga^\al \lp e_\al \rp^i \ve{\pa_i}
$$
and use it to Cliffordize forms, such as $a = \ve{e} \f{a}$ and $\ga^\al = \ve{e} \f{e^\al}$. The frame and coframe satisfy
$$
\ve{e} \f{e} = \ga^\al \lp e_\al \rp^i \ve{\pa_i} \f{dx^j} \lp e_j \rp^\be \ga_\be
= \ga^\al \lp e_\al \rp^i \lp e_i \rp^\be \ga_\be
= \ga^\al \ga_\al
= n
$$
(Since the frame and coframe are used similarly and, as one is the inverse of the other, carry the same information, they are both often collectively referred to as the //''frame''//.)
[[geodesic]]
''momentum'' conserved along [[Killing vector]] directions.
Rovelli p122 for free particle in Minkowski space.
A ''function'' over a [[manifold]], $M$, is usually written as $f(x)$ — a function of the corresponding coordinates, $f \circ x_a^{-} : \Re^n \rightarrow M \rightarrow \Re$. In this case it is a map from the points of the manifold, in the various charts, into the real numbers. The $x$ in $f(x)$ is shorthand for the set of coordinates, $x^i$, over each manifold patch.
A function is more thoroughly described as a section of a [[fiber bundle]].
The rank $2$ exceptional [[Lie group]], [[G2]], is described by its $14$ dimensional [[Lie algebra]], ''g2''. This Lie algebra may be decomposed as a $8$-dimensional [[symmetric|symmetric space]] [[subgroup]], the [[special unitary group]] Lie algebra, [[su(3)]], acting on the standard $3$ representation and its dual, $\bar{3}$,
$$
g2 = su(3) + 3 + \bar{3}
$$
This also relates to
$$
g2 \subset so(7) = so(6) + 6
$$
$$
so(7) = g2 + 7
$$
which gives the fundamental $7$ rep.
An explicit construction of the Lie algebra and the group can be found in:
Ref:
*[[Cerchiai - Euler angles for G2|papers/Cerchiai - Euler angles for G2.pdf]]
A ''gauge transformation'' (//''passive gauge transformation''//) is a transformation of the section (//''gauge''//), $\si(x)$, of a [[fiber bundle]] by an element, $g \in G$, of the structure [[group|Lie group]] to another gauge, $\si'(x) = g \, \si(x)$. A gauge transformation is ''local'' if it has a position dependent $g(x)$ and ''global'' if it isn't. Alternatively, a ''passive coordinate gauge transformation'' may be considered -- and treated equivalently -- that is nothing but a description of how the representation of the section changes under a change of local trivialization.
The [[covariant derivative]], written using a [[connection]], varies covariantly,
\begin{eqnarray}
\f{\na'} \si' &=& g(x) \f{\na} \si\\
\lp \f{d} g \rp \si + g \f{d} \si + \f{A'} g \si &=& g \f{d} \si + g \f{A} \si\\
\end{eqnarray}
giving the transformation law for the connection under a gauge transformation,
$$
\f{A'} = g \f{A} g^- - \lp \f{d} g \rp g^- = g \f{A} g^- + g \lp \f{d} g^- \rp
$$
An infinitesimal transformation, $g \simeq 1 + G^A T_A = 1 + G$, changes the connection to
\begin{eqnarray}
\f{A'} &\simeq& \f{A} - \f{d} G - \f{A} G + G \f{A} = \f{A} - \f{\na} G \\
\de_G \f{A} &=& - \f{\na} G
\end{eqnarray}
Under a gauge transformation, the [[curvature]] changes to
\begin{eqnarray}
\ff{F'} &=& \f{d} \f{A'} + \f{A'} \f{A'} \\
&=& \f{d} \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb + \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb \lb g \f{A} g^- - \lp \f{d} g \rp g^- \rb \\
&=& g \lp \f{d} \f{A} + \f{A} \f{A} \rp g^- \\
&=& g \ff{F} g^- = A_g \ff{F} \\
&\simeq& \ff{F} + G \ff{F} - \ff{F} G = \ff{F} + \lb G , \ff{F} \rb \\
\de_G \ff{F} &=& \lb G , \ff{F} \rb
\end{eqnarray}
All physical, measurable quantities in physics are invariant under gauge transformations.
The above description of gauge transformations presumes the connection to be a field over the base manifold. A gauge transformation may also be described from the viewpoint of structures over the total space of the fiber bundle, as an [[Ehresmann gauge transformation]]. In this space, an ''active gauge transformation'' is a vertical [[autodiffeomorphism|diffeomorphism]] -- this gauge transformation, which transforms the connection over the total space while leaving the local sections fixed, is equivalent to the passive gauge transformation, which transforms the local sections while leaving the connection fixed.
A ''geodesic'' on a [[manifold]] with a [[metric]] or [[frame]] is a [[path]], $x(t)$, of extremal length (or time). This [[proper time]], in temporal [[units]] such as seconds, is the integral of the speed between two points,
$$S = \Delta \ta = \int \f{dt} \ll v \rl = \int \f{dt} \sqrt{\ll v \cdot v \rl} = \int \f{dt} \sqrt{\ll \lp \ve{v} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp \rl} = \int \f{dt} \sqrt{\ll \fr{d x^i}{d t} \fr{d x^j}{d t} g_{ij}(x) \rl} $$
with respect to parameter time, $t$. Extremizing this length with respect to variation of $x^i(t)$ gives a geodesic equation invariant with respect to reparameterization. Alternatively, a geodesic also extremizes the action,
$$S = \int \f{d \ta} \ha v^2 = \int \f{d \ta} \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij}(x) = \int \f{d \ta} \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij}(x) $$
which is a simpler variation and produces the affine geodesic equation, with the parameter set equal to the proper time along the curve, $t=\ta$. The variation is
\begin{eqnarray}
\de S &=& \int \f{d \ta} \lc \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \de x^k \pa_k g_{ij} + \fr{d \de x^i}{d \ta} \fr{d x^j}{d \ta} g_{ij} \rc \\
&=& \int \f{d \ta} \de x^k \lc \ha \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \pa_k g_{ij} - \fr{d}{d \ta} \lp \fr{d x^j}{d \ta} g_{kj} \rp \rc
\end{eqnarray}
and gives the geodesic equation,
$$\fr{d^2 x^k}{d \ta^2} = \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \lp \ha g^{mk} \pa_m g_{ij} - g^{mk} \pa_i g_{mj} \rp
= - \fr{d x^i}{d \ta} \fr{d x^j}{d \ta} \Ga^k{}_{ij}$$
This determines the path on a manifold followed by any freely falling body, given its initial position and velocity. This motion of a free particle also has a [[Hamiltonian formulation|free particle Hamiltonian]]. The geodesic dependends on derivatives of the metric via the [[Christoffel symbols|tangent bundle connection]], and may be expressed by derivatives of the frame via the spin connection. (This relationship also gives a quick way of calculating connection coefficients by varying the metric or frame.) Note that geodesics aren't influenced by [[torsion]] since only the symmetric part of the Christoffel symbols, $\Ga^k{}_{\lp ij \rp}$, enter the geodesic equation.
The [[tangent vector]] to a geodesic is [[parallel transport|tangent bundle parallel transport]]ed along the path,
$$0 = \lp \ve{v} \f{\nabla} \rp \ve{v} = v^i \nabla_i v^k \ve{\pa_k} = \lp \fr{d v^k}{d \ta} + v^i v^j \Ga^k{}_{ij} \rp \ve{\pa_k}$$
by virtue of the geodesic equation. If the velocity is expressed as a [[Clifford element]] using the frame, $v = \ve{v} \f{e}$, then
$$0 = \ve{v} \f{\nabla} v = v^i \nabla_i v^\al \ga_\al = \lp \fr{d v^\al}{d \ta} - v^i v^\be \om_i{}_\be{}^\al \rp \ga_\al$$
in terms of the [[spin connection]]. The speed along a geodesic with respect to proper time is constant, and its square, $v^2$, is either $1$, $0$, or $-1$ depending on whether the velocity is timelike, null, or spacelike.
The component of a geodesic's velocity along any [[Killing vector]] is constant along the path,
$$
...
$$
//add equation//
*<<slider chkSliderhamF hamF 'ham >' 'Hamiltonian dynamics, symplectic geometry'>>
*<<slider chkSliderkkF kkF 'kk >' 'Kaluza-Klein theory, Killing vector fields'>>
*<<slider chkSlidercosmoF cosmoF 'cosmo >' 'cosmology'>>
*<<slider chkSlidergrscalF grscalF 'grscal >' 'gr plus a scalar field, Brans-Dicke theories, conformal transformations'>>
*<<slider chkSliderlqgF lqgF 'lqg >' 'loop quantum gravity, loops, spin foams, spin networks'>>
*<<slider chkSlidertorsF torsF 'tors >' 'torsion, teleparallel gravity'>>
<<ListTagged gr>>
Ref:
*Fabrizio Nesti
**[[Standard Model and Gravity from Spinors|http://arxiv.org/abs/0706.3304]]
***Uses $so(3,1)$ spin connection. Split this into self-dual part for gravity and anti-self-dual part for either electroweak $su(2)_L$ or $su(2)_R$. Nesti breaks both up out of $SO(3,1,\mathbb{C})$. He also goes on to talk about Pati-Salam, but his frame and Higgs are messed up and he doesn't use MacDowell-Mansouri. Ooh, he is getting close to what I'm doing though -- mentions embedding in non-orthogonal groups in a foottiddler. Mirror fermion problem. [[Coleman-Mandula]] doesn't apply because the [[frame]] doesn't have Poincare symmetry.
**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]
***Uses $Cl(4,\mathbb{C})$ for a Pati-Salam model.
*Stephon Alexander
**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]
A ''group'' is a collection of elements, $g \in G$, along with an ordered group product by which one element times a second equals a third, $g_1 g_2 = g_3$. A group has the following properties:
*it includes the ''identity element'', $g 1 = 1 g = g$
*every element has an [[inverse]], $g g^- = g^- g = 1$
*its product is ''associative'', $a(bc)=(ab)c$
*its product is ''closed'', $ab \in G$
One group element, $h$, may act on another, $g$, via three different ''group action''s:
*''left action'': $L_h g = h g$
*''right action'': $R_h g = g h$
*''conjugation'', also known as the inner [[automorphism]] or ''adjoint action'': $A_h g = L_h R_{h^-} g = h g h^-$
A group is ''abelian'' iff $gh = hg$ for all $h, g \in G$.
An example of a group is the set of integers, $G = \left\{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \right\}$, with addition, $+$, as the group product. For this abelian group, $0$ is the identity element.
*James Ryan
**[[A new proposal for group field theory I: the 3d case|http://arxiv.org/abs/gr-qc/0611080]]
The ''helicity'', $h$, of a [[Dirac spinor]], $\Psi$, is the [[spin|spin operator]], $\ve{S}$, in the direction of spatial [[momentum]], $\hat{p} = p_u^\ep \ve{e}_\ep$ (with $p_u^\ep = \fr{1}{|\ve{p}|} p^\ep$),
$$
h \, \Ps = \lp \hat{p} , \ve{S} \rp \Ps = p_u^\ep S_\ep = \ha \lb \si_0 \otimes p_u \rb \Ps
$$
in which $p_u = p_u^\ep \si_\ep$ is the unit spatial momentum direction [[Clifford vector]] using [[Pauli matrices]]. Acting on a [[Weyl spinor]] (the left or right-[[chiral]] half of a Dirac spinor), $\ch = \Ps_{L/R}$, this ''helicity operator'', $p_u$, has [[helicity state]]s, $\ch_\pm$, satisfying
$$
p_u \ch_\pm = p_u^\ep \si_\ep \, {\ch}_\pm = \la_\pm \, {\ch}_\pm
$$
with [[eigen]]value $\la_+ = +1$ corresponding to ''right-handed'' helicity, $h = + \ha$, and $\la_- = -1$ corresponding to ''left-handed'' helicity, $h = - \ha$. Helicity is like [[polarization]], but for spinors.
There are interesting mathematical relationships between [[Cl(3)]] Clifford algebra elements, with [[Pauli matrices]] as representative basis vectors, and [[Weyl spinor]]s. If $p_u = p_u^\va \si_\va$ (often a [[momentum]] direction) is a direction in space, satisfying $p_u \!\!\cdot\! p_u = 1$, then it has [[helicity state]]s, $\ch_\pm$, satisfying
$$
p_u \ch_\pm = \pm \ch_\pm
$$
These [[Weyl spinor]]s are [[Hermitian]]-orthogonal, $\ch_+^\da \ch_- = \ch_-^\da \ch_+ = 0$, and normalized such that $\ch_+^\da \ch_+ = \ch_-^\da \ch_- =1$. Their phases are determined by demanding
$$
\ch_{\pm}^* = \pm \ep \, \ch_\mp
\s \s
\ch_{\pm}(-p) = i \, \ch_{\mp}(p)
$$
The left [[chiral]] representative, $p^0_L$, of the ''null spacetime momentum direction'', $p_0^\mu = (1,p_u^1,p_u^2,p_u^3)$, can be obtained from ''squaring'' the left-helicity eigenstate,
$$
2 \, \ch_- \ch_-^\da =
\lb \begin{array}{cc}
1 - \cos{\th} & - e^{- i \ph} \sin{\th} \\
- e^{ i \ph} \sin{\th} & 1 + \cos{\th}
\end {array} \rb
= \si_0 - p_u = p_0^\mu \bar{\si}_\mu = p^0_L
$$
and the complementary right chiral representative of the null spacetime momentum direction from squaring the right-helicity eigenstate,
$$
2 \, \ch_+ \ch_+^\da = \si_0 + p_u = p_0^\mu \si_\mu = p^0_R
$$
These formulas arise from the fact that $\si_0 \pm p_u$, as a $2 \times 2$ matrix, has a zero eigenvalue, and so is rank $1$ and has a spectral decomposition description using one eigenvector (a Weyl spinor) and its conjugate. These formulas relate to the [[twistor]] program, and the statement that "a spinor is a square root of a vector".
If we outer-multiply opposite pairs of helicity states we get the right and left circular [[polarization]] basis vectors,
$$
2 \, \ch_\pm \ch_\mp^\da =
\lb \begin{array}{cc}
\sin(\th) & e^{-i \ph} \lp \mp 1 - \cos(\th) \rp\\
e^{i \ph} \lp \pm 1 - \cos(\th) \rp & -\sin(\th) \\
\end{array} \rb
= \ep_\pm
$$
These circular polarization vectors satisfy
$$
\ep_\pm \ep_\pm = 4 \, \ch_\pm \ch_\mp^\da \ch_\pm \ch_\mp^\da =0
\s
\ep_\pm \ep_\mp = 4 \, \ch_\pm \ch_\mp^\da \ch_\mp \ch_\pm^\da = 2 (\si_0 \pm p_u)
\s
p_u \ep_\pm = 2 p_u \ch_\pm \ch_\mp^\da = \pm \ep_\pm
$$
and can be factored as $\ep_\pm = \ep_1 \pm i \ep_2$, with these polarization vectors satisfying
\begin{eqnarray}
\ep_1 \ep_1 &=& \fr{1}{4} (\ep_+ + \ep_-)(\ep_+ + \ep_-) = \si_0 \\
\ep_2 \ep_2 &=& - \fr{1}{4} (\ep_+ - \ep_-)(\ep_+ - \ep_-) = \si_0 \\
\ep_1 \ep_2 &=& \fr{1}{4i} (\ep_+ + \ep_-)(\ep_+ - \ep_-) = i p_u = \si \, p_u
\end{eqnarray}
This implies that these three vectors, $\ep_1$, $\ep_2$, and $p_u$, are a right-handed set of orthonormal [[Cl(3)]] vectors, multiplying to give the [[pseudoscalar]], $\ep_1 \ep_2 \, p_u = \si$.
To summarize, choosing a Weyl spinor, $\ch_+$ or $\ch_-$, satisfying $\ch_a^\da \ch_b = \de_{ab}$, with the other one determined by $\ch_\mp = \mp \ep \ch_\pm^*$, uniquely determines three right-handed orthonormal spatial vectors, $\ep_1$, $\ep_2$, and $p_u$.
From the [[helicity identities]] using [[helicity state]]s, there is a nice ''helicity notation'' of [[chiral Clifford vectors|Dirac matrices]] and [[polarization]]. Recall that for any spatial [[momentum]] direction, $p_u = p_u^\va \si_\va$, there are helicity eigenstates, $p_u \ch_\pm = \pm \ch_\pm$. From a spectral decomposition, this gives
$$
2 \ch_\pm \ch_\pm^\da = \si_0 \pm p_u = p_{0 \, R/L}
$$
which are real null momentum directions, with $p_0^0 = 1$, satisfying $p_{0 \, R/L} \, \ch_\pm = 2 \ch_\pm$ and $p_{0 \, R/L} \, \ch_\mp = 0$. For any (non-normalized) [[Weyl spinor]] we choose, it determines, through the above formula, a null momentum of some energy. (It's nice to be able to specify a null vector without imposing a constraint.) We now generalize this to complex null momentum, $p^\mu \in \mathbb{C}$, with
$$
p_{R/L} = p^0 \si_0 \pm p^\va \si_\va = E \, (\si_0 \pm p_u^\va \si_\va)
$$
(with complex $p_u p_u = p_u^1 p_u^1 + p_u^2 p_u^2 + p_u^3 p_u^3 = 1$) satisfying
$$
0 = | p_{R/L} | = \det \lp p_{R/L} \rp = p^\mu p^\nu \et_{\mu\nu} = \bar{p}_{R/L} p_{R/L} = p_{L/R} p_{R/L} = \ha {\rm Tr} \lp p_R \, p_L \rp = \li p_R \, p_L \ri
$$
using the [[determinant]], in which the [[scalar part]] operator $\li A \ri$, is equivalent to taking half the [[trace]]. As a $2 \times 2$ rank $1$ matrix, $p_{R/L}$ has an eigensystem,
$$
p_{R/L} \, \ch_\pm = 2 E \, \ch_\pm
\s
p_{R/L} \, \ch_\mp = 0
\s
\tilde{\ch}_\pm \, p_{R/L} = 2 E \, \tilde{\ch}_\pm
\s
\tilde{\ch}_\mp \, p_{R/L} = 0
$$
in which the left eigenvectors, $\tilde{\ch}_\pm$, are row vectors also satisfying $\tilde{\ch}_a \ch_b = \de_{ab}$. If $\tilde{\ch}$ is the $2 \times 2$ matrix with rows $\tilde{\ch}_\pm$, and $\ch$ is the matrix with columns $\ch_\pm$, then $\tilde{\ch} = \ch^-$ -- so the $\tilde{\ch}_\pm$ are determined by the $\ch_\pm$. Explicitly,
$$
| \ch | = \ch_-^T \ep \ch_+
\s \;\;\;
\tilde{\ch}_+ = \fr{1}{| \ch |} \ch_-^T \ep
\s \;\;\;
\tilde{\ch}_- = - \fr{1}{| \ch |} \ch_+^T \ep
$$
using the [[skew]], $\ep$. If $\tilde{\ch}_\pm = \ch_\pm^\da$ then $p$ is real and this matches the helicity eigensystem above, scaled by $E$, with $\ch_\mp = \pm \ep \ch^*_\pm = (\ch_\pm)^C$ [[charge conjugate]]s, and so $\os{\ch}_\pm = \fr{1}{|\ch|} \ch_\pm^\da$ with $|\ch| = \ch_+^\da \ch_+ = \ch_-^\da \ch_-$. However, for complex $p$, having four complex degrees of freedom (constrained by being null), the Weyl spinors $\ch_+$ and $\ch_-$ (or another pair, such as $\ch_+$ and $\tilde{\ch}_+$) are independent. We partially normalize these eigenvectors by restricting them to have determinant one, $| \ch | = 1$, so we have
$$
1 = | \ch | = \ch_-^T \ep \ch_+ = - \ch_+^T \ep \ch_-
\s \;\;\;
\tilde{\ch}_+ = \ch_-^T \ep
\s \;\;\;
\tilde{\ch}_- = - \ch_+^T \ep
$$
Then, to acommodate the $2E\,$'s floating around, we define $\ps_\pm = \sqrt{2E} \ch_\pm$ and $\tilde{\ps}_\pm = \sqrt{2E} \tilde{\ch}_\pm$, so we have spectral decompositions,
$$
p_R = 2E \, \ch_+ \tilde{\ch}_+ = \ps_+ \tilde{\ps}_+ = a ] \l a
\s \s
p_L = 2E \, \ch_- \tilde{\ch}_- = \ps_- \tilde{\ps}_- = a \g [ a
$$
in which we use a snazzy helicity notation for the column and row vector representatives,
$$
\ba{ccc}
\ba{rcl}
a \g \!\!&\!\!=\!\!&\!\! \ps_- = \lb \ba{c} \ps_-^\wedge \\ \ps_-^\vee \ea \rb = \ep \os{\ps}{}^T_+ = \ep \l a {}^T \\
a ] \!\!&\!\!=\!\!&\!\! \ps_+ = \lb \ba{c} \ps_-^\wedge \\ \ps_-^\vee \ea \rb = - \ep \os{\ps}{}^T_- = - \ep [ a{}^T \\
[ a \!\!&\!\!=\!\!&\!\! \os{\ps}_- = \lb \ba{cc} \os{\ps}{}_-^\wedge & \os{\ps}{}_-^\vee \ea \rb = - \ps_+^T \ep = - a ]^T \ep \\
\l a \!\!&\!\!=\!\!&\!\! \os{\ps}_+ = \lb \ba{cc} \os{\ps}{}_+^\wedge & \os{\ps}{}_+^\vee \ea \rb = \ps_-^T \ep = a \g^T \ep
\ea
& \s &
\ba{rcl}
\bar{a} \g \!\!&\!\!=\!\!&\!\! \bar{\ps}_- = \lp \ps_+ \rp^C = \ep \, \ps^*_+ = \os{\ps}{}^\da_- = [ a {}^\da \\
\bar{a} ] \!\!&\!\!=\!\!&\!\! \bar{\ps}_+ = \lp \ps_- \rp^C = - \ep \, \ps^*_- = \os{\ps}{}^\da_+ = \l a{}^\da \\
[ \bar{a} \!\!&\!\!=\!\!&\!\! \bar{\os{\ps}}_- = \lp \os{\ps}_+ \rp^C = - \os{\ps}{}^*_+ \ep = \ps_-^\da = a \g^\da \\
\l \bar{a} \!\!&\!\!=\!\!&\!\! \bar{\os{\ps}}_+ = \lp \os{\ps}_- \rp^C = \os{\ps}{}^*_- \ep = \ps_+^\da = a ]^\da
\ea
\ea
$$
for a particle labeled "$a$", and its [[charge conjugate]], "$\bar{a}$". This is consistent with $p_L$ being the [[2D matrix conjugate|determinant]] of $p_R$,
$$
\bar{p}_R = - \ep \, p_R^T \ep = (\ep \os{\ps}{}^T_+ )(- \ps^T_+ \ep) = \ps_- \tilde{\ps}_- = p_L
$$
The self-contractions (including a non-Lorentz invariant one) satisfy
$$
\l a \, a \g \, = [a \, a] = 0 \s \s \l a \, a ] = [a \, a \g = 2 E_a
$$
For two different Weyl spinors contractions satisfy
$$
\l a \, b \g = - \l b \, a \g
\s \;\;\;
[ a \, b ] = - [ b \, a ]
\s \;\;\;
\l a \, b ] = [ b \, a \g
\;\;\;\;\; \;\;\;
\l a \, b \g^\da = [ \bar{b} \, \bar{a} ]
$$
and for four, the ''Schouten helicity identities'',
\begin{eqnarray}
\l a \, b \g \l c \, d \g &=& \l a \, c \g \l b \, d \g - \l a \, d \g \l b \, c \g \\
[ a \, b ] [ c \, d ] &=& [ a \, c ] [ b \, d ] - [ a \, d ] [ b \, c ] \\
\end{eqnarray}
We can use these to calculate fun Lorentz invariant scalars, like the contraction of two null momenta,
$$
p_a \cdot p_b = p_a^\mu p_b^\nu \et_{\mu \nu} = \ha {\rm Tr} \lp \, p_{aL} p_{bR} \, \rp = \ha {\rm Tr} \lp \, a \g [ a \, b ] \l b \, \rp = \ha [ a \, b ] \l b \, a \g
$$
which is half the length squared of their sum,
$$
{\rm det}\lp p_{aR} + p_{bR} \rp
= \ha {\rm Tr} \lp \, (p_{aL} + p_{bL}) (p_{aR} + p_{bR}) \, \rp
= \ha {\rm Tr} \lp \, \lp a \g [ a + b \g [ b \rp \lp a ] \l a + b ] \l b \rp \, \rp
= [a \, b] \l b \, a \g
$$
Using helicity notation, the ''complex circular [[polarization]] basis vectors'',
$$
\ep_+ = \ep^\va_+ \si_\va = \fr{1}{E_a} a ] [ a
\s \s
\ep_- = \ep^\va_- \si_\va = \fr{1}{E_a} a \g \l a
$$
are traceless and therefore purely spatial, and satisfy
$$
p_{R/L} \, \ep_{\pm} = 2 E \, \ep_{\pm}
\s
p_{R/L} \, \ep_{\mp} = 0
\s
\ep_\pm \, \ep_\pm = 0
\s
\ep_\pm \, \ep_\mp = \fr{2}{E} \, p_{R/L}
\s
\ep_1 \ep_2 = i p_s
$$
A ''helicity phase change'',
$$
a \g \to e^{i \fr{\ph}{2}} a \g
\s \s
a] \to e^{-i \fr{\ph}{2}} a];\;
\s \s
\l a \to e^{i \fr{\ph}{2}} \l a
\s \;\;\;\;\
[ a \to e^{- i \fr{\ph}{2}} [ a
$$
leaves momentum invariant but changes the polarization phase, $\ep_\pm \to e^{\pm i \ph} \ep_\pm$.
All this works for complex momentum. If we wish to restrict to real momentum in Minkowski spacetime, then we restrict the momentum to be invariant under the Hermitian conjugate, corresponding to a [[Majorana spinor]] restriction,
$$
p_{R/L} = p_{R/L}^\da \s \ps_\pm = (\ps_\mp)^C = \mp \ep \, \ps^*_\mp = \os{\ps}{}^\da_\pm \s a \g = \bar{a} \g \s a ] = \bar{a} ]
$$
We get a split signature momentum if we restrict the momentum to be invariant under the complex and 2D matrix conjugate,
$$
p_{R/L} = \bar{p}_{R/L}^* = -\ep \, p_{R/L}^\da \ep = p_{L/R}^* \s \ps_\pm = \ps^*_\mp \s a ] = a^* \g \s \l a = [ a {}^*
$$
If we wish to restrict to real momentum in Euclidean spacetime, then we require the momentum to be invariant under the Hermitian and 2D matrix conjugate,
$$
p_{R/L} = \bar{p}_{R/L}^\da = p_{L/R}^\da \s \ps_\pm = (\ps_\pm)^C = \pm \ep \, \ps^*_\pm = \os{\ps}{}^\da_\mp \s a ] = \bar{a} \g \s \l a = [ \bar{a}
$$
which gives a funky spinor restriction with no solution. This reflects the fact that a non-zero $p$ cannot be null in Euclidean spacetime.
The philosophy behind helicity notation is that one can choose, say, $\ps_-$ and $\tilde{\ps}_-$, as four complex numbers, which specify the complex momentum and spin (polarization) of a massless particle, and can be used to simplify scattering amplitude calculations. If we restrict to some real spacetime signature, then choosing two complex numbers determines the momentum and spin.
If, using polar coordinates and [[Pauli matrices]], the [[momentum]] direction is
$$
p_u =
p_u^\va \si_\va
= \si_1 \sin{\th} \cos{\ph}
+ \si_2 \sin{\th} \sin{\ph}
+ \si_3 \cos{\th}
$$
then the normalized [[eigen]]states ($\ch^\da_a \ch_b = \de_{ab}$) of the [[helicity]] operator, $p_u \ch_{\pm} = \pm \ch_\pm$, are the right and left ''helicity states'' (or ''helicity eigenspinors'', or ''phase-balanced helicity states''), $\ch_\pm$, which can be written as
$$
p_u =
\lb \begin{array}{cc}
\cos{\th} & e^{- i \ph} \sin{\th} \\
e^{ i \ph} \sin{\th} & - \cos{\th}
\end {array} \rb
\s\;
\ch_+ =
\lb \begin{array}{c}
e^{-i \ph / 2} \cos{\fr{\th}{2}} \\
e^{i \ph / 2} \sin{\fr{\th}{2}}
\end {array} \rb
\s\;
\ch_- =
\lb \begin{array}{c}
e^{- i \ph / 2} \sin{\fr{\th}{2}} \\
- e^{i \ph / 2} \cos{\fr{\th}{2}}
\end {array} \rb
$$
highlighting the property of a [[Weyl spinor]] needing to be rotated through $4 \pi$ to return to itself. Or, if one prefers cartesian momentum coordinates, these are
$$
p_u =
\lb \begin{array}{cc}
p_u^3 & p_u^1 - i \, p_u^2 \\
p_u^1 + i \, p_u^2 & - p_u^3
\end {array} \rb
\s\;
\ch_+ = \fr{\sqrt{p_u^1 - i \, p_u^2}}{ \lp (p_u^1)^2 + (p_u^2)^2 \rp^{\fr{1}{4}} \sqrt{2 \lp 1 + p_u^3 \rp } }
\lb \begin{array}{c}
1 + p_u^3 \\
p_u^1 + i \, p_u^2
\end {array} \rb
\s\;
\ch_- = \fr{\sqrt{p_u^1 - i \, p_u^2}}{ \lp (p_u^1)^2 + (p_u^2)^2 \rp^{\fr{1}{4}} \sqrt{2 \lp 1 - p_u^3 \rp } }
\lb \begin{array}{c}
1 - p_u^3 \\
- p_u^1 - i \, p_u^2
\end {array} \rb
$$
These helicity states, with these chosen phases, satisfy [[helicity identities]],
$$
\ch_{\pm}^* = \pm \ep \, \ch_\mp
\s \s
\ch_{\pm}(-p) = \pm i \, \ch_{\mp}(p)
$$
using the [[skew]].
There are other possible phase choices for $\ch_\pm$, but they result in less nice relationships under conjugation and momentum reversal. It is sometimes convenient to consider four other ''phase-related helicity eigenspinors'',
$$
\ch'{}_+^{\wedge/\vee} = (1 + p_u) \ch^{\wedge/\vee}
\s \;\;\;\;\;
\ch'{}_-^{\wedge/\vee} = (1 - p_u) \ch^{\wedge/\vee}
$$
related to $\ch_\pm$ by normalization and phases. Explicitly, the normalized versions of these are
$$
\ba{lcl}
\ch_+^\wedge =
\lb \ba{c} \cos{\fr{\th}{2}} \\ e^{i \ph} \sin{\fr{\th}{2}} \ea \rb =
\fr{1}{\sqrt{2(1 + p_u^3)}} \lb \ba{c} 1 + p_u^3 \\ p_u^1 + i \, p_u^2 \ea \rb
& \s \s &
\ch_-^\wedge =
\lb \ba{c} \sin{\fr{\th}{2}} \\ -e^{i \ph} \cos{\fr{\th}{2}} \ea \rb =
\fr{1}{\sqrt{2(1 - p_u^3)}} \lb \ba{c} 1 - p_u^3 \\ -p_u^1 - i \, p_u^2 \ea \rb
\\
\ch_+^\vee =
\lb \ba{c} e^{-i \ph} \cos{\fr{\th}{2}} \\ \sin{\fr{\th}{2}} \ea \rb =
\fr{1}{\sqrt{2(1 - p_u^3)}} \lb \ba{c} p_u^1 - i \, p_u^2 \\ 1 - p_u^3 \ea \rb
& \s \s &
\ch_-^\vee =
\lb \ba{c} -e^{-i \ph} \sin{\fr{\th}{2}} \\ \cos{\fr{\th}{2}} \ea \rb =
\fr{1}{\sqrt{2(1 + p_u^3)}} \lb \ba{c} -p_u^1 + i \, p_u^2 \\ 1 + p_u^3 \ea \rb
\ea
$$
which are interrelated by
$$
\ch_{\pm}^\wedge{}^* = - \ep \, \ch_\mp^\vee
\s \s
\ch_{\pm}^\vee{}^* = + \ep \, \ch_\mp^\wedge
\s \s
\ch_{\pm}^\wedge (-p) = \ch_{\mp}^\wedge (p)
\s \s
\ch_{\pm}^\vee (-p) = \ch_{\mp}^\vee (p)
$$
The ''holonomy'' is the [[path holonomy]], $U$, for an arbitrary closed path on the base manifold of a [[fiber bundle]]. It may be written heuristically as
$$
U = Pe^{-\oint \f{A}}
$$
in which the [[connection]] is integrated all the way around the path. (The only real meaning of this expression is that it is a solution for the path holonomy at the end point (which equals the initial point) of the path.)
It is enlightening to calculate the approximate holonomy for a small, square-ish path. Such a path may be specified by choosing two orthonormal vectors, $\ve{u}$ and $\ve{v}$, at a point $x_{0}$ and making a closed path by going $\va$ in the $\ve{u}$ direction, then $\va$ along $\ve{v}$, $\varepsilon$ along $-\ve{u}$, then $\va$ along $-\ve{v}$ back to $x_{0}$. These four path segments, each parameterized by $0 \leq t \leq \va$, are given by
$$
\va_{1}^{i}=tu^{i}, \quad
\va_{2}^{i}=\va u^{i}+tv^{i}, \quad
\va_{3}^{i}=\va u^{i}+\va v^{i}-tu^{i},\quad
\va_{4}^{i}=\va v^{i}-tv^{i}
$$
and produce an anti-symmetric [[second order path dependence|path holonomy]],
$$
\va^{ij} = \int_{0}^{\va}\f{dt}\,\fr{d\va_{1}^{i}}{dt}\va_{1}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{2}^{i}}{dt}\va_{2}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{3}^{i}}{dt}\va_{3}^{j}
+\int_{0}^{\va}\f{dt}\,\fr{d\va_{4}^{i}}{dt}\va_{4}^{j}
=\va^{2} \lp v^{i}u^{j}-v^{j}u^{i} \rp
$$
implying a [[loop|vector-form algebra]] described by a tangent 2-vector,
$$
\vv{L}=\ha L^{ij}\ve{\pa_i}\ve{\pa_j}
=\ha \va^{ij}\ve{\pa_i}\ve{\pa_j}
=\va^{2}v^{i}u^{j}\ve{\pa_i}\ve{\pa_j}
=\va^{2}\ve{v}\,\ve{u}
$$
The holonomy around this small loop is approximately the [[path holonomy]] to second order,
$$
U \simeq 1 + \va^{ij} \lb - \pa_{j} A_{i} + A_{i} A_{j}\rb
=1 + \ha \va^{ij} \lb \pa_i A_j - \pa_j A_i + 2 A_i \times A_j \rb
=1 + \ha \va^{ij} F_{ij}
=1 - \vv{L} \ff{F}
$$
with the (defining) appearance of the [[curvature]],
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A} = \f{dx^i} \f{dx^j} \lp \pa_i A_j + A_i \times A_j \rp = \ha \f{dx^i} \f{dx^j} F_{ij}
$$
The contraction of the loop with the curvature, $\vv{L} \ff{F}$, is a nice example of [[vector-form algebra]]. Any fiber element, $C$, parallel transported around a small loop, $\vv{L}$, is transformed to
$$
C \mapsto C' = UC \simeq C - \vv{L} \ff{F} C
$$
to first order in loop area, $\va^2$. This provides a nice alternative definition of curvature in terms of parallel transport around small closed paths.
A ''homogeneous space'' (a.k.a. //''coset space''//, //''quotient space''//, or //''Klein geometry''//), $S=G/H$, is both a left [[coset]] and [[manifold]] formed by modding a [[Lie group geometry]], $G$, by a [[subgroup]], $H$. The points, $x \in S$, of a homogeneous space are specified by their coset representatives, $r(x) \in G$, up to right action by $H$,
$$
x \sim \lb r(x) \rb = \lb r(x) \, h(y) \rb = r(x) \, H = \left\{ r(x) \, h(y) : \forall \, h(y) \in H \right\}
$$
The homogeneous space is the base space, $M = S = G/H$, of a [[principal bundle]] with total space $E=G$ and $F=H$ as the structure group and fiber. The defining map is $\pi : g \mapsto [g]$ and the choice of ''coset representative section'' (//''homogeneous reference section''// or //''transversal''//), $r : S \rightarrow G$ (a function of points on the base space, valued in the total space), serves as a [[reference section|Ehresmann principal bundle connection]] and provides the [[local trivialization|fiber bundle]], $\ph : (x,h) \mapsto r(x) \, h \in G$. A homogeneous space has a natural ''zero point'' corresponding to the equivalence class of the identity in $G$, so the coset representative section and coordinates are chosen so $r(0) = 1$. The [[Maurer-Cartan form]], $\f{\cal I} = g^- \f{d} g$, over $G$ [[pulls back|pullback]] along the reference section, $g=r(x)$, to give the ''Maurer-Cartan frame'' over the homogeneous space,
$$
\f{I}(x) = r^* \f{\cal I} = r^- \f{d} r \in \f{\mathfrak{g}}
$$
which leads to a description of the [[homogeneous space geometry]] over $S$ or to the [[Ehresmann homogeneous space geometry]] over $G$. Note that a homogeneous space, $S=G/H$, is itself a Lie group if $H$ is a [[normal subgroup]] of $G$.
A [[principal bundle]] gauge transformation corresponds to a transformation of a reference section, $r(x) \in G$ of a [[homogeneous space]],
$$
r'(x) = r(x) \, h(x)
$$
by $h(x) \in H$. This transformation doesn't move the points of the homogeneous space, since $[r'(x)] = [r(x) \, h(x)] = [r(x)]$, but just moves the section up or down the fibers at those points by a [[diffeomorphism]], $\ph(x,y)=(x,y_\ph(x,y))$, in accordance with the point of view of the [[Ehresmann principal bundle gauge transformation]]. The [[Maurer-Cartan frame|homogeneous space]] over $S=G/H$ transforms to
$$
\f{I'}(x) = \lp r h \rp^* \f{\cal I} = h^- r^- \f{d} \lp r h\rp = h^- \f{I} h + h^- \f{d} h = \f{e'_S} + \f{A'_S}
$$
corresponding to the [[homogeneous space frame|homogeneous space geometry]], $\f{e_S} = \f{e_S^A} K_A$, and [[homogeneous H-connection|homogeneous space geometry]], $\f{A_S} = \f{A_S^P} H_P$, changing to
\begin{eqnarray}
\f{e'_S} &=& h^- \f{e_S} h \\
\f{A'_S} &=& h^- \f{A_S} h + h^- \f{d} h
\end{eqnarray}
since $H$ is taken to be [[reductive]] in $G$, which implies $A_{h^-} {\mathfrak{g}/\mathfrak{h}} \in {\mathfrak{g}/\mathfrak{h}}$, $A_{h^-} {\mathfrak{h}} \in {\mathfrak{h}}$, and $h^- \pa_a h \in {\mathfrak{h}}$. The gauge transformation of the H-connection is familiar as the [[gauge transformation]] of a principal bundle connection, and the transformation of the homogeneous space frame, $\f{{e'}_S^A} = \lp K^A , h^- K_B h \rp \f{e_S^B} = L^A{}_B \f{e_S^B}$, is a [[Lorentz rotation]], familiar as the co[[tangent bundle gauge transformation]].
For an infinitesimal gauge transformation, $h \simeq 1 + h^P H_P = 1 + H$, these transformations are
\begin{eqnarray}
\f{e'_S} &\simeq& (1 - H) \f{e_S^B} K_B (1 + H) \simeq \f{e_S} + \lb \f{e_S} , H \rb \\
\f{A'_S} &\simeq& (1 - H) \f{A_S^R} H_R (1 + H) + (1 - H) \f{d} (1 + H) \simeq \f{A_S} + \f{d} H + \lb \f{A_S} , H \rb
\end{eqnarray}
to first order in the gauge parameters, $h^P$.
A more general possible gauge transformation is
$$
r'(x) = r(x) \, g(x)
$$
by $g(x) \in G$. This transformation may move the points of the homogeneous space, with diffeomorphism $\ph(x,y)=(x_\ph(x,y),y_\ph(x,y))$. The Maurer-Cartan frame over $S$ transforms to
$$
\f{I'}(x) = g^- \f{I} g + g^- \f{d} g = \f{e'_S} + \f{A'_S}
$$
which possibly mixes the homogeneous space frame and H-connection. For an infinitesimal gauge transformation,
$$
g \simeq 1 + g^A K_A + g^P H_P = 1 + K + H
$$
this transformation is
\begin{eqnarray}
\f{I'} &\simeq& \lp 1 - K - H \rp \lp \f{e_S^B} K_B + \f{A_S^Q} H_Q \rp \lp 1 + K + H \rp + \f{d} K + \f{d} H \\
&\simeq& \f{I} + \lb \f{e_S} , K \rb + \lb \f{A_S} , K \rb + \lb \f{e_S} , H \rb + \lb \f{A_S} , H \rb + \f{d} K + \f{d} H \\
&=& \f{I} - g^A \f{e_S^B} \lp C_{AB}{}^C K_C + C_{AB}{}^P H_P \rp - g^A \f{A_S^Q} C_{AQ}{}^C K_C - g^P \f{e_S^B} C_{PB}{}^C K_C - g^P \f{A_S^Q} C_{PQ}{}^R H_R + \f{d} g^A K_A + \f{d} g^P H_P
\end{eqnarray}
giving
\begin{eqnarray}
\f{e'_S} &\simeq& \f{e_S} + \lp - g^A \f{e_S^B} C_{AB}{}^C - g^A \f{A_S^Q} C_{AQ}{}^C - g^P \f{e_S^B} C_{PB}{}^C + \f{d} g^C \rp K_C \\
&=& \f{e_S} + \f{d} K + \lb \f{A_S} , K \rb + \lb \f{e_S} , H \rb + \lb \f{e_S} , K \rb_K \\
\f{A'_S} &\simeq& \f{A_S} + \lp - g^A \f{e_S^B} C_{AB}{}^R - g^P \f{A_S^Q} C_{PQ}{}^R + \f{d} g^R \rp H_R \\
&=& \f{A_S} + \f{d} H + \lb \f{A_S} , H \rb + \lb \f{e_S} , K \rb_H
\end{eqnarray}
to first order in the gauge parameters, $g^I$. There is, though, a potential problem with this type of gauge transformation: If we choose $g(x)=r^-(x)$ then the section transforms to $r'(x) = r(x) \, g(x) = r \, r^- = 1$, which is no longer a section since $\pi \circ r'$ is not the identity map on $S$. It is still interesting to consider though, as this choice results in $\f{I'} = r'^- \f{d} r' = 0$, but such a gauge transformation may not be allowed as it may not come from a diffeomorphism, unless the space is contractable.
A [[homogeneous space]], $S=G/H$, built from a [[Lie group]], $G$, and a [[subgroup]], $H$, inherits a geometry from the [[Lie group tangent bundle geometry]] of $G$ and how the $H$ subgroup -- a [[submanifold]] of $G$ -- sits in $G$. The flows on the [[Lie group manifold|Lie group geometry]] are described by the [[Lie algebra]] generators, $T_I \in \mathfrak{g}$, with [[index|indices]] $I$ running from $1$ to $n_G$ -- the dimension of $G$. These generators are presumed to be rotated so the [[Killing form]],
$$
\lp T_I, T_J \rp = g_{IJ} = C_{IK}{}^L C_{JL}{}^K
$$
is diagonal and the structure constants satisfy $C_{IKL} = - C_{ILK}$. The subgroup, $H$, is taken to be [[reductive]] in $G$, with generators $H_P = T_P \in \mathfrak{h}$, (with $P$-series indices running from $1$ to $n_H$). The $n_S = (n_G - n_H)$ remaining generators are the ''coset generators'', $K_A = T_A$, spanning the [[vector space]], $\mathfrak{g}/\mathfrak{h}$. We take $z^i$ to be coordinates for $G$, $y^p$ to be coordinates for $H$, and $x^a$ to be coordinates for $S$. A good choice for [[coset representative|homogeneous space]]s is the [[exponentiation]] of the coset generators,
$$
r(x) = e^{x^a K_a} \in G
$$
A coset representative is not necessarily a subgroup of $G$, but it is a section and a [[submanifold]]. If $G$ has a matrix representation, the exponentiation above gives the explicit form of $r(x)$ as a matrix -- a good way to think of it. Whatever the choice of coset representative, the [[Maurer-Cartan form]] over $G$ [[pulls back|pullback]] to give the [[Maurer-Cartan frame|homogeneous space]] over $S$,
$$
\f{I}(x) = \f{e_S} + \f{A_S} = r^- \f{d} r
$$
which splits into the ''homogeneous space frame'', $\f{e_S} = \f{e_S^A} K_A$, and the ''homogeneous H-connection'', $\f{A_S} = \f{A_S^P} H_P$. The coefficients are determined by the Lie group geometry and may be computed explicitly using the Killing form,
\begin{eqnarray}
\f{e_S^A} &=& \lp K^A, \f{I} \rp \\
\f{A_S^P} &=& \lp H^P, \f{I} \rp
\end{eqnarray}
These homogeneous space frame 1-forms may be used as the [[frame]] 1-forms for the [[tangent bundle]] over $S$, with resulting ''homogeneous space [[metric]]'',
$$
\lp \ve{u}, \ve{v} \rp = \lp \ve{u} \f{e_S}, \ve{v} \f{e_S} \rp = u^A v^B \lp K_A, K_B \rp = u^A v^B g_{AB} = u^a v^b \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB} = u^a v^b g_{ab}
$$
Note that this metric, $g_{ab} = \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB}$, over $S$ does NOT correspond to the natural [[submanifold geometry]] of the coset representative in $G$ -- that metric would be:
$$
g'_{ab} = \lp e^S_a \rp^A \lp e^S_b \rp^B g_{AB} + \lp A^S_a \rp^P \lp A^S_b \rp^Q g_{PQ}
$$
Rather, the homogeneous space metric is independent of the choice of coset representative, and thus necessarily independent of the homogeneous H-connection. The homogeneous H-connection, $\f{A_S}(x)$, is a particular (constrained to be part of the Maurer-Cartan frame) [[principal bundle]] connection when the homogeneous space is viewed as the base of a principal $H$-bundle.
With the homogeneous space frame in hand it is straightforward to calculate the [[homogeneous space tangent bundle geometry]] based on a choice of [[torsion]]; and to calculate the [[homogeneous space geometry symmetries]] -- the [[Killing vector]]s of the homogeneous space frame.
Refs:
*Roberto Camporesi
**http://calvino.polito.it/~camporesi/
*Leonardo Castellani
**http://www.mfn.unipmn.it/%7ecastella/
**[[On G/H geometry and its use in M-theory compactifications|papers/9912277.pdf]]
**[[Symmetries of Coset Spaces and Kaluza-Klein Supergravity|papers/Symmetries of Coset Spaces and Kaluza-Klein Supergravity.pdf]]
*[[Super coset space geometry|papers/0610039.pdf]]
*Excellent new paper:
**[[Heat Kernel on Homogeneous Bundles over Symmetric Spaces|papers/0701489.pdf]]
Not surprisingly, a [[homogeneous space geometry]] has a large symmetry group, described by [[Killing vector]] fields, $\ve{\xi}(x)$, over $S=G/H$. Most of the symmetries, $\ve{\xi_I}$, correspond to the [[left action|group]] of the [[Lie group]], $G$, through its [[Lie algebra]] generators, $T_I$. However, more symmetries, $\ve{\xi'_X}$, come from the right action of a different group, though some of these may correspond to some of the $\ve{\xi_I}$.
An element $g \in G$, acts from the left on the coset element, $[r(x)] \in S$, to give another coset element, $[g \, r(x)] = [r(x')]$. This implies that the left action of $g$ on the coset representative, $r(x)$, is
$$
g \, r(x) = r(x') h
$$
for some $h \in H$. If the group element is approximated near the identity by $g \simeq (1 + \ep^I T_I)$, with a corresponding [[flow]] on the coset manifold of $x' \simeq x + \ep^I \xi_I$, and $h \simeq (1 + \ep^I h_I^P(x) H_P)$, the above equation,
$$
(1 + \ep^I T_I) \, r \simeq (r + \ep^I \ve{\xi_I} \f{d} r) (1 + \ep^I h_I^P H_P)
$$
gives, to first order in $\ep^I$,
$$
T_I r = \ve{\xi_I} \f{d} r + h_I^P r H_P
$$
Multiplying on the left by $r^-$ and using the Maurer-Cartan frame over $S$,
$$
\f{I} = r^- \f{d} r = \f{e_S^A} K_A + \f{A_S^P} H_P
$$
gives
$$
r^- T_I r = \lp \xi_I \rp^a \lp e^S_a \rp^A K_A + \lp \xi_I \rp^a \lp A^S_a \rp^P H_P + h_I^P H_P
$$
plugging this into the $\mathfrak{g}$ [[Killing form]] with the generator duals of the $K_A \in \mathfrak{g}/\mathfrak{h}$ and $H_P \in \mathfrak{h}$ gives explicit expressions for the coefficients of the ''left Killing vector fields on the homogeneous space geometry'',
$$
\lp \xi_I \rp^a(x) = \lp e^S_A \rp^a \lp K^A , r^- T_I r \rp
$$
and the ''H-compensator'',
$$
h_I{}^P(x) = -\lp \xi_I \rp^a \lp A^S_a \rp^P + \lp H^P , r^- T_I r \rp
$$
These $n$ Killing vector fields comprise the left action flows of $G$ on $S$. They are demonstrably Killing. Using the definition of the [[Lie derivative]],
$$
{\cal L}_{\ve{\xi_I}} \f{e_S^A} = \ve{\xi_I} \lp \f{d} \f{e_S^A} \rp +\f{d} \lp \ve{\xi_I} \f{e_S^A} \rp
$$
and an expression from the [[homogeneous space tangent bundle geometry]],
\begin{eqnarray}
\ve{\xi_I} \lp \f{d} \f{e_S^A} \rp &=& \ve{\xi_I} \lp - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A - \f{A_S^P} \f{e_S^B} C_{PB}{}^A \rp \\
&=& - \lp \xi_I \rp^C \f{e_S^B} C_{CB}{}^A - \lp \ve{\xi_I} \f{A_S^P} \rp \f{e_S^B} C_{PB}{}^A + \f{A_S^P} \lp \xi_I \rp^B C_{PB}{}^A
\end{eqnarray}
combined with the coset representative relation, a Killing form identity, and the commutation relations for the [[reductive]] coset,
\begin{eqnarray}
\f{d} \lp \ve{\xi_I} \f{e_S^A} \rp &=& \f{d} \lp K^A , r^- T_I r \rp = \lp K^A , \lb r^- T_I r, \f{I} \rb \rp = - \lp \lb K^A , \f{I} \rb, r^- T_I r \rp \\
&=& - \f{e_S^B} C^A{}_B{}^P \lp H_P, r^- T_I r \rp - \lp \f{e_S^B} C^A{}_B{}^C + \f{A_S^P} C^A{}_P{}^C \rp \lp K_C, r^- T_I r \rp \\
&=& - \f{e_S^B} C^A{}_B{}^P \lp h_{IP} + \lp \xi_I \rp^a \lp A^S_a \rp_P \rp - \lp \f{e_S^B} C^A{}_B{}^C + \f{A_S^P} C^A{}_P{}^C \rp \lp \xi_I \rp_C
\end{eqnarray}
gives, after happy cancellation,
$$
{\cal L}_{\ve{\xi_I}} \f{e_S^A} = - \f{e_S^B} h_I{}^P C_P{}^A{}_B
$$
with nice Killing rotation coefficients, $\lp B_I \rp_B{}^A = h_I{}^P C_{PB}{}^A$.
The right action of $g \in G$ on a coset element, $[r] = r H$, only makes sense if $g$ "gets past" the $H$ so that $R_g \, [r] = r H g = r g H = [r g]$, which is true iff $g$ is in the [[normalizer]], $g \in N_G(H) = N(H)$. Of course, if $g \in H$ its right action has no effect on $[r]$, so the ''right action'' group of symmetries is the group $N(H)/H$. The right Killing vectors, $\ve{\xi'_X}$, and H-compensators, $h'$, corresponding to a $g \in N(H)/H$ come from
$$
\begin{array}{rcl}
r \, g &=& r(x') h' \\
r \, (1 + \ep^X K_X) &\simeq& (r + \ep^X \ve{\xi'_X} \f{d} r)(1 + \ep^X {h'}_X^P H_P) \\
K_X &=& \lp \xi'_X \rp^a \lp e^S_a \rp^A K_A + \lp \xi'_X \rp^a \lp A^S_a \rp^P H_P + {h'}_X^P H_P
\end{array}
$$
(in which $K_X \in {\rm Lie}(N(H)/H) \subset {\rm Lie}(G/H) = \mathfrak{g}/\mathfrak{h}$ are a reduced linear combination of $K_A$) and are
\begin{eqnarray}
\lp \xi'_X \rp^a(x) &=& \lp e^S_X \rp^a \\
h'_X{}^P(x) &=& -\lp \xi'_X \rp^a \lp A^S_a \rp^P + \de_I^P
\end{eqnarray}
Since the right Killing vectors are a reduced linear combination of the orthonormal basis vectors, the Killing equation is
\begin{eqnarray}
{\cal L}_{\ve{\xi'_X}} \f{e_S^A} &=& {\cal L}_{\ve{e_X}} \f{e_S^A} = \ve{e_X} \lp \f{d} \f{e_S^A} \rp \\
&=& \ve{e^S_X} \lp - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A - \f{A_S^P} \f{e_S^B} C_{PB}{}^A \rp \\
&=& \f{e_S^B} \lp -C_{XB}{}^A - \lp A^S_X \rp^P C_{PB}{}^A + \lp A^S_B \rp^P C_{PX}{}^A \rp
\end{eqnarray}
showing that $\ve{\xi'_X}$ is Killing, with Killing rotation coefficients $\lp B'_X \rp_B{}^A = \lp -C_{XB}{}^A - \lp A_X \rp^P C_{PB}{}^A \rp$, if the last term above vanishes -- which it does since the reductivity of $H$ in $G$ implies $\lb {\rm Lie}(H), {\rm Lie}(G/H) \rb \subset {\rm Lie}(G/H) = \mathfrak{g}/\mathfrak{h}$, and that $H$ is normal in $N(H)/H$ implies $\lb {\rm Lie}(H), {\rm Lie}(N(H)/H) \rb \subset {\rm Lie}(H)$, and together these imply $\lb {\rm Lie}(H), {\rm Lie}(N(H)/H) \rb = 0$, which gives $ C_{PX}{}^A = 0$.
After this analysis one might think the full symmetry group of the homogeneous space geometry is $G \times N(H)/H$, and that is the biggest it could be, but it might be smaller since some of the right Killing vectors, $\ve{\xi'_X}$, may not be independent of the left Killing vectors, $\ve{\xi_I}$.
An analysis of [[reductive]] [[homogeneous space geometry]] resulted in the Maurer-Cartan frame,
$$
\f{I} = \f{e_S} + \f{A_S} = r^- \f{d} r
$$
over the [[homogeneous space]], $S=G/H$, which split into the homogeneous space frame, $\f{e_S} = \f{dx^a} \lp e^S_a \rp^A K_A$, and homogeneous H-connection, $\f{A_S} = \f{dx^a} \lp A^S_a \rp^P H_P$. As the [[pullback]] of the [[Maurer-Cartan form]] [[curvature]], the ''homogeneous space curvature'' vanishes,
$$
0 = \ff{F}(x) = \f{d} \f{I} + \ha \lb \f{I}, \f{I} \rb = \f{d} \f{e_S} + \f{d} \f{A_S} + \ha \lb \f{e_S}, \f{e_S} \rb + \lb \f{A_S}, \f{e_S} \rb + \ha \lb \f{A_S}, \f{A_S} \rb
$$
which, via the [[reductive]] homogeneous space commutation relations, gives
\begin{eqnarray}
0 &=& \f{d} \f{e_S^A} + \f{A_S^P} \f{e_S^B} C_{PB}{}^A + \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A \\
0 &=& \f{d} \f{A_S^P} + \ha \f{A_S^Q} \f{A_S^R} C_{QR}{}^P + \ha \f{e_S^C} \f{e_S^D} C_{CD}{}^P
\end{eqnarray}
This implies the ''[[principal bundle]] curvature of the homogeneous H-connection'' is
$$
\ff{F_H} = \ff{F_H^P} H_P = \f{d} \f{A_S} + \ha \lb \f{A_S}, \f{A_S} \rb = - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P H_P
$$
But what [[tangent bundle connection]], $\f{w}^A{}_B$, do we choose to build over $S$ when treating $\f{e_S^A}$ as the [[frame]] 1-forms? I.e., what do we choose for the [[torsion]],
$$
\ff{T^A} = \f{d} \f{e_S^A} + \f{\om}^A{}_B \f{e_S^B} = ?
$$
There are two decent looking choices:
If we choose a torsion of $\ff{T^A} = - \ha \f{e_S^C} \f{e_S^B} C_{CB}{}^A$, the ''torsionful homogeneous space connection'' is $\f{w}^A{}_B = \f{A_S^P} C_{PB}{}^A$, which produces a nice [[Riemann curvature]] of
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \f{d} \f{A_S^P} C_{PB}{}^A + \f{A_S^Q} C_{QC}{}^A \f{A_S^R} C_{RB}{}^C \\
&=& - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P C_{PB}{}^A + \f{A_S^Q} \f{A_S^R} \lp - \ha C_{QR}{}^P C_{PB}{}^A + C_{QC}{}^A C_{RB}{}^C \rp \\
&=& - \ha \f{e_S^F} \f{e_S^E} C_{FE}{}^P C_{PB}{}^A
\end{eqnarray}
by the [[Jacobi identity|Lie algebra]]. This is nice because it matches the H-connection curvature,
$$
\ff{R}^A{}_B = \ff{F_H^P} C_{PB}{}^A
$$
But we often want torsion to vanish, $\ff{T^A} = 0$. This choice results in the ''torsionless homogeneous space connection'',
$$
\f{w}^A{}_B = \ha \f{e_S^C} C_{CB}{}^A + \f{A_S^P} C_{PB}{}^A
$$
which produces the Riemann curvature,
\begin{eqnarray}
\ff{R}^A{}_B &=& \f{d} \f{w}^A{}_B + \f{w}^A{}_C \f{w}^C{}_B \\
&=& \ha \f{d} \f{e_S^C} C_{CB}{}^A + \f{d} \f{A_S^P} C_{PB}{}^A + \lp \ha \f{e_S^F} C_{FC}{}^A + \f{A_S^Q} C_{QC}{}^A \rp \lp \ha \f{e_S^E} C_{EB}{}^C + \f{A_S^R} C_{RB}{}^C \rp \\
&=& \ha \f{e_S^F} \f{e_S^E} \lp - \fr{1}{4} C_{FE}{}^C C_{CB}{}^A - C_{FE}{}^P C_{PB}{}^A \rp \\
&=& \ha \f{e_S^F} \f{e_S^E} R_{FE}{}^A{}_B
\end{eqnarray}
and a [[Ricci curvature]] of
\begin{eqnarray}
\f{R}{}_B &=& \ve{e^S_A} \ff{R}^A{}_B = \f{e_S^E} R_{AE}{}^A{}_B \\
&=& \f{e_S^E} \lp - \fr{1}{4} C_{AE}{}^C C_{CB}{}^A - C_{AE}{}^P C_{PB}{}^A \rp \\
&=& - \ha \f{e_S^E} \lp g_{EB} - \ha C_{EA}{}^C C_{BC}{}^A \rp
\end{eqnarray}
which gives a [[curvature scalar]] of
$$
R = \ve{e^S_B} \f{R}{}^B = - \ha \lp n_S - \ha C_{BA}{}^C C^B{}_C{}^A \rp
$$
For a [[symmetric space]], $C_{AB}{}^C=0$, there is no choice -- the torsion vanishes and the ''symmetric space connection'' is $\f{w}^A{}_B = \f{A_S^P} C_{PB}{}^A$.
The analogue of [[spherical coordinates]] in $n$ dimensions are given in terms of $r$ and the $(n-1)$ ''angular coordinates'', $a^w$, by
\begin{eqnarray}
x^1 &=& r \cos(a^1) \\
x^2 &=& r \sin(a^1) \cos(a^2) \\
x^3 &=& r \sin(a^1) \sin(a^2) \cos(a^3) \\
& \vdots \\
x^{n-1} &=& r \sin(a^1) \dots \sin(a^{n-2}) \cos(a^{n-1}) \\
x^n &=& r \sin(a^1) \dots \sin(a^{n-2}) \sin(a^{n-1})
\end{eqnarray}
Ref:
*http://en.wikipedia.org/wiki/Hypersphere#Hyperspherical_coordinates
An ''antisymmetric index bracket'' is used to produce an indexed quantity (tensor) that is antisymmetric in its indices. For example, for a tensor with two [[indices]], $a_{ij}$, the antisymmetric part of this tensor is
\[ a_{\lb ij \rb} = \ha \lp a_{ij} - a_{ji} \rp \]
A tensor is antisymmetric in its indices iff it equals its corresponding antisymmetric part, $a_{ij} = a_{\lb ij \rb}$. Such an antisymmetric tensor changes sign under the interchange of any two neighboring indices, $a_{ij}=-a_{ji}$.
For a tensor with three indices, $b_{ijk}$, the antisymmeterized part is
\[ b_{\lb ijk \rb} = \fr{1}{3!} \lp b_{ijk}+b_{jki}+b_{kij}-b_{jik}-b_{ikj}-b_{kji} \rp \]
The ''antisymmeterized index bracket'' is similar in operation to the [[antisymmetric bracket]].
A ''symmetric index bracket'' is used to produce an indexed quantity (tensor) that is symmetric in its indices. For example, for a tensor with two indices, $a_{ij}$, the symmetric part of this tensor is
\[ a_{\lp ij \rp} = \ha \lp a_{ij} + a_{ji} \rp \]
A tensor is symmetric in its indices iff it equals its corresponding symmetric part, $a_{ij} = a_{\lp ij \rp}$. Such a symmetric tensor is invariant under the interchange of any two neighboring indices, $a_{ij}=a_{ji}$.
For a tensor with three indices, $b_{ijk}$, the symmeterized part is
\[ b_{\lp ijk \rp} = \fr{1}{3!} \lp b_{ijk}+b_{jki}+b_{kij}+b_{jik}+b_{ikj}+b_{kji} \rp \]
Unless stated otherwise, repeated indices in expressions are summed — for example,
\[ \ve{v} \f{f} = v^i f_i = \sum_{i=0}^{n-1} v^i f_i \]
This, Einstein's summation convention, loses the information on the range of the sum. To remedy this deficiency, indices from different parts of the alphabets are taken to range over different integers corresponding to the spaces they coordinatize or label:
| !Latin index | !Greek index | !Range over |!For |
| $i,j,k,l,m,n$ | $\al,\be,\ga,\de,\ep,\up$ | $0 \dots (n-1) \, {\rm or} \, 1 \dots n$ |all $n$, or any appropriate subset |
| $a,b,c,d$ | $\mu,\nu,\ka,\la$ | $1,2,3, (4 \, {\rm or} \, 0)$ |[[spacetime]] |
| $e,f,g,h$ | $\va,\ze,\ta,\io$ | $1,2,3$ |space |
| $w,x,y,z$ | $\pi,\rh,\si,\xi$ | $1 \dots (n-1)$ |spatial |
| $p,q,r,u$ | $\th,\ph,\ch,\ps$ | $4 \dots (n-1) \, {\rm or} \, 5 \dots n$ |Kaluza-Klein or fiber coordinates — i.e. other than spacetime |
| $A,B,C$ | | $1 \dots$ ? |Lie algebra elements or general indices |
Lower case Latin indices are for [[coordinates|manifold]], Greek indices are [[Clifford algebra]] basis element labels, and upper case Latin indices are [[Lie algebra]] generator element labels.
Since points, $x \in V$, of a [[representation space]], $V$, are acted on by Lie group elements, $g \in G$, this induces an ''infinite-dimensional representation'' of the Lie group acting on the infinite-dimensional representation space of functions of $x$,
$$
(g \cdot f)(x) = f(g^- x)
$$
This relation gives, for example, the correspondence between rotations in space, spherical harmonics, and angular momenta as partial differential operators. The [[inverse]] is needed so that
$$
(g_1 g_2 \cdot f)(x) = (g_1 \cdot (g_2 \cdot f))(x) = f( g_2^- g_1^- x) = f((g_1 g_2)^- x)
$$
If a classical field, $\Ps$, transforms under some [[Lie group]] element, $g$, including a possible [[Lorentz transformation]], as
$$
\Ps(x) \mapsto \Ps'(x) = O(g) \Ps(g^- x)
$$
in which $O$ is a linear operator acting on $\Ps$, then the corresponding [[quantum field]] operator, $\hat{\Ps}$, transforms under the corresponding ''infinite-dimensional unitary representation'',
$$
\hat{\Ps}(x) \mapsto \hat{\Ps'}(x) = \hat{U}(g) \hat{\Ps}(x) \hat{U}(g)^- = O(g) \hat{\Ps}(g^- x)
$$
in which the [[unitary representation]] operator, $\hat{U}(g)$, which acts on an [[infinite-dimensional representation]] space, [[Fock space]], is determined above. Note that, acting on classical fields, a group element, $g$, typically acts via linear action on field components along with [[diffeomorphism]]s, whereas the corresponding unitary operators act only on the creation and annihilation operators of the corresponding quantum field. An [[antiunitary]] representation operator, $\hat{U}'(g) = \hat{U}(g) \hat{K}$, acts on creation and annihilation operators and as complex conjugation.
We can consider transformations by group elements infinitesimally close to the identity, $g=1+\ep G$, in which $G$ is a [[Lie algebra]] element. With $U(1+\ep G) \simeq 1+\ep \hat{O}_G$, $O(1+\ep G) \simeq 1 + \ep O_G$, and $g^- x^\mu \simeq x^\mu - \ep (G^\mu + G^\mu{}_\nu x^\nu)$, the relationship between the transformations of the field operators and field (above) gives the relationship between infinite dimensional unitary Lie algebra operator brackets and linear operators on the field,
$$
\lb \hat{O}_G, \hat{\Ps}(x) \rb = O_G \hat{\Ps}(x) - (G^\mu + G^\mu{}_\nu x^\nu) \pa_\mu \hat{\Ps}(x)
$$
with $\hat{O}_G$ usually anti-[[Hermitian]].
For understanding the [[weight structure|Lie algebra structure]] of an [[infinite-dimensional unitary representation]] of a noncompact [[Lie group]], we choose a maximal compact subgroup of that Lie group, and a Cartan subalgebra of the compact subgroup's subalgebra.
We can construct an ''infinite-dimensional unitary representation of sl(2)'', by considering the isomorphism of [[sl(2)]] to so(2,1) and the action of the corresponding angular momentum and boost operators on complex functions of three real dimensions. Specifically, the basis generators are
$$
\begin{array}{ccc}
\hat{T}_1 = 2 \lp x_0 \pa_2 + x_2 \pa_0 \rp
&
\hat{T}_2 = 2 \lp x_1 \pa_2 - x_2 \pa_1 \rp
&
\hat{T}_3 = 2 \lp - x_0 \pa_1 - x_1 \pa_0 \rp
\end{array}
$$
which satisfy sl(2) generator brackets. The [[Hermitian form]] on the infinite-dimensional space of functions of three variables is
$$
\langle f | g \rangle = \int{\!\! d^3 \! x \, f^* g}
$$
Transformations corresponding to Sl(2) [[Lie group]] elements, $\hat{U}$, are [[unitary]], $\langle \hat{U} f | \hat{U} g \rangle = \langle f | g \rangle$. Near the identity the unitary operators are $\hat{U} f \simeq (1 + \ep^\pi \hat{T}_\pi) f$, which satisfy, for example,
$$
\begin{array}{rcl}
0 \!\!&\!\!=\!\!&\!\! \int{\! d^3 \! x \, \lp (\hat{T}_1 f)^* g + f^* (\hat{T}_1 g) \rp} \\
\!\!&\!\!=\!\!&\!\! \int{\! d^3 \! x \, \lp 2 g \lp x_0 \pa_2 - x_2 \pa_0 \rp f^* + 2 f^* \lp x_0 \pa_2 - x_2 \pa_0 \rp g \rp} \\
\!\!&\!\!=\!\!&\!\! \int{\! d^3 \! x \, \lp 2 x_0 \pa_2 \lp f^* g \rp - 2 x_2 \pa_0 \lp f^* g \rp \rp} \\
\end{array}
$$
for all $f$ and $g$ satisfying $f^* g |_\infty = 0$. Thus these generators are all anti-[[Hermitian]], $\hat{T}_\pi^\da = - \hat{T}_\pi$. We can simplify these generators in cylindrical-hyperbolic coordinates,
$$
\begin{array}{ccc}
x_0 = r \sinh{\ka}
&
x_1 = r \cosh{\ka} \cos{\ph}
&
x_2 = r \cosh{\ka} \sin{\ph}
\end{array}
$$
in which they become
$$
\begin{array}{ccc}
\hat{T}_1 = 2 \lp \tanh{\ka} \cos{\ph} \, \pa_\ph + 2 \sin{\ph} \, \pa_\ka \rp
&
\hat{T}_2 = 2 \, \pa_\ph
&
\hat{T}_3 = 2 \lp \tanh{\ka} \sin{\ph} \, \pa_\ph - 2 \cos{\ph} \, \pa_\ka \rp
\end{array}
$$
which we see do not depend on $r$. We can use these to construct eigenvectors of $\hat{\rm Ad}_{T_2}$,
$$
\hat{V}_\pm = \ha \lp \hat{T}_3 \mp i \hat{T}_1 \rp = - e^{\pm i \ph} \lp \pa_\ka \pm i \tanh{\ka} \, \pa_\ph \rp
= - e^{\pm i \ph} \lp (1-x^2) \,\pa_x \pm i x \, \pa_\ph \rp
$$
in which we've changed coordinates again by defining $x = \tanh{\ka}$. These satisfy the commutation relations
$$
\lb \hat{T}_2 , \hat{V}_\pm \rb = \pm 2 i \hat{V}_\pm \s \lb \hat{V}_- , \hat{V}_+ \rb = i \hat{T}_2
$$
The $\hat{V}_\pm$ act as raising and lowering operators on [[eigen]]functions of $\hat{T}_2$. If $w_m \sim e^{im\ph}$, then $\hat{T}_2 w_n = 2 i m w_n$, and
$$
\hat{T}_2 (\hat{V}_\pm w_n) = \hat{V}_\pm \hat{T}_2 w_n \pm 2 i \hat{V}_\pm w_n
= 2 i (m \pm 1 ) ( \hat{V}_\pm w_n)
$$
Nice basis eigenfunctions in the representation space are $w_{n,m} = x^n e^{i m \ph}$, for which
$$
\hat{T}_2 \, w_{n,m} = 2 i m \, w_{n,m}
\s
\hat{V}_\pm \, w_{n,m} = - n \, w_{n - 1 ,m \pm 1} + ( n \pm m) \, w_{n + 1 ,m \pm 1}
$$
A [[Casimir operator]] is
$$
\hat{\Om} = \hat{T}_1^2 - \hat{T}_2^2 + \hat{T}_3^2
= - \hat{T}_2^2 + 2 \lp \hat{V}_+ \hat{V}_- + \hat{V}_- \hat{V}_+ \rp
= 4 (x^2-1) \lp \pa_\ph^2 + (x^2-1) \, \pa_x^2 + x \, \pa_x \rp
$$
which acts on these basis functions... but not as its eigenfunctions, which are uglier. If we assume Casimir eigenfunctions, $\Om w_{\la,m} = \la^2 w_{\la,m}$, then we do have
$$
\hat{V}_\pm \hat{V}_\mp = \fr{1}{4} \lp \hat{\Om} + \hat{T}_2^2 \pm 2 i \hat{T}_2 \rp
$$
and so $\hat{V}_\pm \hat{V}_\mp w_{\la,m} = (\fr{1}{4} \la^2 - m(m \mp 1) ) w_{\la,m}$.
Another way of describing sl(2) Lie algebra elements is as $2 \times 2$ matrices acting on 2D coordinates, $\lb \begin{array}{c} x^1 \\ x^2 \end{array} \rb$. To understand an [[infinite-dimensional unitary representation of a noncompact Lie group]] we choose a compact Lie subalgebra element to span a [[Cartan subalgebra|Lie algebra structure]], $C = c T_2$, with resulting $V_\pm$ eigenvectors,
$$
T_2 = \left[\begin{array}{cc}
0 & -1\\
1 & 0\end{array}\right]
\s
V_\pm = \ha \lp T_3 \mp i T_1 \rp = \ha
\left[\begin{array}{cc}
1 & \mp i \\
\mp i & -1\end{array}\right]
$$
which satisfy
$$
\lb T_2, V_\pm \rb = \pm 2 i \, V_\pm
\s
\lb V_-, V_+ \rb = i \, T_2
$$
We construct the [[adjoint action|adjoint representation]] of the corresponding Lie algebra generators in an [[infinite-dimensional unitary representation]] from
$$
\lb \hat{O}_G, \hat{\Ps}(x) \rb = - G^i{}_j x^j \pa_i \hat{\Ps}(x)
$$
obtaining
$$
\hat{\rm Ad}_{T_2} = x^1 \pa_2 - x^2 \pa_1
\s
\hat{\rm Ad}_{V_\pm} = \ha \lp - x^1 \pa_1 \pm i x^1 \pa_2 + x^2 \pa_2 \pm i x^2 \pa_1 \rp
$$
This [[representation space]], of complex functions of two variables (minus the origin), is reducible. We can break it into irreducible representation spaces spanned by basis functions -- eigenfunctions of our Cartan subalgebra operator, $\hat{\rm Ad}_{T_2}$. These will be easier to find after switching to polar coordinates, $(x^1,x^2) = (r \cos(\th), r \sin(\th))$, in which
$$
\hat{\rm Ad}_{T_2} = \pa_\th
\s
\hat{\rm Ad}_{V_\pm} = \ha e^{\mp 2 i \th} \lp \pm i \pa_\th - r \pa_r \rp
$$
We then see that nice basis eigenfunctions are $w_{n,m} = r^n e^{i m \th}$, for which
$$
\hat{\rm Ad}_{T_2} w_{n,m} = i m \, w_{n,m}
\s
\hat{\rm Ad}_{V_\pm} \, w_{n,m} = \mp \fr{1}{2} m \, w_{n,m \mp 2} - \ha n \, w_{n+1,m \mp 2}
$$
A [[Casimir operator]] is
$$
\Om = T_1^2 - T_2^2 + T_3^2 + 1
\sim \hat{\rm Ad}_{T_2} \hat{\rm Ad}_{T_2} + 2 \lp \hat{\rm Ad}_{V_+} \hat{\rm Ad}_{V_-} + \hat{\rm Ad}_{V_-} \hat{\rm Ad}_{V_+} \rp + 1
= - 2 \pa_\th^2 + 3 r \pa_r + r^2 \pa_r^2 + 1
$$
which acts on these basis functions to give $\hat{\Om} \, w_{n,m} = \lp n(n+2) - 2 m^2 \rp w_{n,m}$.
Ref:
Borcherds textbook notes p185
The integral over a volume, $V$, of the [[exterior derivative]] of a [[differential form]] equals the integral of that form over the boundary, $\pa V$, of that volume,
$$
\int_V \f{d} \nf{F} = \int_{\pa V} \nf{F}
$$
This, ''Stoke's theorem'', may be used to evaluate integrals by finding an ''antiderivative'' of the integrand. For example,
$$
\int_{\lb 0,1 \rb} \f{dx} \, x = \lb \ha x^2 \rl_0^1 = \ha
$$
in which $F = \ha x^2$ is the antiderivative of $\f{d} F = \f{dx} \, x$, and $\pa V$ consists of the boundary points $0$ and $1$. The "integral" over two points is simply the ordered sum of the integrand evaluated at those points.
Stoke's theorem is a generalization of the ''fundamental theorem of calculus''.
//need to patch together simply connected regions for this definition to work. Betti number? DeRham chains?//
[[Donald Knuth|http://en.wikipedia.org/wiki/Knuth]] suggested the use of a minus sign for group inverses,
\begin{eqnarray}
g^- &=& g^{-1}\\
gg^- &=& g^- g = 1
\end{eqnarray}
during a talk on notation, http://scpd.stanford.edu/scpd/students/Dam_ui/pages/ArchivedVideoList56K.asp?Include=musings. It's more compact and makes sense, since raising a group element to a power is not always natural, but the inverse is.
Using the definition for the [[determinant]] of a matrix, such as the [[frame]] matrix, $\lp e_i\rp^\al$, and its [[matrix inverse]], $\lp e^-_\al \rp^i = \lp e_\al \rp^i$, with the [[permutation symbol]] gives
\begin{eqnarray}
\ep^{\al \be \dots \ga} &=& \ep^{ij \dots k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_k\rp^\ga\\
\ep^{\al \be \dots \de \ga} \lp e_\ga \rp^k &=& \ep^{ij \dots m k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_m\rp^\de\\
\ep^{\al \be \dots \ep \de \ga} \lp e_\de \rp^m \lp e_\ga \rp^k &=& \ep^{ij \dots n m k} \lp e_i\rp^\al \lp e_j\rp^\be \dots \lp e_n\rp^\ep
\end{eqnarray}
Combining these identities with the [[permutation identities]] allows the [[matrix inverse]] to be written explicitly, as well as giving other expressions, such as
\begin{eqnarray}
\lp e_{\lb \ga \rd}\rp^k \lp e_{\ld \de \rb} \rp^m &=& \fr{\ll \et \rl}{2 \lp n-2 \rp!} \ep^{ij\dots nmk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_n \rp^\ep \ep_{\al \be \dots \ep \ga \de} \\
&=& \fr{1}{2 \ll e \rl \lp n-2 \rp!} \va^{ij\dots nmk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_n \rp^\ep \ep_{\al \be \dots \ep \ga \de}
\end{eqnarray}
Jet spaces are spaces of derivatives of sections.
Refs:
*Gennadi Sardanashvily
**[[Ten Lectures on Jet Manifolds in Classical and Quantum Field Theory|papers/0203040.pdf]]
A ''ket'', $| v \rangle \in \mathcal{H}$, is an element of a [[Hilbert space]]. The ''bra-ket'' notation is inspired by the [[Hermitian form]] on this vector space, which gives the bra-ket of a ket with some ''bra'', $\langle u | = (| u \rangle)^\da$, to produce a complex number, $\langle u | v \rangle \in \mathbb{C}$. Kets transform under Lie group or Lie algebra operators in a [[unitary representation]], $\hat{A} \, | v \rangle = | \hat{A} v \rangle$, and bras are their [[Hermitian]] conjugates, $\langle u | \, \hat{A}^\da = \langle \hat{A} u | = ( \hat{A} | u \rangle )^\da$. Hilbert space is often spanned by a set of ''basis kets'', such as ''particle position basis kets'', $| x \rangle$, which allow us to conjugate and contract to get a [[wavefunction]].
The [[left action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\ph_h(x)$ on the [[Lie group manifold|Lie group geometry]],
$$
L_h g(x) = h g(x) = g(\ph_h(x))
$$
A vector field on the Lie group manifold is ''left invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,
$$
L_{h*} \ve{v}(x) = \ve{v} \f{\pa} \ph_h(x) = \ve{v}(\ph_h(x))
$$
The partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],
$$
\f{\pa} g(\ph_h(x)) = \lp \f{\pa} \ve{\ph_h}(x) \rp \f{\pa} g(\ph_h) = h \f{\pa} g(x)
$$
and the defining equation for the [[right action vector fields and 1-forms|Lie group geometry]],
$$
\f{\pa} g = g T_B \f{\xi_R^B}
$$
to write
$$
\lp \f{\pa} \ve{\ph_h}(x) \rp h g T_B \f{\xi_R^B}(\ph_h) = h g T_B \f{\xi_R^B}(x)
$$
and get
$$
\f{\pa} \ve{\ph_h}(x) = \f{\xi_R^B}(x) \ve{\xi^R_B}(\ph_h(x))
$$
This implies the right action vector fields are left invariant,
$$
L_{h*} \ve{\xi^R_C}(x) = \ve{\xi^R_C}(x) \f{\pa} \ph_h(x) = \ve{\xi^R_C}(x) \f{\xi_R^B}(x) \ve{\xi^R_B}(\ph_h) = \ve{\xi^R_C}(\ph_h(x))
$$
A [[differential form]] is left invariant iff it is invariant under the [[pullback]], $L_h^* \nf{F}(\phi_h(x)) = \nf{F}(x)$. The 1-form duals to left invariant vector fields, such as the duals to the right action vector fields, are left invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be left invariant iff it is invariant under the pushforward, $L_{h*} \nf{\ve{K}}(x) = \nf{\ve{K}}(\phi_h(x))$, or pullback.
The defining equations for the left and right action Killing vector fields, $\ve{\xi^L_B}$ and $\ve{\xi^R_B}$, over a [[Lie group geometry]],
\begin{eqnarray}
\ve{\xi^L_B} \f{d} g &=& T_B g \\
\ve{\xi^R_B} \f{d} g &=& g T_B
\end{eqnarray}
and for their [[1-form]] duals,
\begin{eqnarray}
\f{\xi_L^B} T_B &=& \lp \f{d} g \rp g^- \\
\f{\xi_R^B} T_B &=& g^- \f{d} g
\end{eqnarray}
satisfying $\ve{\xi^L_B} \f{\xi_L^C} = \de_B^C$ and $\ve{\xi^R_B} \f{\xi_R^C} = \de_B^C$, combine with the [[Killing form]] to give the ''left-right rotator'',
$$
L^C{}_B = \ve{\xi^L_B} \f{\xi_R^C} = \ve{\xi^L_B} \lp T^C , g^- \f{d} g \rp = \lp T^C, g^- T_B g \rp
$$
This is a [[Lorentz rotation]],
\begin{eqnarray}
L^A{}_B L^C{}_D g_{AC} &=& \lp T^A, g^- T_B g \rp \lp T^C, g^- T_D g \rp g_{AC} \\
&=& \lp g T^A g^-, T_B \rp \lp g T_A g^-, T_D \rp \\
&=& \lp T'^A, T_B \rp \lp T'{}_A, T_D \rp \\
&=& \lp T_B, T_D \rp
= g_{BD}
\end{eqnarray}
giving one set of Killing vector fields in terms of the other,
\begin{eqnarray}
\ve{\xi^L_B} &=& L^C{}_B \ve{\xi^R_C} \\
\ve{\xi^R_B} &=& L_B{}^C \ve{\xi^L_C}
\end{eqnarray}
A Lorentz rotation of the structure constants by the left-right rotator leaves them invariant,
\begin{eqnarray}
L^C{}_D L^B{}_E C_{CB}{}^A L_A{}^F &=& \lp T^C, g^- T_D g \rp \lp T^B, g^- T_E g \rp \lp \lb T_C,T_B \rb, T^A \rp \lp T_A, g^- T^F g \rp \\
&=& \lp \lb g^- T_D g,g^- T_E g \rb, g^- T^F g \rp = \lp g^- \lb T_D , T_E \rb g, g^- T^F g \rp \\
&=& C_{DE}{}^F
\end{eqnarray}
The [[exterior derivative]] of the left-right rotator is
\begin{eqnarray}
\f{d} L^C{}_B &=& \f{d} \lp T^C, g^- T_B g \rp \\
&=& \lp T^C, \lb g^- T_B g \, , \, g^- \f{d} g \rb \rp \\
&=& \lp \lb \f{\xi_R^A} T_A , T^C \rb , g^- T_B g \rp \\
&=& \f{\xi_R^A} C_A{}^C{}_D L^D{}_B
\end{eqnarray}
The ''left/right [[chiral]]ity projector''s,
$$
P_{L/R} = \ha \lp 1 \pm i \ga \rp
$$
are built using the spacetime Clifford algebra, [[Cl(1,3)]], [[pseudoscalar]], $\ga = \ga_0 \ga_1 \ga_2 \ga_3$. In the [[Weyl representation|Dirac matrices]], they are
\begin{eqnarray}
P_L &=&
\lb \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array} \rb \\
P_R &=&
\lb \begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array} \rb
\end{eqnarray}
A ''linear operator'', $A$, maps vectors from one [[vector space]] to another or itself, $A : v \to v'=Av$, such that $A (a u + b v) = a A u + b A v$ (for any two vectors, and scalars, $a$ and $b$). If the origin space is spanned by bases $e_a$ and the target space is spanned by $e'_b$, then the action of the operator is described by
$$
A: e_a \to e'_a = A \, e_a = e'_b A^b{}_a
$$
corresponding to a ''matrix'', $A^b{}_a$, with a sum always implied over repeated up and down [[indices]].
$$
v'^b e'_b = v' = Av =A v^a e_a = e'_b A^b{}_a v^a \s \text{so, for the components,} \s v'^b = A^b{}_a v^a
$$
If we think of the vector as a column of numbers, the first (upper) index indicates the row and the last (lower) index the column, so
$$
\lb \ba{c} v'^1 \\ v'^2 \ea \rb = \lb \ba{ccc} A^1{}_1 & A^1{}_2 & A^1{}_3 \\ A^2{}_1 & A^2{}_2 & A^2{}_3 \ea \rb \lb \ba{c} v^1 \\ v^2 \\ v^3 \ea \rb
$$
If the target space has a [[metric]], $g'_{ab} = (e'_a, e'_b)$, then a linear operator's ''lowered'' matrix components are
$$
A_{ba} = ( e'_b , A e_a ) = ( e'_b , e'_c A^c{}_a ) = g'_{bc} A^c{}_a
$$
with any ''tensor'' indices lowered by a metric or raised by its inverse, such as $A_b{}^a = g_{bc} A^c{}_d g^{da}$.
If a linear operator maps between $n$ dimensional spaces, it corresponds to an $n \times n$ dimensional matrix. The space of $n \times n$ dimensional matrices that have inverses (non-vanishing [[determinant]]) is the ''general linear group'' -- the [[Lie group]] $GL(n)$.
ref:
[[Little Higgs Review|papers/0502182.pdf]]
nice review
A ''local Clifford rotation'', or ''spin gauge transformation'', is a position-dependent [[Clifford rotation]] of the [[Clifford basis elements]] for a [[Clifford vector bundle]] or [[graded Clifford bundle|Clifford vector bundle]]. The change may be expressed using a position-dependent rotor, $U(x)$, as
$$
\ga'_\al = U \ga_\al U^- = \ga_\be L^\be{}_\al
$$
This is an active transformation of bundle elements. A [[Dirac spinor]] and [[frame]] rotate as
$$
\Ps(x) \to \Ps'(x) = U(x) \Ps(x) \s\s \f{e} \to \f{e}' = U \f{e} \, U^-
$$
The transformation law for the [[spin connection]] under this [[Clifford gauge transformation]] is
$$
\f{\om}' = U \f{\om} \, U^- + 2 U \f{d} \, U^-
$$
Using the Weyl representation of [[Dirac matrices]] and the expression of a rotor as the [[exponentiation]] of a bivector, we have $U^- = \ga^0 U^\da \ga_0$, which gives $\bar{\Ps} \to \bar{\Ps}' = \Ps^\da U^\da \ga^0 = \bar{\Ps} U^-$ for the Clifford rotation of the [[Dirac adjoint]].
For an infinitesimal gauge transformation, $U \simeq 1 + \ha B$, parameterized by an arbitrary Clifford bivector valued field, $B(x)$, the spin connection changes to
$$
\f{\om}' \simeq \f{\om} - \f{d} B - \ha \f{\om} B + \ha B \f{\om} = \f{\om} - \f{\na} B
$$
giving the change $\de \f{\om} = - \f{\na} B$. The [[frame]] infinitesimally rotates to $\f{e'} = U \f{e} \, U^- \simeq \f{e} + B \times \f{e}$.
Physics in curved spacetime is independently invariant under [[local Clifford rotation]]s and auto[[diffeomorphism]]s. These are rotations and changes of shape of the fields. An ''isometry'', or ''local Lorentz transformation'', is a restricted autodiffeomorphism, $x^i \to x'^i = \ph^i(x)$, that leaves the [[metric]] unchanged,
$$
g_{ij} \to g'_{ij}(x') = \pa_i \ph^m(x) \, \pa_j \ph^n(x) \, g_{mn}(x) = g_{ij}(x)
$$
Equivalently, an isometry is an autodiffeomorphism that corresponds to a local Clifford rotation of the [[frame]],
$$
\f{e} \to \f{e}' = \f{\ve{L}} \f{e} (x'(x)) = U \f{e} (x'(x)) U^-
$$
for some [[rotor|Clifford rotation]], $U(x)$, with the [[vector valued form]] ''Lorentz transformation operator'' defined as
$$
\f{\ve{L}}(x) = \f{dx^i} L_i{}^j \ve{\pa'_j} \s\s L_i{}^j(x) = \pa_i \ph^j(x)
$$
As a [[Lorentz rotation]], the inverse of the Lorentz transform matrix is its transpose, $L^-_i{}^j = L^j{}_i$. Note, however, that it is not necessarily the case that $\ph^i(x)$ is $L^-_j{}^i(x) x^j$ -- unless the manifold is flat [[Minkowski|Minkowski metric]] spacetime, a [[rest frame]], in which case $L^-_j{}^i$ is a constant matrix and the local Lorentz transformation is thus a global [[Lorentz transformation]].
For a general [[manifold]] with metric, local Lorentz transformations don't always exist over manifold regions, even though [[local Clifford rotation]]s do. However, one can always create an autodiffeomorphism, $\ph^\mu(x) = L^-_\nu{}^\mu x^\nu$, that is a [[Lorentz transformation]] in a [[rest frame]] near some point on the manifold.
A ''local conformal transformation'', is a restricted auto[[diffeomorphism]], $x^i \to x'^i = \ph^i(x)$, that leaves the [[metric]] unchanged up to a scaling,
$$
g_{ij} \to g'_{ij}(x') = \pa_i \ph^m(x) \, \pa_j \ph^n(x) \, g_{mn}(x) = \La(x)^2 g_{ij}(x)
$$
(A [[local Lorentz transformation]] is a local conformal transformation with unit scaling, $\La=1$.) Equivalently, a local conformal transformation is an autodiffeomorphism that corresponds to a local Clifford rotation and scaling of the [[frame]],
$$
\f{e} \to \f{e}' = \f{\ve{L}} \f{e} (x'(x)) = \La(x) \, U \f{e} (x'(x)) U^-
$$
for some [[rotor|Clifford rotation]], $U(x)$, with the [[vector valued form]] ''Lorentz transformation operator'' defined as
$$
\f{\ve{L}}(x) = \f{dx^i} L_i{}^j \ve{\pa'_j} \s\s L_i{}^j(x) = \pa_i \ph^j(x)
$$
A local conformal transformation of [[Minkowski spacetime|rest frame]] is a global [[conformal transformation]].
[<img[images/png/manifold.png]]An oriented $n$ dimensional differentiable ''manifold'', $M$, may be visualized as a curved $n$ dimensional surface embedded in a higher dimensional, pseudo-Euclidean space. A manifold is described mathematically by a collection of coordinate charts (patches), $\left\{ \left( U_a, \: x_a \right) \right\}$, with the open sets, $U_a$, labeled by $a$, covering $M$, and the coordinates, $x_a : U_a \rightarrow \mathbb{R}^n$, homeomorphic maps into open subsets of $\mathbb{R}^n$ such that overlap maps, $x_a \circ x_b^{-} : \mathbb{R}^n \rightarrow M \rightarrow \mathbb{R}^n$, defined on $x_b( U_a \cap U_b)$, are infinitely differentiable. So, every point, $x$, on the manifold is labeled by a set of $n$ real ''coordinates'', $x_a^i(x)$, in some chart, $U_a$, with coordinate [[indices]], $i$, typically running from $1$ to $n$ or from $0$ to $(n-1)$. In most practical cases the chart label, $a$, is not written and the coordinates are simply written as $x^i$ with some chart implied.
For more on manifolds, see http://en.wikipedia.org/wiki/Manifold
[[Massless Dirac solutions|massless Dirac solutions]] can be interrelated by several conjugation identities, similar to [[Dirac solution identities]]. Because the $CPT$ interrelations mess with phases in interesting ways, it's nice to keep track of these by using phase-related [[helicity state]]s, $\ch_\pm^{\wedge/\vee}$, or, alternatively, using the phase-balanced [[helicity state]]s, $\ch_\pm$, in
$$
u_p^{L \, \wedge/\vee} = \lb \ba{c} \ch_-^{\wedge/\vee} \\ 0 \ea \rb
\s \s
u_p^{R \, \wedge/\vee} = \lb \ba{c} 0 \\ \ch_+^{\wedge/\vee} \ea \rb
\s \s
v_p^{L \, \wedge/\vee} = \lb \ba{c} \xi_-^{\wedge/\vee} \\ 0 \ea \rb
\s \s
v_p^{R \, \wedge/\vee} = \lb \ba{c} 0 \\ \xi_+^{\wedge/\vee} \ea \rb
$$
$$
u_p^L = \lb \ba{c} \ch_- \\ 0 \ea \rb
\s \s \s
u_p^R = \lb \ba{c} 0 \\ \ch_+ \ea \rb
\s \s \s
v_p^L = \lb \ba{c} \xi_- \\ 0 \ea \rb
\s \s \s
v_p^R = \lb \ba{c} 0 \\ \xi_+ \ea \rb
$$
in which $\xi_+^{\wedge/\vee} = - \ep \ch_-^{\wedge/\vee \, *} = \mp \ch_+^{\vee/\wedge}$, $\xi_-^{\wedge/\vee} = + \ep \ch_+^{\wedge/\vee \, *} = \pm \ch_-^{\vee/\wedge}$, and $\xi_\pm = \mp \ep \ch_\mp^* = - \ch_\pm$ are charge conjugate helicity states. We use helicity identities,
$$
\ep \, \ch_{\pm}^\wedge{}^* = \ch_\mp^\vee
\s
\ep \, \ch_{\pm}^\vee{}^* = - \ch_\mp^\wedge
\s
\ep \, \ch_{\pm}^* = \mp \ch_\mp
\s \s
\ch_{\pm}^{\wedge/\vee} (-p) = \ch_{\mp}^{\wedge/\vee} (p)
\s
\xi_{\pm}^{\wedge/\vee} (-p) = - \xi_{\mp}^{\wedge/\vee} (p)
\s
\ch_\pm (-p) = \pm i \, \ch_\mp (p)
\s
\xi_\pm (-p) = \mp i \, \xi_\mp (p)
$$
and the Weyl representation of the [[Dirac matrices]], giving
$$
i \ga_2 = \lb \ba{cc} 0 & \ep \\ -\ep & 0 \ea \rb
\s \s
i \ga_0 = \lb \ba{cc} 0 & i \\ i & 0 \ea \rb
\s \s
i \ga_0 \ga = \lb \ba{cc} 0 & -1 \\ 1 & 0 \ea \rb
\s \s
\ga_{13} = \lb \ba{cc} - \ep & 0 \\ 0 & - \ep \ea \rb
\s \s
\ga = \lb \ba{cc} -i & 0 \\ 0 & i \ea \rb
$$
Mapping particles to antiparticles involves identities related to [[charge conjugation|charge conjugate]],
$$
i \ga_2 \lp u_p^{L \, \wedge/\vee} \rp^* = v_p^{R \, \wedge/\vee}
\s
i \ga_2 \lp u_p^{R \, \wedge/\vee} \rp^* = v_p^{L \, \wedge/\vee}
\s
i \ga_2 \lp v_p^{L \, \wedge/\vee} \rp^* = u_p^{R \, \vee/\wedge}
\s
i \ga_2 \lp v_p^{R \, \wedge/\vee} \rp^* = u_p^{L \, \vee/\wedge}
$$
$$
i \ga_2 u_p^{L \, *} = v_p^R
\s \s \s
i \ga_2 u_p^{R \, *} = v_p^L
\s \s \s
i \ga_2 v_p^{L \, *} = u_p^R
\s \s \s
i \ga_2 v_p^{R \, *} = u_p^L
$$
Reversing the momentum involves identities related to [[parity conjugation|parity conjugate]],
$$
i \ga_0 \, {u}_{-p}^{L \, \wedge/\vee} = + i \, u_{p}^{R \, \wedge/\vee}
\s
i \ga_0 \, {u}_{-p}^{R \, \wedge/\vee} = + i \, u_{p}^{L \, \wedge/\vee}
\s
i \ga_0 \, {v}_{-p}^{L \, \wedge/\vee} = - i \, v_{p}^{R \, \wedge/\vee}
\s
i \ga_0 \, {v}_{-p}^{R \, \wedge/\vee} = - i \, v_{p}^{L \, \wedge/\vee}
$$
$$
i \ga_0 \, u_{-p}^L = + u_{p}^R
\s \s
i \ga_0 \, u_{-p}^R = - u_{p}^L
\s \s
i \ga_0 \, v_{-p}^L = - v_{p}^R
\s \s
i \ga_0 \, v_{-p}^R = + v_{p}^L
$$
Reversing the momentum and spin involves identities related to [[time conjugation|time conjugate]],
$$
i \ga_0 \ga \, u_{-p}^{L \, \wedge/\vee} = \pm \, v_{p}^{R \, \vee/\wedge}
\s
i \ga_0 \ga \, u_{-p}^{R \, \wedge/\vee} = \pm \, v_{p}^{L \, \vee/\wedge}
\s
i \ga_0 \ga \, v_{-p}^{L \, \wedge/\vee} = \pm \, u_{p}^{R \, \vee/\wedge}
\s
i \ga_0 \ga \, v_{-p}^{R \, \wedge/\vee} = \pm \, u_{p}^{L \, \vee/\wedge}
$$
$$
i \ga_0 \ga \, u_{-p}^L = +i \, v_{p}^R
\s \s
i \ga_0 \ga \, u_{-p}^R = +i \, v_{p}^L
\s \s
i \ga_0 \ga \, v_{-p}^L = -i \, u_{p}^R
\s \s
i \ga_0 \ga \, v_{-p}^R = -i \, u_{p}^L
$$
$$
\ga_{13} \, u_{-p}^{L \, \wedge/\vee} = \mp \, \lp u_{p}^{L \, \vee/\wedge} \rp^*
\s
\ga_{13} \, u_{-p}^{R \, \wedge/\vee} = \mp \, \lp u_{p}^{R \, \vee/\wedge} \rp^*
\s
\ga_{13} \, v_{-p}^{L \, \wedge/\vee} = \mp \, \lp v_{p}^{L \, \vee/\wedge} \rp^*
\s
\ga_{13} \, v_{-p}^{R \, \wedge/\vee} = \mp \, \lp v_{p}^{R \, \vee/\wedge} \rp^*
$$
$$
\ga_{13} \, u_{-p}^L = -i \, u_{p}^{L \, *}
\s \s
\ga_{13} \, u_{-p}^R = -i \, u_{p}^{R \, *}
\s \s
\ga_{13} \, v_{-p}^L = +i \, v_{p}^{L \, *}
\s \s
\ga_{13} \, v_{-p}^R = +i \, v_{p}^{R \, *}
$$
$$
\ga_{13} \, u_{-p}^{L \, \wedge/\vee \, *} = - \, v_{p}^{L \, \wedge/\vee}
\s
\ga_{13} \, u_{-p}^{R \, \wedge/\vee \, *} = + \, v_{p}^{R \, \wedge/\vee}
\s
\ga_{13} \, v_{-p}^{L \, \wedge/\vee \, *} = + \, u_{p}^{L \, \wedge/\vee}
\s
\ga_{13} \, v_{-p}^{R \, \wedge/\vee \, *} = - \, u_{p}^{R \, \wedge/\vee}
$$
$$
\ga_{13} \, u_{-p}^{L \, *} = -i \, v_{p}^{L}
\s \s
\ga_{13} \, u_{-p}^{R \, *} = -i \, v_{p}^{R}
\s \s
\ga_{13} \, v_{-p}^{L \, *} = +i \, u_{p}^{L}
\s \s
\ga_{13} \, v_{-p}^{R \, *} = +i \, u_{p}^{R}
$$
which is kind of subtle since the order matters, $u_{-p}^{R \, *} = u^{R \, *}(-p) = - \lp u_{-p}^R \rp^*$.
Using the chiral [[Dirac matrices]], the [[Dirac equation]] with mass $m$ has two positive energy, $\Ps = u_p^{\wedge/\vee} e^{- i p_\mu x^\mu}$, and two negative energy, $\Ps = v_p^{\wedge/\vee} e^{+ i p_\mu x^\mu}$, [[Dirac solutions]] that relate to eigenvectors of the [[spin operator]] in a rest frame. When the mass is zero there is no rest frame, and it is more convenient to work with positive and negative energy Dirac [[helicity]] eigenspinors, $\Ps = u_p^{L/R} e^{- i p_\mu x^\mu}$ and $\Ps = v_p^{L/R} e^{+ i p_\mu x^\mu}$, with
$$
u_p^L = \lb \ba{c} \ch_-(\hat{p}) \\ 0 \ea \rb
\s \s
u_p^R = \lb \ba{c} 0 \\ \ch_+(\hat{p}) \ea \rb
\s \s
v_p^L = \lb \ba{c} \xi_-(\hat{p}) \\ 0 \ea \rb
\s \s
v_p^R = \lb \ba{c} 0 \\ \xi_+(\hat{p}) \ea \rb
$$
satisfying $u_p^{L/R} = P_{L/R} u_p^{L/R}$ and $v_p^{L/R} = P_{L/R} v_p^{L/R}$ under the [[left/right chirality projector]] and $h \, u_p^{L/R} = \mp \ha u_p^{L/R}$ and $h \, v_p^{L/R} = \mp \ha v_p^{L/R}$ under the helicity operator, in which $\hat{p}$ is the [[momentum]] direction, and $\ch_\pm$ are [[helicity state]]s. The $v_p^{L/R}$ are constructed to be [[charge conjugate]]s,
$$
(u_p^L)^C = i \ga_2 u_p^{L \, *} =
\lb \ba{cc} & \ep \\ -\ep & \ea \rb
\lb \ba{c} \ch_- \\ 0 \ea \rb^*
= \lb \ba{c} 0 \\ - \ep \ch_-^* \ea \rb
= \lb \ba{c} 0 \\ \xi_+ \ea \rb
= v_p^R
\s \s
(u_p^R)^C = i \ga_2 u_p^{R \, *} =
\lb \ba{cc} & \ep \\ -\ep & \ea \rb
\lb \ba{c} 0 \\ \ch_+ \ea \rb^*
= \lb \ba{c} \ep \ch_+^* \\ 0 \ea \rb
= \lb \ba{c} \xi_- \\ 0 \ea \rb
= v_p^L
$$
with charge conjugate Weyl spinors,
$$
\xi_\pm = \mp \ep \ch_\mp^* = - \ch_\pm
$$
related by a [[helicity state]] identity. These are solutions of the massless Dirac equation, with
$$
\ps_L = \ch_- e^{- i p_\mu x^\mu} + \xi_- e^{+ i p_\mu x^\mu}
\s \s
\ps_R = \ch_+ e^{- i p_\mu x^\mu} + \xi_+ e^{+ i p_\mu x^\mu}
$$
left and right handed [[Weyl solutions]] including positive and negative energy. The usual Dirac solution normalization doesn't work, since $\bar{u}_p^L u_p^L = u_p^{L \, \da} \ga_0 u_p^L = 0$, but these massless solutions are normalized such that $u_p^{L/R}{}^\da u_p^{L/R} = 1$ and $v_p^{L/R}{}^\da v_p^{L/R} = 1$.
These solutions match Dirac solutions for $m=0$, up to normalization and phase. To get the match, including phases, we must use phase-related helicity spinors,
$$
\ba{rclcrcl}
u'{}_{pL}^{\wedge/\vee} = \ch_-^{\wedge/\vee} \!\!&\!\!=\!\!&\!\! \lp 1 - p_u \rp \ch^{\wedge/\vee}
& \s &
v'{}_{pL}^{\wedge/\vee} = \xi_-^{\wedge/\vee} \!\!&\!\!=\!\!&\!\! + \ep \ch_+^{\wedge/\vee \, *} = \pm \ch_-^{\vee/\wedge} = \lp 1 - p_u \rp \xi^{\wedge/\vee}
\\
u'{}_{pR}^{\wedge/\vee} = \ch_+^{\wedge/\vee} \!\!&\!\!=\!\!&\!\! \lp 1 + p_u \rp \ch^{\wedge/\vee}
& \s &
v'{}_{pR}^{\wedge/\vee} = \xi_+^{\wedge/\vee} \!\!&\!\!=\!\!&\!\! - \ep \ch_-^{\wedge/\vee \, *} = \mp \ch_+^{\vee/\wedge} = - \lp 1 + p_u \rp \xi^{\wedge/\vee}
\ea
$$
which works for either choice of ''seed spinor'', $\ch^\wedge$ or $\ch^\vee$. These may be thought of as [[Pauli spinor|Weyl spinor]]s [[Lorentz boost]]ed to lightspeed in the $p_u$ direction.
A ''massless quantum Dirac spinor'', similar to a [[quantum Dirac spinor]], corresponds to two [[quantum Weyl spinor]]s of opposite chirality, and can be written as
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{L/R} u_p^{L/R} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{R/L \, \da} v_p^{L/R} e^{+i p_\mu x^\mu} \rp
$$
in which the [[massless Dirac solutions]], incorporating $u_p^{L/R}$ and $v_p^{L/R}$, are [[helicity]] eigenstates.
The [[creation and annihilation operators of a massless quantum Dirac spinor]], $\ud{\hat{b}}_{\, p}^{R/L \, \da}$ and $\ud{\hat{a}}_{\, p}^{L/R}$, create and annihilate massless antiparticles and particles of right or left chirality. (For massless fermions, helicity and chirality match.) There are charge, parity, and time [[conjugates of a massless quantum Dirac spinor]].
The [[Dirac adjoint]] of a [[massless quantum Dirac spinor]] field is the ''quantum Dirac spinor adjoint'',
$$
\ud{\hat{\bar{\Psi}}} = \ud{\hat{\Psi}}^\da \ga^0
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{L/R \, \da} {\bar{u}}_p^{L/R} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{R/L} {\bar{v}}_p^{L/R} e^{-i p_\mu x^\mu} \rp
$$
in which $\ud{\hat{a}}_p^{L/R \, \da}$ and $\ud{\hat{b}}_p^{L/R}$ are [[creation and annihilation operators]] for the corresponding fermion and anti-fermion, and the adjoints of the [[massless Dirac solutions]] are
$$
{\bar{u}}_p^L = u_p^{L \, \da} \ga^0 =
\lb \begin{array}{cc} 0 & \ch_-^\da\end{array} \rb
= {\bar{v}}_p^L
\s \s
{\bar{u}}_p^R = u_p^{R \, \da} \ga^0 =
\lb \begin{array}{cc} \ch_+^\da & 0 \end{array} \rb
= - {\bar{v}}_p^R
$$
The complex ''matrix elements'' of a linear operator on a [[Hilbert space]] are its bra-[[ket]] between normalized basis vectors,
$$
A^j{}_i{} = \langle u^j | \hat{A} | u_i \rangle
$$
This allows Linear operations to be computed using matrix algebra,
$$
\hat{A} | v \rangle = | u_j \rangle \langle u^j | \hat{A} | u_i \rangle \langle u^i | v \rangle
= | u_j \rangle A^j{}_i{} v^i
$$
The [[inverse]] of a square matrix, such as the [[frame]] matrix, $\lp e_i \rp^\al$, may be written explicitly as
\begin{eqnarray}
\lp e^-_\ga \rp^k &=& \fr{\ll \et \rl}{\lp n-1 \rp!} \ep^{ij\dots mk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_m \rp^\de \ep_{\al \be \dots \de \ga} \\
&=& \fr{1}{\ll e \rl \lp n-1 \rp!} \va^{ij\dots mk} \lp e_i \rp^\al \lp e_j \rp^\be \dots \lp e_m \rp^\de \ep_{\al \be \dots \de \ga}
\end{eqnarray}
by using the [[inverse matrix identities]] and [[permutation identities]]. It satisfies $\lp e^-_\al \rp^i \lp e_i \rp^\be = \de_\al^\be$ and $\lp e_i \rp^\al \lp e^-_\al \rp^j = \de_i^j$.
The [[structure constants|Lie algebra]], $C_{AB}{}^C$, of [[f4]] can be expressed as a matrix of Lie brackets between basis generators,
$$
\lb
\begin{array}{cccc}
\ga_{\al \be} &
\ga_{0 \al} &
Q^-_a &
Q^+_b
\end{array}
\rb
\;\;\; \textrm{and} \;\;\;
\lb
\begin{array}{cccc}
\ga_{\ga \de} &
\ga_{0 \ga} &
Q^-_c &
Q^+_d
\end{array}
\rb
$$
as
$$
\lb
\begin{array}{cccc}
2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} & 2 \left\{ \et_{\be \ga} \ga_{0 \al} - \et_{\al \ga} \ga_{0 \be} \right\} & Q^-_e ( \overline{\Ga}_\al \Ga_\be )^e{}_c & Q^+_e ( \Ga_\al \overline{\Ga}_\be )^e{}_d \\
2 \left\{ \et_{\ga \al} \ga_{0 \de} - \et_{\de \al} \ga_{0 \ga} \right\} & -2 \, \ga_{\al \ga} & Q^+_e ( - \Ga_\al )^e{}_c & Q^-_e ( \bar{\Ga}_\al )^e{}_d \\
-Q^-_e ( \overline{\Ga}_\ga \Ga_\de )^e{}_a & -Q^+_e ( - \Ga_\ga )^e{}_a & - \ga_{\al \be} ( \overline{\Ga}{}^\al \Ga^\be )_{ac} & -\ga_{0 \al} ( \overline{\Ga}{}^\al )_{ad} \\
-Q^+_e ( \Ga_\ga \overline{\Ga}_\de )^e{}_b & -Q^-_e ( \bar{\Ga}_\ga )^e{}_b & \ga_{0 \al} ( \overline{\Ga}{}^\al )_{bc} & - \ga_{\al \be} ( \Ga^\al \overline{\Ga}{}^\be )_{bd} \\
\end{array}
\rb
$$
A [[Lie group]] ''matrix representation'', $\pi$, is a [[representation]] that is a map from Lie group elements into $n \times n$ complex matrices,
$$
\Pi : G \mapsto GL(n,\mathbb{C})
$$
The corresponding ''representation space'' (sometimes also lazily referred to as the representation) is the real or complex vector space on which those matrices act. Via [[exponentiation]], each Lie group representation gives rise to a [[Lie algebra]] ''representation'',
$$
\pi(X) = \fr{d}{dt} \Pi(e^{tX}) |_{t=0} \in GL(n,\mathbb{C})
$$
for all $X \in {\frak g}$. We also have $\Pi(e^X) = e^{\pi(X)}$. Matrix representatives of Lie algebra elements act on the same representation space as representatives of the corresponding Lie group. And they allow an explicit expression of the [[Lie bracket|Lie algebra]] via matrix multiplication,
$$
\pi(\lb T_A , T_B \rb) = \pi(T_A) \, \pi(T_B) - \pi(T_B) \, \pi(T_A)
$$
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<<ListTagged meta>>
The metaplectic representation...
Metaplectic Lie algebra is same as symplectic Lie algebra. As a representation, it acts as the Schrodinger representation, on wave functions. The metaplectic algebra in quantum mechanics corresponds to the symplectic algebra in classical mechanics. Time evolution operator acting on a wavefunction is an element of the metaplectic group, and I think this means it is an element of the [[infinite-dimensional unitary representation]].
Lie algebra basis elments... {a*a*,aa, H}
[aa,a*a*] = 2 H
Derivation of the Heisenberg algebra
[a*a*,a]=a*
Woit p268
Metaplectic rep is unitary, acting on Fock space. after changing to opps
i(a*a* + aa)
a*a* - aa
a*a + aa*
Metaplectic rep is double cover of SL(2,R)
Doesn't have a rep as finite dim matrices.
A ''metric'', $g$, for a [[vector space]] is an object that takes two vectors and spits out a number -- it determines the symmetric ''scalar product'',
$$
\lp \ve{u}, \ve{v} \rp = u^i v^j \lp \ve{\pa_i}, \ve{\pa_j} \rp = v^i u^j g_{ij} \in \mathbb{R}
$$
A metric on a manifold gives the scalar product between any two [[coordinate basis vectors]] at any point, $g_{ij} = \lp \ve{\pa_i}, \ve{\pa_j} \rp$. When a [[frame]] exists on the manifold it determines this scalar product and metric, using the [[Clifford algebra]] dot product and the [[vector-form algebra]], as
\[ \lp \ve{u}, \ve{v} \rp = \lp \ve{u} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp = u^\al \ga_\al \cdot v^\be \ga_\be = u^\al v^\be \et_{\al \be} = u^i \lp e_i \rp^\al v^j \lp e_j \rp^\be \et_{\al \be} = u^i v^j g_{ij} \]
with the use of frame coefficients and the [[Minkowski metric]] replacing the use of a metric if desired. Using component [[indices]], the ''metric matrix'' (often just abbreviated as "metric") in terms of the frame matrix is
\[ g_{ij} = \lp e_i \rp^\al \lp e_j \rp^\be \et_{\al \be} \]
The metric is invariant under [[Lorentz transformations|Lorentz rotation]] of the frame,
\[ g'_{ij} = \lp e'_i \rp^\al \lp e'_j \rp^\be \et_{\al \be} = \lp e_i \rp^\ga L^\al{}_\ga \lp e_j \rp^\de L^\be{}_\de \et_{\al \be} = \lp e_i \rp^\ga \lp e_j \rp^\de \et_{\ga \de} = g_{ij} \]
Another way of seeing this is that the scalar product of two tangent vectors is invariant under [[Clifford adjoint]] transformations of the frame,
$$\f{e} \mapsto \f{e'} = U \f{e} U^-$$
$$\lp \ve{u} \f{e'} \rp \cdot \lp \ve{v} \f{e'} \rp = \lp \ve{u} U \f{e} U^- \rp \cdot \lp \ve{v} U \f{e} U^- \rp =
\lp \ve{u} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp$$
with Lorentz transformations forming a subset of these.
A ''norm'' on a vector space, $| u |^2 \in \mathbb{R}$, induces a metric via the ''polarization identity'',
$$
( u , v ) = \fr{1}{4} \lp |u+v|^2 - |u-v|^2 \rp = \ha \lp |x+y|^2 - |x|^2 - |y|^2 \rp
$$
The combined spacetime curvature is:
$$
\ff{F_s} = \ha \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp
$$
in which $\ff{R}$ is the [[Clifford vector bundle]] curvature, $\La$ is the [[cosmological constant|Einstein's equation]], and $\f{e}$ is the [[frame]]. The ''modified BF action for gravity'' over a four dimensional base [[spacetime]] is:
$$
S_s = \int \li \ff{B_s} \ff{F_s} - \fr{g \La}{48} \ff{B_s} \ff{B_s}) \ga \ri
$$
in which $\ff{B_s}$ is the ''dual bivector valued 2-form'', $g$ is some small coupling constant, and $\ga$ is the spacetime [[pseudoscalar]]. Insisting that $\de S_s =0$ under $\de \ff{B_s}$ gives:
$$
\ff{B_s} = \fr{12}{g \La} \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \ga^-
$$
Plugging this back into the action gives:
$$
S_s = \fr{3}{g \La} \int \li \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \lp \ff{R} + \fr{\La}{6} \f{e} \f{e} \rp \ga^- \ri
$$
Multiplying this out gives a Chern-Simons boundary term,
$$
\li \ff{R} \ff{R} \ga^- \ri = \f{d} \li \lp \f{\om} \f{d} \f{\om} + \fr{1}{3} \f{\om} \f{\om} \f{\om} \rp \ga^- \ri
$$
as well as the [[Clifford curvature scalar]],
$$
\li \f{e}\f{e} \ff{R} \, \ga^- \ri = \nf{e} R
$$
and a [[volume form]] term,
$$
\li \f{e}\f{e} \f{e} \f{e} \ga^- \ri = 4! \, \nf{e}
$$
Dropping the boundary term, the action is the Einstein-Hilbert action,
$$
S_s = \fr{1}{g} \int \nf{e} \lp R + 2 \La \rp
$$
Varying the frame in the modified BF action for gravity gives the equation of motion:
$$
0 = \f{e} \cdot \ff{B_s} = \ha \lp \f{e} \cdot \ff{R} + \fr{\La}{6} \f{e} \f{e} \f{e} \rp
$$
Taking the [[Hodge dual]] and [[Clifford dual]] of these trivector valued 3-forms gives
$$
\begin{eqnarray}
* \lp \f{e} \cdot \ff{R} \rp \ga^- &=& \lp * \f{e^\mu} \f{e^\nu} \f{e^\rh} \rp \fr{1}{4} R_{\nu\rh}{}^{\ka\la} \lp \ga_{\mu\ka\la} \ga^- \rp \\
&=& \lp \fr{1}{3!} \ep^{\mu\nu\rh\de} \f{e_\de} \rp \fr{1}{4} R_{\nu\rh}{}^{\ka\la} \lp \fr{1}{3!} \ep_{\mu\ka\la\ga} \ga^\ga \rp \\
&=& \f{e_\de} \fr{\ll \et \rl}{3! \,4} \de^{\nu\rh\de}_{\lb \ka\la\ga \rb} R_{\nu\rh}{}^{\ka\la} \ga^\ga \\
&=& - \fr{\ll \et \rl}{3! \, 3!} \lp \f{R} - \fr{1}{2}\f{e} R \rp
\end{eqnarray}
$$
and
$$
* \lp \f{e} \f{e} \f{e} \rp \ga^- = \lp \fr{1}{3!} \ep^{\mu\nu\rh\de} \f{e_\de} \rp \lp \fr{1}{3!} \ep_{\mu\ka\la\ga} \ga^\ga \rp
= \fr{\ll \et \rl}{3!} \f{e}
$$
and we see that this equation of motion is [[Einstein's equation]],
$$
\f{R} - \fr{1}{2} \f{e} R = \La \f{e}
$$
Ref:
*K. Krasnov
**[[Non-metric gravity: A status report|http://arxiv.org/abs/0711.0697]]
***This looks interesting, have only read some and need to finish.
If we consider a relativistic classical field or [[quantum field]] in a [[spacetime]] [[rest frame]], it has normalized plane wave solutions, such as $\Ph^\pm_p(x) = e^{\pm i p_\mu x^\mu} \!$, in which $x^\mu$ are rest frame coordinates (with temporal [[units]], $[x^\mu]=T$) and the ''spacetime momentum components'' (with units $[p_\mu]=1/T$) are
$$
p_\mu = (E,-p^1,-p^2,-p^3)
$$
with ''energy''
$$
E = \sqrt{p_s \!\!\cdot\! p_s \, c^2 + m^2 c^4 } \simeq m c^2 + \fr{1}{2m} p_s \!\!\cdot\! p_s
$$
for particle mass $m$ and (spatial) ''momentum components'', $p_s^\va = \fr{1}{c} p^\va = \fr{1}{c} p_\xi \eta^{\va \xi}$, with the approximation above for large $m$ giving the ''kinetic energy''. We are here using a mostly-negative [[Minkowski metric]] for spacetime, but a positive metric for our space, so we need to be careful with how signs flop around.
For a plane wave, the momentum components are related to ''phase velocity'', $v^\va_p = \fr{E}{p_s^\va}$, and ''group velocity'',
$$
v^\va_g = \fr{\pa E}{\pa p_s^\va} = \fr{p_s^\va \, c^2}{E} \simeq \fr{1}{m} p_s^\va
$$
The group velocity is the velocity of a wave packet, and so associated with the velocity of the corresponding particle. This gives the familiar relation, $p_s = m \, v_g$, in the nonrelativistic limit.
In the rest frame, using [[Clifford algebra]], the ''spacetime momentum'' is $p = p_\mu \ga^\mu$, satisfying
$$
p^2 = p_\mu p^\mu = E^2 - p_s^2 = m^2
$$
and the ''momentum'' is $p_s = p_s^\va \si_\va$. (Sometimes the "s" subscript is dropped when we're clearly talking about spatial momentum.) So we have
$$
p \!\cdot\! x = p_\mu x^\mu = E \, x^0 + p_\va x^\va = E \, t - p_s^\va \, x_s^\va = E \, t - p_s \!\!\cdot\! x_s
$$
We will also often refer to the unit-vector ''momentum direction'', $p_u = \fr{1}{|p_s|} p_s = \fr{1}{\sqrt{p_s \cdot p_s}} p_s$.
In four dimensional [[spacetime]] the [[Cl(1,3)]] Clifford basis vectors in the [[Weyl representation|Dirac matrices]] are $\ga_0 = \si_1 \otimes \si_0$ and $\ga_\va = -i \si_2 \otimes \si_\va$, in which $\si_\va$ are the [[Pauli matrices]]; so the momentum is represented as
$$
p = p^\mu \ga_\mu =
\lb
\begin{array}{cc}
0 & p_L \\
p_R \ & 0
\end{array}
\rb
=
\lb
\begin{array}{cc}
0 & p^\mu \bar{\si}_\mu \\
p^\mu \si_\mu \ & 0
\end{array}
\rb
=
\lb
\begin{array}{cc}
0 & E - c \, p_s \\
E + c \, p_s & 0
\end{array}
\rb
$$
In the above expression the momentum is $p_s = |p_s| p_u$ and the left and right ''chiral momentum'' is $p_{L/R}$. The momentum direction can be expressed in polar coordinates, using the Pauli matrix representation, as
$$
p_u =
p_u^\va \si_\va =
\si_1 \sin{\th} \cos{\ph}
+ \si_2 \sin{\th} \sin{\ph}
+ \si_3 \cos{\th}
=
\lb \begin{array}{cc}
\cos{\th} & e^{- i \ph} \sin{\th} \\
e^{ i \ph} \sin{\th} & - \cos{\th}
\end {array} \rb
$$
Momentum in a rest frame can be described using two [[Weyl spinor]]s in a [[twistor]], or by [[helicity state]]s using [[helicity identities]].
Momentum in curved spacetime is a [[1-form]] associated by [[Cliffordization]], $\f{p} = p_\mu \f{e}^\mu = p \!\cdot\! \f{e}$. This can be related to a particle [[path]].
When we consider a [[manifold]] there are ''natural'' geometric objects that arise "for free" -- without the addition of any further algebraic structure. [[Path|path]]s on the manifold lead to the definition of [[tangent vector]]s and the [[tangent bundle]], their dual [[1-form]]s and the [[cotangent bundle]] lead to [[differential form]]s and [[vector valued form]]s. Such objects are invariant under [[coordinate change]]. Explicitly, a natural object has coordinate indexed components that all transform as a [[tensor|coordinate change]]. Since coordinates and algebraic objects are a computational artifice, only natural objects may be expected to be physically meaningful.
A ''natural operator'', such as the [[exterior derivative]], [[Lie derivative]], [[FuN derivative]], [[FuN curvature]], [[covariant derivative]], or [[vector-form algebra]] product acts on natural objects and produces natural objects as a result. Another way of understanding this is that natural operators commute with [[diffeomorphism]]s. A physical theory satisfies ''geometric naturalness'' to the degree that it is constructed of natural operators.
To be more explicit, a tangent vector is natural since between two coordinate systems
$$
\ve{v} = v^j \ve{\pa^x_j} = v^j \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} = v'^p \ve{\pa^y_p} = \ve{v'}
$$
An example of an operation that is not natural (an ''unatural'' operator) is the [[partial derivative]] acting on tangent vectors, since between two coordinate systems
\begin{eqnarray}
\f{\pa} \ve{v} &=& \f{dx^i} \pa^x_i v^j \ve{\pa_j} = \f{dy^m} \fr{\pa x^i}{\pa y^m} \lp \fr{\pa y^k}{\pa x^i} \pa^y_k v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p}
= \f{dy^m} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} \\
\neq \f{\pa'} \ve{v'} &=& \f{dy^m} \pa^y_m \lp v^j \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p} = \f{dy^m} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + \f{dy^m} v^j \lp \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
\end{eqnarray}
The resulting object, $\f{\pa} \ve{v}$, is not natural because that last term does not vanish. However, unnatural objects can sometimes be assembled into natural objects if such terms are made to cancel. For example, subtracting
$$
\ve{u'} \f{\pa'} \ve{v'} = u^q \fr{\pa y^m}{\pa x^q} \lp \pa^y_m v^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + u^q v^j \lp \fr{\pa y^m}{\pa x^q} \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
$$
from
$$
\ve{v'} \f{\pa'} \ve{u'} = v^q \fr{\pa y^m}{\pa x^q} \lp \pa^y_m u^j \rp \fr{\pa y^p}{\pa x^j} \ve{\pa^y_p} + v^q u^j \lp \fr{\pa y^m}{\pa x^q} \pa^y_m \fr{\pa y^p}{\pa x^j} \rp \ve{\pa^y_p}
$$
the last terms of each cancel to give a natural object, the Lie bracket,
$$
\ve{v'} \f{\pa'} \ve{u'} - \ve{u'} \f{\pa'} \ve{v'} = \ve{v} \f{\pa} \ve{u} - \ve{u} \f{\pa} \ve{v} = \lb \ve{v} , \ve{u} \rb_L
$$
Geometry over a disconnected manifold?
This seems to give an elementary picture:
http://arxiv.org/abs/hep-th/9401145
A [[subgroup]], $N \subset G$, of a [[group]], $G$, is called a ''normal subgroup'', $N \triangleleft G$, iff it is invariant under [[conjugation|group]]; that is, for each $n \in N$ and all $g \in G$ the element $A_g n = g n g^- \in N$. Another way of saying this is that for each $n \in N$ and all $g \in G$ there is an $n' \in N$ such that $gn = n'g$. Or, equivalently, $gN=Ng$ for all $g \in G$.
If $N$ and $G$ are [[Lie group]]s, their elements near the identity may be approximated by [[Lie algebra]] elements,
\begin{eqnarray}
g &\simeq& 1 + x^I T_I \\
n &\simeq& 1 + n^P N_P
\end{eqnarray}
in which $T_I$ and $N_P$ are the $\mathfrak{g}$ and $\mathfrak{n}$ generators. Iff $N$ is a normal subgroup, $N \triangleleft G$, then, collecting orders of $x^I$ gives
\begin{eqnarray}
g n g^- &=& n' \\
\lp 1 + x^I T_I \rp \lp 1 + n^P N_P \rp \lp 1 - x^J T_J \rp &\simeq& \lp 1 + n^P N_P + x^I n_I^P N_P \rp \\
\lb T_I, n^P N_P \rb &=& n_I^P N_P
\end{eqnarray}
and so the Lie algebras satisfy
$$
\lb \mathfrak{g}, \mathfrak{n} \rb \subset \mathfrak{n}
$$
The ''normalizer'' of a subset, $S$, in $G$ is the [[subgroup]] consisting of all elements of $G$ that leave $S$ invariant under conjugation,
$$
N_G(S) = \lc n \in G \; | \; nSn^- = S \rc
$$
Another way of thinking of this is that $N_G(S)$ consists of all elements, $n \in G$, satisfying $ns=sn'$ for each $s \in S$ and some $n' \in N_G(S)$. The normalizer of a single element, $N_G(a) = C_G(a)$, is the [[centralizer]], and in general the centralizer is a [[normal subgroup]] of the normalizer, $C_G(S) \triangleleft N_G(S)$. If $H$ is a subgroup of $G$, the normalizer, $N_G(H)$, is the largest subgroup of $G$ having $H$ as a normal subgroup, $H \triangleleft N_G(H)$.
If $H$ and $N = N_G(H)$ are [[Lie group]]s, their [[Lie algebra]] generators satisfy
$$
\lb \mathfrak{n}, \mathfrak{h} \rb \subset \mathfrak{h}
$$
I am happy to announce that Deferential Geometry has been chosen as a sponsored project by the [[Foundational Questions Institute|http://www.fqxi.org/aw-lisi.html]].
The mathematical basis for the project are being incrementally loaded into the wiki, from the bottom up. It's currently about half way there. From this partial foundation now in place, several physics ideas are being actively extended and pursued. Since this is an ongoing process, with changes occurring daily, things may look a bit chaotic. For a traditional, linear introduction to the physics content of the Deferential Geometry project, interested readers may wish to look at two recent papers:
*[[An Exceptionally Simple Theory of Everything]]
*[[Quantum mechanics from a universal action reservoir|http://arxiv.org/abs/physics/0605068]]
I recently attended the very fun [[FQXi 07 conference]], where I had a great time and managed to give this [[talk for FQXi 07]] in ten minutes.
The ''octonions'', $v^a e_a \in \mathbb{O}$, are an eight-dimensional [[division algebra]], spanned by eight basis elements, $e_0,...,e_7$. The octonion identity element is $e_0=1$, and the other seven can be thought of as different imaginary directions, squaring to $-1$. Octonionic multiplication is non-commutative and, unusually, non-associative. A multiplication table for the basis elements is
| $\;e_0\;$ | $e_1$ | $e_2$ | $e_3$ | $e_4$ | $e_5$ | $e_6$ | $e_7$ |
| $\;e_1\;$ | $-e_0$ | $e_4$ | $e_7$ | $-e_2$ | $e_6$ | $-e_5$ | $-e_3$ |
| $\;e_2\;$ | $-e_4$ | $-e_0$ | $e_5$ | $e_1$ | $-e_3$ | $e_7$ | $-e_6$ |
| $\;e_3\;$ | $-e_7$ | $-e_5$ | $-e_0$ | $e_6$ | $e_2$ | $-e_4$ | $e_1$ |
| $\;e_4\;$ | $e_2$ | $-e_1$ | $-e_6$ | $-e_0$ | $e_7$ | $e_3$ | $-e_5$ |
| $\;e_5\;$ | $-e_6$ | $e_3$ | $-e_2$ | $-e_7$ | $-e_0$ | $e_1$ | $e_4$ |
| $\;e_6\;$ | $e_5$ | $-e_7$ | $e_4$ | $-e_3$ | $-e_1$ | $-e_0$ | $e_2$ |
| $\;e_7\;$ | $e_3$ | $e_6$ | $-e_1$ | $e_5$ | $-e_4$ | $-e_2$ | $-e_0$ |
which can be written using an ''octonion multiplication coefficient matrix'' as
$$
e_a e_b = M_{ab}{}^c e_c
$$
so, for example, $e_1 e_2 = e_4$ and $M_{12}{}^4 = 1$. ''Octonionic conjugation'' is given by
$$
\os{e}_0 = e_{\os{0}} = e_0 \;\;\;\;\; \os{e}_1 = e_{\os{1}} = -e_1 \;\;\;\; ... \;\;\;\; \os{e}_7 = e_{\os{7}} = -e_7
$$
and satisfies $\widetilde{(e_a e_b)} = e_{\os{b}} e_{\os{a}}$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$.
Multiplying an octonion by its conjugate gives a real, its norm,
$$
\os{v} v = v^a v^b \os{e}_a e_b =
v^0 v^0 + v^1 v^1 + v^2 v^2 \, + \;\;\;\; ... \;\;\;\; + \,\, v^7 v^7
= v \, \tilde{v} = | v |^2
$$
allowing us to calculate the inverse of any nonzero octonion, $v^- = \frac{\tilde{v}}{|v|^2}$. Note that $v v$ is real only if $v$'s real part or vector part is zero. The ''octonion [[metric]]'' is defined as
$$
(u,v) = \ha \lp \os{u} v + \os{v} u \rp = u^0 v^0 + u^1 v^1 + u^2 v^2 \, + \;\;\;\; ... \;\;\;\; + \,\, u^7 v^7
$$
so
$$
(e_a, e_b) = \ha ( \os{e}_a e_b + \os{e}_b e_a ) = n_{ab}
$$
with $n_{ab} = \de_{ab}$ for the octonions. This metric can be used to raise or lower octonion indices (which has no effect for the octonions, but matters for the [[split-octonion]]s, for which $n'_{ab} = \text{diag}(+1,+1,+1,+1,-1,-1,-1,-1)$). The combination of metric and octonionic conjugates, $\os{e}^a = n^{ab} \os{e}_b \in \tilde{\mathbb{O}}$, are the [[duals|dual space]] to the octonions, $\os{e}^a e_b = \de^a_b$. The matrix $\de^\os{b}_a = \text{diag}(+1,-1,-1,-1,-1,-1,-1,-1)$ can be used to twiddle or untwiddle indices, such as $M^\os{b}{}_{ac} = \de^\os{b}_d M^d{}_{ac} = n^{\os{b}d} M_{dac}$.
The octonions are not associative and so do not have a faithful matrix representation; however, a nonfaithful real $8 \times 8$ representation comes from their multiplication table,
$$
(e_c)^b{}_a = M^\os{b}{}_{ac}
$$
which satisfies the metric expression, $(e_a, e_b) = n_{ab}$, but $e_a e_b \ne M_{ab}{}^c e_c$.
The [[octonion]]s and [[split-octonion]]s have a few useful identities.
Their multiplication satisfies the [[Moufang identities]]. Moreover, the octonions are not associative but they are ''alternative'':
$$
\begin{array}{rcl}
(zz)x & = & z(zx) \\
(xz)z & = & x(zz) \\
(zx)z & = & z(xz) \;\; = \;\; z x z
\end{array}
$$
The basis elements, $e_c$, anticommute with each other, $e_a e_b = - e_b e_a$ , except for the identity element, $e_0 = 1$, which commutes with all. Octonion conjugation flips the signs of the non-identity basis elements,
$$
e_{\tilde{0}} = e_0 \s e_{\tilde{c} > 0} = - e_c
$$
so an octonion splits into real and ''octonionic imaginary'' parts, $z = z_{\mathbb R} + z_{\mathbb I}$, with its conjugate equal to $\tilde{z} = z_{\mathbb R} - z_{\mathbb I}$. This allows the octonion metric to be defined via
$$
(e_a, e_b) = \ha (e_\os{a} e_b + e_\os{b} e_a) = n_{ab}
$$
which is $n_{ab} = \de_{ab}$ for the [[octonion]]s and $n_{ab} = n'_{ab} = diag(+1,+1,+1,+1,-1,-1,-1,-1)$ for the [[split-octonion]]s. Under conjugation this results in ''metric identities'',
$$
(\tilde{x},\tilde{y}) = (x, y) \s (x,\tilde{y}) = (\tilde{x},y)
$$
The combination of metric and octonionic congugates, $\os{e}^a = n^{ab} \os{e}_b \in \tilde{\mathbb{O}}$, are the [[duals|dual space]] to the octonions, $\os{e}^a e_b = \de^a_b$.
Conjugation mixes easily with alternativity, so, for example,
$$
(x \tilde{z}) z = (x z_{\mathbb R} - x z_{\mathbb I})(z_{\mathbb R} + z_{\mathbb I}) = x (z_{\mathbb R} z_{\mathbb R} - z_{\mathbb I} z_{\mathbb I}) = x (\tilde{z} z)
$$
Also, we have
$$
v \tilde{v} = v_{\mathbb R} v_{\mathbb R} - v_{\mathbb I} v_{\mathbb I} = \tilde{v} v = (v,v)
$$
so octonions are easily invertible,
$$
v^- = \fr{ \tilde{v} }{v \tilde{v}}
$$
If we choose a unit length octonion, $u \tilde{u} = 1$, then we can define an ''octonionic reflection'' along $u$,
$$
R_u v = v_\perp - v_\parallel = v - 2 (v,u) u = v - (v \tilde{u}) u - (u \tilde{v}) u = - u \tilde{v} u
$$
similar to a [[Clifford reflection]]. This leaves the metric invariant, by alternativity,
$$
(R_u x, R_u y) = (- u \tilde{x} u, - u \tilde{y} u) = \ha ( (u \tilde{x} u) (\tilde{u} y \tilde{u}) + (u \tilde{y} u) (\tilde{u} x \tilde{u}) ) = u (\os{x},\os{y}) \tilde{u} = (x , y)
$$
The octonionic product coefficients in $e_a e_b = M_{ab}{}^c e_c$, and octonionic conjugation, are used to define
$$
\Ga_c{}^a{}_b = M_c{}^a{}_\os{b}
$$
related to the [[octonionic representation of Cl(8)]], which can have its index lowered by the octonion metric to $\Ga_{cab} = M_{ca\os{b}} = M_c{}^e{}_\os{b} n_{ae}$. We also have the ''conjugation identity'', $M_{\os{a}\os{b}c} = M_{ba\os{c}}$, and the transpose, $\overline{\Ga}_{cab} = \Ga_{cba}$. These gamma matrices satisfy a ''cyclic identity'', related to [[triality]],
$$
\begin{array}{c}
\Ga_{abc} = \Ga_{bca} = \Ga_{cab} \\
= M_{\os{b}\os{a}c} = M_{\os{a}\os{c}b} = M_{\os{c}\os{b}a} \\
= M_{ab\os{c}} = M_{ca\os{b}} = M_{bc\os{a}} \\
(\tilde{x},yz) = (\tilde{y},zx) = (\tilde{z},xy) \\
\end{array}
$$
This can all be summarized by saying we can always cyclicly permute $\Ga$ and $\overline{\Ga}$ indices, and $M$ indices with some conjugating.
There are real, $16 \times 16$, [[chiral]] [[representation|Clifford matrix representation]]s of the [[Cl(8)]] [[Clifford algebra]] related to [[octonion]]s, via [[Clifford division algebra representation]]. Explicitly, from the octonion multiplication coefficient matrix, the cyclic octonionic representation chiral dirac matrix is given by
$$
v^c \Ga_c =
\lb \begin{array}{cccccccc}
v^0 & -v^1 & -v^2 & -v^3 & -v^4 & -v^5 & -v^6 & -v^7 \\
-v^1 & -v^0 & v^4 & v^7 & -v^2 & v^6 & -v^5 & -v^3 \\
-v^2 & -v^4 & -v^0 & v^5 & v^1 & -v^3 & v^7 & -v^6 \\
-v^3 & -v^7 & -v^5 & -v^0 & v^6 & v^2 & -v^4 & v^1 \\
-v^4 & v^2 & -v^1 & -v^6 & -v^0 & v^7 & v^3 & -v^5 \\
-v^5 & -v^6 & v^3 & -v^2 & -v^7 & -v^0 & v^1 & v^4 \\
-v^6 & v^5 & -v^7 & v^4 & -v^3 & -v^1 & -v^0 & v^2 \\
-v^7 & v^3 & v^6 & -v^1 & v^5 & -v^4 & -v^2 & -v^0 \\
\end{array} \rb
$$
The resulting [[pseudoscalar]] is $\ga' = \ga'_0 \ga'_1 \ga'_2 \ga'_3 \ga'_4 \ga'_6 \ga'_7 = \si_3 \otimes 1$.
One generation of [[standard model]] fermions consists of left-[[chiral]] [[Weyl spinor]] electron neutrinos, $\nu_{e}$, electrons, $e$, up quarks, $u$, and down quarks, $d$, as well as their anti-particles, $\bar{e}, \bar{u}, \bar{d}$ (and possibly the anti-neutrino, $\bar{\nu}_e$, which might not exist). These particles have the following ''hypercharge'', $Y$, and ''weak charge'', $W$, which combine as their ''electric charge'', $Q = (Y+W)/2$:
$$
\begin{array}{|l|ccc|}
\hline
& Y & W & Q \\
\hline
\nu_{eL} & -1 & +1 & 0 \\
\bar{\nu}_{eL } & 0 & 0 & 0 \\
e_L & -1 & -1 & -1 \\
\bar{e}_L & +2 & 0 & +1 \\
u_L & +\fr{1}{3} & +1 & +\fr{2}{3} \\
\bar{u}_L & -\fr{4}{3} & 0 & -\fr{2}{3} \\
d_L & +\fr{1}{3} & -1 & -\fr{1}{3} \\
\bar{d}_L & +\fr{2}{3} & 0 & +\fr{1}{3} \\
\hline
\end{array}
$$
The up and down quarks also have pairs of nonzero [[su(3)]] ''color'' charges. This full set of charges is replicated for each of the other two generations of fermions in the [[Elementary particle zoo]], plus their [[charge-parity conjugates|conjugates of a massless quantum Dirac spinor]].
A matrix, $L$, is ''orthogonal'', iff its [[inverse]], $L^- = L^T$, is its [[transpose]].
*<<slider chkSliderpersonF personF 'person >' 'people with interesting physics, usually with links to papers'>>
<<ListTagged paper>>
A geometric object is parallel transported if it is perceived to be unmoving by an observer traveling along with it. Equivalently, a [[fiber bundle]] section, $C(x)$, having values along a [[path]], $x(t)$, is ''parallel transport''ed iff it is [[horizontal|covariant derivative]] along the path,
$$
0 = \ve{v} \f{\na} C = v^i \pa_i C + v^i A_i{}^B T_B C = \fr{d}{d t} C(x(t)) + \ve{v} \f{A} C
$$
In this equation $\ve{v} = \fr{dx^i}{dt} \ve{\pa_i}$ is the path velocity and $\f{A}C$ represents the [[connection]] acting via the left action on the fiber section -- in practice the connection will act appropriately to the specific case.
The solution to the parallel transport equation may be written via the [[path holonomy]].
The ''parity conjugate'' of a [[Dirac spinor]] field, $\Ps$, using the Weyl representation of the [[Dirac matrices]], comes from a [[unitary]] operation,
$$
\Ps(t,x)^P = i \ga_0 \Ps(t,-x)
$$
in which the ''parity operator'', $P$, is an element of the [[Lorentz group]] that flips [[chiral]]ity and squares to $P^2 = -1$. The [[Dirac equation]] with a parity conjugate [[gauge field|principal bundle]],
$$
\f{A} = \f{dt} A^0(t,x) + \f{dx^\ep} A_\ep(t,x) \to \f{A}^P = \f{dt} A^0(t,-x) - \f{dx^\ep} A_\ep(t,-x)
$$
is satisfied by the parity conjugate spinor,
$$
\begin{array}{rcl}
\lp i \ga^\mu (\pa_\mu + A_\mu^P ) - m \rp \Ps^P
\!\!&\!\!=\!\!&\!\! \lp i \ga^\mu ( \pa_\mu + A_\mu^P ) - m \rp i \ga_0 \Ps(t,-x) \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \lp i \ga^0 ( \pa_0 + A_0(t,-x) ) - i \ga^\va ( \pa_\va - A_\va(t,-x) ) - m \rp \Ps(t,-x) \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \lp i \ga^0 ( \pa_0 + A_0(t,x') ) - i \ga^\va ( - \pa'_\va - A_\va(t,x') ) - m \rp \Ps(t,x') \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \lp i \ga^\mu ( \pa'_\mu + A_\mu(t,x') ) - m \rp \Ps(t,x') = 0
\end{array}
$$
Note that if there is a [[chiral]] coupling between gauge field and spinor, such as in the weak interaction, $W_\mu P_L \Ps$, in which $P_{L/R}$ is the [[left/right chirality projector]], this will not be invariant under parity conjugation since $\ga_0 P_L \ga_0 = P_R$.
Parity conjugation of a [[quantum Dirac spinor]] results from the action of a corresponding [[unitary]] operator in the [[infinite-dimensional unitary representation]],
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^P = \hat{\cal{P}} \ud{\hat{\Ps}} \hat{\cal{P}}^-
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp ( \ud{\hat{a}}_p^{\wedge/\vee})^P {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + (\ud{\hat{b}}_p^{\wedge/\vee \, \da})^P {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{-p}^{\wedge/\vee} u_{-p}^{\wedge/\vee} e^{-i (Et-p \cdot x_s)} + \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} v_{-p}^{\wedge/\vee} e^{+i (Et - p \cdot x_s)} \rp \\
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp i \ud{\hat{a}}_{-p}^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i (Et- p \cdot x_s)} - i \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} v_{p}^{\wedge/\vee} e^{+i (Et- p \cdot x_s)} \rp
\end{array}
$$
using some [[Dirac solution identities]], $i \ga_0 \, u_{-p}^{\wedge/\vee} = + i \, u_p^{\wedge/\vee}$ and $i \ga_0 \, v_{-p}^{\wedge/\vee} = - i \, v_p^{\wedge/\vee}$. The corresponding parity conjugation transformation of the creation and annihilation operators is thus
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^P = + i \, \ud{\hat{a}}_{-p}^{\wedge/\vee}
\s\; \s
\lp \ud{\hat{b}}_p^{\wedge/\vee \, \da} \rp^P = -i \, \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da}
$$
The parity conjugate of a particle with [[momentum]] $p$ is the same particle with momentum $-p$.
For massless particles, represented by a [[massless quantum Dirac spinor]], the parity conjugate of a left-handed fermion is the right-handed fermion field with opposite momentum.
The ''partial derivative'', $\partial_i$, of a [[function]], $f(x)$, over a [[manifold]] is a [[derivative|derivation]] taken with respect to one manifold coordinate, while holding the other coordinates constant,
$$
\pa_i f = \fr{\pa f}{\pa x^i} = \pa^x_i f
$$
This derivative is explicitly dependent on the choice of coordinates, and must be glued together over different manifold patches.
Partial derivatives may also be taken of geometric objects (sections of [[fiber bundle]]s), with care taken to keep track of which elements are coordinate dependent and which are constant. As an example,
$$
\pa_i \f{A} = \pa_i \f{dx^j} A_j{}^B T_B = \f{dx^j} \lp \pa_i A_j{}^B \rp T_B
$$
The partial derivatives of [[fiber basis elements|vector bundle]], including [[coordinate basis vectors]], [[coordinate basis 1-forms]], and [[Lie algebra]] basis elements, vanish,
$$
\pa_i \ve{\pa_j}=0 \qquad \pa_i \f{dx^j} = 0 \qquad \pa_i T_A=0
$$
The partial derivative may be combined with [[coordinate basis 1-forms]] to produce the ''partial derivative operator'',
$$
\f{\pa}=\f{dx^i} \pa_i = \f{dx^i} \fr{\pa}{\pa x^i}
$$
(usually also just referred to as the //''partial derivative''//). This is NOT a [[natural]] operator on [[vector valued form]]s (or [[vectors|tangent bundle]]) but it is a natural operator on [[differential form]]s -- for which it is the [[exterior derivative]],
$$
\f{\pa} \f{f} = \f{dx^i} \f{dx^j} \pa_i f_j = \f{d} \f{f}
$$
Even though it is not a natural operator on VVF's, and therefore does not by itself produce geometrically meaningful objects, it may still be used on them,
$$
\f{\pa} \f{\ve{A}} = \f{dx^i} \f{dx^j} \lp \pa_i A_j{}^k \rp \ve{\pa_k}
$$
and provides a useful, coordinate dependent but index free calculational device when combined with other terms using [[vector-form algebra]] to build coordinate invariant geometric objects.
As a useful example, the partial derivative operator satisfies the ''chain rule'',
$$
\f{\pa} f(y(x)) = \f{dx^i} \lp \pa_i y^j(x) \rp \pa_j f(y) = \lp \f{\pa} \ve{y}(x) \rp \f{\pa} f(y)
$$
with the partial derivatives taken with respect to the functional dependencies as written.
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A ''path'' or //''curve''//, $c$, on a [[manifold]] is a connected set of manifold points, with these path points usually labeled by a continuous, monotonically increasing real parameter, $t$.
$$
c:\mathbb{R} \to M
$$
$$
c(t)=x(t) \in M
$$
Paths are often written in terms of the $n$ coordinates (in some manifold patch) corresponding to the path points, $x^i(t)=x_c^i(t)=c^i(t)$, for any parameter value. Physics is invariant under arbitrary reparameterizations of the path. Each path point, labeled by $t$, has a [[tangent vector]], $\ve{v}(t)$, with amplitude dependent on the parameter. Any "initial" path point, $x(0)$, corresponding to parameter $t=0$, has nearby path points given approximately by
$$
x^i(t) \simeq x^i(0) + t v^i(0)
$$
to first order in $t$. A path, or technically a path segment, can be defined to exist for a subset of the reals, such as $t \in \lb 0,1 \rb$. Such a path is called a ''closed path'' or //''loop''// iff its ends meet, $x(0) = x(1)$, and otherwise is called an ''open path''. Paths, other than loops which intersect in one place, are usually restricted to be non-self-intersecting.
Any [[typical fiber|fiber bundle]] element at an initial point, $C_0 = C(x(0))$, may be [[parallel transport]]ed along a path, $x(t)$, by solving the (set of) first order ODE's,
$$
\fr{d}{dt} C = - \ve{v} \f{A} C
$$
Since the [[connection]] is in the Lie algebra of the structure group the solution may be expressed as $C(x(t)) = U(t) C_0$, in which $U(t) \in G$ is the ''path holonomy''. (We use the left action throughout this example, which may be adapted for the appropriate structure group action.) $U(t)$ is an element of the structure group that acts on any initial fiber element to give the solution to the parallel transport equation along a path. Plugging this form for the solution into the parallel transport equation, the path holonomy is the solution to the resulting ''holonomy equation'',
$$
\fr{d}{d t} U(t) = - \ve{v} \f{A} U
$$
from an initial condition of $U(0)=1$. (once again, the actual action of the connection on the holonomy will depend on the specific group action -- here taken to be the left action.) This equation may be readily converted to an integral equation,
$$
U(t) - 1 = - \int_0^t \f{dt} \fr{dx^i}{dt} A_i U(t)
$$
For small displacements along the path, $x^i = x^i_0 + \va^i(t)$, the solution may be found to any order. To first order,
$$
U(t) \simeq 1 - \int_0^t \f{dt} \fr{d \va^i}{dt} A_i(x_0) U(0) = 1 - \va^i A_i
$$
and to second order,
\begin{eqnarray}
U(t) &\simeq& 1 - \int_0^t \f{dt} \fr{d \va^i}{dt} \lb A_i + \va^j \pa_j A_i \rb \lb 1 - \va^k A_k \rb \\
&\simeq& 1 - \va^i A_i + \va^{ij} \lb - \pa_j A_i + A_i A_j \rb
\end{eqnarray}
with the ''second order path dependence'' above defined as
$$
\va^{ij} = \lb \int_0^t \f{dt} \fr{d \va^i}{dt} \va^j \rb
$$
The solution to the holonomy equation may be written heuristically as
$$
U(t) = Pe^{-\int \f{A}} = Pe^{-\int_0^t \f{dt} v^i A_i}
$$
in which $P$ stands for "''path ordered''", and is there to make sure it's understood that this isn't a proper [[exponentiation]], but rather a way of heuristically writing the solution to the holonomy equation.
The path holonomy is the [[holonomy]] for a path that isn't necessarily closed.
A [[path]] in a [[rest frame]], $x(t)$, parameterized by $t$ (which can be unitless or can be equated with $x^0$ and have temporal units), the [[velocity|tangent vector]] along the path, with respect to this parameter, is
$$
v = \fr{dx}{dt} = \fr{dx^\al}{dt} \ga_\al = v^\al \ga_\al
$$
Along a path segment, the [[proper time]], $\tau$, in seconds or other time units, steps forward as
$$
\f{d\tau} = \sqrt{ \f{dx^\al} \f{dx^\be} \et_{\al \be} } = \sqrt{ \f{dx^0} \f{dx^0} - \fr{1}{c^2} \f{dx_s^\pi} \f{dx_s^\rh} \de_{\pi \rh} }
$$
in which $\et_{\al \be}$ is a mostly-negative [[Minkowski metric]]. In terms of parameter time, the proper time changes as
$$
\fr{d\tau}{dt} = \sqrt{\fr{dx^\al}{dt} \fr{dx^\be}{dt} \et_{\al \be}} = \sqrt{v^\al v^\be \et_{\al \be}} = \sqrt{v \!\cdot\! v} = \ll v \rl
$$
If the path is parameterized (or reparameterized) by proper time, $t(\tau)=\tau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$. The proper time describes how time is experienced by a particle or observer moving along that path. The time coordinate, $x^0$, in the rest frame is the proper time for a path along that coordinate line — a line having constant velocity $v=\ga_0$. ''Null lines'', paths followed by massless particles such as light, are ''null paths'', $|v|=0$, of constant velocity, $\fr{dv}{dt}=0$.
Contracting components of $n$ dimensional [[permutation symbol]]s gives
\begin{eqnarray}
\ep^{\al \be \dots \ga} \ep_{\al \be \dots \ga} &=& \ll \et \rl n! \\
\ep^{\al \be \dots \ga} \ep_{\de \be \dots \ga} &=& \ll \et \rl \lp n-1 \rp! \de^\al_\de \\
\ep^{\al \be \ga \dots \de} \ep_{\ep \up \ga \dots \de} &=& \ll \et \rl 2! \lp n-2 \rp! \de^{\al \be}_{\lb \ep \up \rb} \\
\ep^{\al \dots \be \ga \dots \de} \ep_{\ep \dots \up \ga \dots \de} &=& \ll \et \rl p! \lp n-p \rp! \de^{\al \dots \be}_{\lb \ep \dots \up \rb}
\end{eqnarray}
The ''permutation symbol'' (or ''Levi-Civita symbol'', or ''antisymmetric symbol'') for label [[indices]] ranging over $n$ dimension is
$$
\ep_{\al \dots \de} = n! \de^0_{\lb \al \rd} \dots \de^{(n-1)}_{\ld \de \rb} = n! \de^{0 \dots (n-1)}_{\lb \al \dots \de \rb}
$$
using the antisymmetric [[index bracket]]. Alternatively, the indices may range from $1$ to $n$, or over any collection of $n$ numbers. It is antisymmetric in all indices — returning $1$ for positive permutations, $-1$ for negative permutations, and $0$ if any indices are repeated. For example, $\ep_{12}=1$. The indices may be raised with a [[Minkowski metric]] to get
$$
\ep^{\al \be \dots \ga} = \et^{\al \de} \et^{\be \ep} \dots \et^{\ga \up} \ep_{\de \ep \dots \up} = \ll \et \rl \ep_{\al \be \dots \ga}
$$
Since [[Clifford basis vectors]] anti-commute, the permutation symbol arises geometrically as
\[ \ep_{\al \be \dots \ga} = \li \ga_\al \ga_\be \dots \ga_\ga \ga^- \ri \]
using the inverse of the [[pseudoscalar]], $\ga^- = \ga^{n-1} \dots \ga^1 \ga^0$, and the [[scalar part]] operator. These satisfy several [[permutation identities]].
The ''permutation symbol for coordinate [[indices]]'' ranging over $n$ dimension is
$$
\va^{i \dots j} = n! \de_{0 \dots (n-1)}^{\lb i \dots j \rb}
$$
These indices may be lowered with a [[metric]] to get
$$
\va_{i \dots j} = g_{ik} \dots g_{jl} \va^{k \dots l}
$$
Label and coordinate indices may be changed using the [[frame]] and its inverse,
$$
\begin{array}{rclcrcl}
\ep^{i \dots j} &=& \ep^{\al \dots \be} \lp e_\al \rp^i \dots \lp e_\be \rp^j & \s &
\va^{\al \dots \be} &=& \va^{i \dots j} \lp e_i \rp^\al \dots \lp e_j \rp^\be \\
\ep_{i \dots j} &=& \lp e_i \rp^\al \dots \lp e_j \rp^\be \ep_{\al \dots \be} & \s &
\va_{\al \dots \be} &=& \lp e_\al \rp^i \dots \lp e_\be \rp^j \va_{i \dots j}
\end{array}
$$
The two different permutation tensors contract to give [[determinant]]s,
\begin{eqnarray}
\ep^{\al \dots \be} \ep_{\al \dots \be} &=& \ll \et \rl n! \\
\va^{i \dots j} \va_{i \dots j} &=& \ll g \rl n! = \ll e \rl^2 \ll \et \rl n! \\
\ep^{\al \dots \be} \va_{\al \dots \be} &=& \ep^{i \dots j} \va_{i \dots j}
= \ep_{\al \dots \be} \va^{\al \dots \be} = \ep_{i \dots j} \va^{i \dots j}
= \ll e \rl n!
\end{eqnarray}
and they are related by:
$$
\va_{i \dots j} = \ll e \rl \ll \et \rl \ep_{i \dots j}
$$
//Yes, it's non-standard to have two permutation symbols -- but I didn't like all the factors of $\ll e \rl$ floating around. With the way I've defined them, the permutation tensors, $\ep_{\al \dots \be}$ and $\va^{i \dots j}$, actually ARE position-independent permutation symbols in these indices, which may be raised, lowered, or converted at will.//
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<<ListTagged physics>>
A ''pin group'', $G=Pin(n)$, is the double cover of the [[orthogonal group|special orthogonal group]]. The two groups have the same [[spin Lie algebra]], $pin(n) = so(n)$. The ''generalized pin group'', $Pin(p,q)$, is the double cover of the corresponding generalized orthogonal group. The generalized pin group is a [[subgroup]] of the corresponding [[Clifford group]],
$$
Pin(p,q) = \{ U \in Cl(p,q) \ni U \tilde{U} = \pm 1 \}
$$
in which $\tilde{U}$ is the [[Clifford reverse|Clifford conjugate]]. Pin group elements act on Clifford elements and spinors as [[Clifford reflection]]s, or, by combining reflections, as [[Clifford rotation]]s.
For a [[plane wave solution|Maxwell solutions]] to [[Maxwell's equations]], the [[electromagnetic field]], $A$, (with spatial [[momentum]], $p_s$) oscillates in a direction called the ''polarization'', $\ep'$, a [[Cl(1,3)]] [[Clifford vector]]. If we restrict to [[Weyl gauge|Maxwell solutions]], $A_0 = \ph = \ep^0 = 0$, then the polarization is purely spatial and we work within [[Cl(3)]], using [[Pauli matrices]] for the representation. The polarization must then be perpendicular to the momentum, and we can span this space of solutions by choosing two spatial orthonormal ''polarization basis vectors'', $\ep_1$ and $\ep_2$, satisfying $\ep_1 \ep_1 = 1$, $\ep_2 \ep_2 = 1$, $\ep_1 \cdot \ep_2 = 0$, and
$$
\ep_1 \ep_2 \, p_u = \si
$$
with $\si$ the spatial [[pseudoscalar]], equal to $i$ in the Pauli rep. In this way, $\ep_1$, $\ep_2$, and the ''momentum unit vector'', $p_u = \fr{1}{E} p_s$ (with energy $E=\sqrt{p_s \!\!\cdot\! p_s}$), form a right-handed set of spatial orthonormal Clifford vectors. We can then define the [[Hermitian]] conjugate pair, $\ep_\pm = \ep_1 \pm i \ep_2$, as ''circular polarization basis vectors'', and use them to describe right and left ''circularly polarized electromagnetic waves''. These complex basis vectors are uniquely defined by $p_u$ up to phase, $\ep_\pm \sim e^{\pm i \ph} \ep_\pm$, and satisfy
$$
\ep_- = \ep_+^\da
\s \;\;\;
p_u \ep_\pm = \mp i \si \ep_\pm = \pm \ep_\pm
\s \;\;\;
\ep_\pm \ep_\mp = 2(1 \mp i \si p_u) = 2(1 \pm p_u)
$$
Solving from these restrictions explicitly, from
$$
p_u = \si_1 \sin{\th} \cos{\ph} + \si_2 \sin{\th} \sin{\ph} + \si_3 \cos{\th}
$$
we get (up to phase),
$$
\ep_\pm = \ep^\va_\pm \si_\va =
\lb \begin{array}{cc}
\sin(\th) & e^{-i \ph} \lp \mp 1 - \cos(\th) \rp\\
e^{i \ph} \lp \pm 1 - \cos(\th) \rp & -\sin(\th) \\
\end{array} \rb
= 2 \, \ch_\pm \ch_\mp^\da
$$
in which we see (using [[helicity identities]]) that $\ep_\pm$ can be related to right and left-[[helicity state]]s. Breaking this polarization back into its real components, we get
\begin{eqnarray}
\ep_1 &=& - \si_1 \cos{\th} \cos{\ph} - \si_2 \cos{\th} \sin{\ph} + \si_3 \sin{\th} \\
\ep_2 &=& \si_1 \sin{\ph} - \si_2 \cos{\ph}
\end{eqnarray}
Returning to Cl(1,3) using the [[Weyl representation|Dirac matrices]], we have
$$
\ep'_\pm = \ep'_\pm{}^{\!\mu} \ga_\mu = \ep_\pm^\va \ga_\va = -i \si_2 \otimes \ep_\pm
$$
A ''principal bundle'' or //''principal $G$-bundle''// is a [[fiber bundle]] with arbitrary base, $M$, and structure group, $G$, acting via [[left action|Lie group]] on typical fibers -- [[Lie group geometries|Lie group geometry]] homeomorphic to $G$. Unlike the case for other kinds of bundles, the [[Lie group]], $G$, may also act [[transitive]]ly on the fibers via the [[right action|Lie group]]. For a section, $C(x)$, transforming under the left action [[gauge transformation]], $C \mapsto C' = g(x) C$, the [[covariant derivative]] is
$$
\f{\na} C = \f{d} C + \f{A} C
$$
with the ''principal bundle [[connection]]'' (//gauge field//), $\f{A} = \f{dx^i} A_i{}^B T_B$, a 1-form over $M$ valued in the [[Lie algebra]] of $G$.
Any fiber element, $C$, at $t=0$ may be [[parallel transport]]ed to $C(t) = U(t) C$ along a [[path]] on the base by a parameter dependent element $U \in G$, the [[path holonomy]], $U=Pe^{-\int_0^t \f{A}}$, satisfying the path holonomy equation,
$$
\fr{d}{dt} U(t) = - \ve{v} \f{A} U
$$
Applying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),
$$
\f{\na} \f{\na} C = \f{d} \lp \f{d} C + \f{A} C \rp + \f{A} \lp \f{d} C + \f{A} C \rp
= \lp \f{d} \f{A} + \f{A} \f{A} \rp C
= \ff{F} C
$$
gives the ''principal bundle [[curvature]]'',
$$
\ff{F} = \f{d} \f{A} + \f{A} \f{A} = \f{d} \f{A} + \ha \lb \f{A}, \f{A} \rb
$$
a Lie algebra valued 2-form with components,
$$
\ff{F^B} = \f{d} \f{A^B} + \ha \f{A^C} \f{A^D} C_{CD}{}^B
$$
This expression for the curvature may alternatively be obtained from the [[holonomy]].
Under a gauge transformation, $C(x) \mapsto C'(x) = g(x) C(x)$, the covariant derivative changes to
\begin{eqnarray}
\f{\na'} C' &=& g \lp \f{\na} C \rp\\
\lp \f{d} g \rp C + g \f{d} C + \f{A'} g C &=& g \f{d} C + g \f{A} C\\
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'} = g \f{A} g^- + g \f{d} g^-
$$
An infinitesimal transformation, $g \simeq 1 + G^A T_A = 1 + G$, changes the connection to
$$
\f{A'} \simeq \f{A} - \f{d} G - \f{A} G + G \f{A} = \f{A} - \f{\na} G
$$
The curvature consequently transforms under a gauge transformation to
$$
\ff{F'} = \f{d} \f{A'} + \f{A'} \f{A'} = g \ff{F} g^- \simeq \ff{F} + \lb G, \ff{F} \rb
$$
The covariant derivative acting on a Lie algebra valued field (rather than a section) such as the curvature, transforming under the adjoint action, $\ff{F'} = g \ff{F} g^-$, is
$$
\f{\na} \ff{F} = \f{d} \ff{F} + \f{A} \ff{F} - \ff{F} \f{A} = \f{d} \ff{F} + \lb \f{A}, \ff{F} \rb
$$
A ''principle bundle'' is a [[principal bundle]] with strong moral fiber.
For a [[path]] designated by coordinates, $x^i(t)$, and parametrized by $t$ the [[velocity|tangent vector]] along the path with respect to this parameter is
\[ \ve{v} = v^i \ve{\pa_i} = \fr{dx^i}{dt} \ve{\pa_i} \]
and the magnitude of the velocity is
\[ \ll v \rl = \sqrt{\ll v \cdot v \rl} = \sqrt{\ll \lp \ve{v} \f{e} \rp \cdot \lp \ve{v} \f{e} \rp \rl} = \sqrt{\ll v^i v^j g_{ij} \rl} \]
using the [[frame]], $\f{e}$, to map the velocity into a [[rest frame]]. The change in ''proper time'', in seconds or other time [[units]], $T$, along the path is the integral along the path,
\[ \De \ta = \int \f{dt} \ll v \rl \]
The proper time describes how much time passes for a particle or observer moving along that path — in contrast, parameter time is not necessarily physically meaningful. In terms of parameter time, the proper time changes as $\fr{d\tau}{dt} = \ll v \rl$. If the path is parameterized (or reparameterized) by proper time so that $t(\tau)=\tau$, then the magnitude of the velocity with respect to proper time is one, $|v|=1$, and the parameter value, in $T$ units, marks out how a clock would read, traveling that path.
It is only possible to parameterize paths by proper time if their velocity is never null. For positive [[signature|Minkowski metric]], $\et_{00} = +1$, a ''timelike'' path satisfies $v \cdot v > 0$, a ''null'' path satisfies $v \cdot v = 0$, and a ''spacelike'' path satisfies $v \cdot v < 0$. This is reversed for negative signature. A null path, $|v|=0$, is lightlike and time does not pass for a particle or observer on that path. For a spacelike path, the proper time gives the ''spatial distance'' along the path, in "light seconds" or other spatial distance units, such as meters if the proper time is multiplied by the speed of light, $c$. Massive particles and observers only travel timelike paths, and massless particles only travel null paths.
A path between two points that extremizes proper time is a [[geodesic]].
The Clifford ''pseudoscalar'', or ''//Clifford volume element//'', is the grade $n$ [[Clifford basis element|Clifford basis elements]], formed by the (antisymmetric) product of the $n$ [[Clifford basis vectors]],
$$
\ga = \ga_0 \ga_1 \dots \ga_{n-1} = \fr{\ll \et \rl}{n!} \ep^{\al \be \dots \ga} \ga_{\al \be \dots \ga}
$$
(giving the relation to the [[permutation symbol]]). For a [[Clifford algebra]] of [[signature|Minkowski metric]] $(p,q)$ the pseudoscalar squares to
$$
\ga \ga = \lp -1 \rp^q \lp -1 \rp^{\fr{n \lp n-1 \rp}{2}} = \lp -1 \rp^{\ha (p-q)(p-q-1)}
$$
and so has the inverse
$$
\ga^- = \lp -1 \rp^{\ha (p-q)(p-q-1)} \ga
$$
The pseudoscalar commutes with all even [[Clifford grade]]d elements, $A^e \ga = \ga A^e$, and commutes or anticommutes with odd graded elements, dependent on overall dimension, $A^o \ga = (-1)^{n+1} \ga A^o$.
For [[Cl(1,3)]], $\ga \ga = -1$ and so $\ga^- = -\ga$. And, since $n=1+3=4$, this ''[[spacetime]] pseudoscalar'' anticommutes with odd Clifford grade elements.
A map, $\phi:x \mapsto y$, takes a point $x$ on a [[manifold]] $M$ to a point $y$ on the manifold $N$ (which may be $M$ itself). These points have coordinates $x^i$ and $y^j = \ph^j(x)$ in some local patches. When the map is smooth (continuously differentiable) the [[partial derivative]], $\fr{\pa y^j}{\pa x^i} = \pa_i \ph^j(x)$, is well defined. In this way $\phi$ induces a map, $\phi^*$, from any [[differential form]], $\f{a}$, at $y$ to a form at $x$ -- the ''pullback'' of $\f{a}$ along $\phi$,
$$
\f{\phi^*a} = \lb \f{\phi^*a} \rl_x = \phi^* \lb \f{a} \rl_y = \phi^* \f{a}= \phi^* \f{dy^j} a_j(y) = \f{dx^i} \lb \fr{\pa y^j}{\pa x^i} \rl_x a_j(y) = \lp \f{\pa} \ve{\ph} \rp \f{a}
$$
using the [[vector-form algebra]]. This generalizes to forms of any grade. $\phi$ also induces a map, $\phi_*$, from any [[tangent vector]], $\ve{v}$, at $x$ to a vector at $y$ -- the ''pushforward'' of $\ve{v}$ along $\phi$,
$$
\ve{\phi_*v} = \lb \ve{\phi_*v} \rl_y = \phi_* \lb \ve{v} \rl_x = \phi_* \ve{v}= \phi_* \ve{\fr{\pa}{\pa x^i}} v_i(x) = \ve{\fr{\pa}{\pa y^j}} \lb \fr{\pa y^j}{\pa x^i} \rl_x v_i(x) = \lp \ve{v} \f{\pa} \rp \ve{\ph}
$$
It is easy to confirm that $\ve{v} \lp \phi^* \f{a} \rp = \lp \phi_* \ve{v} \rp \f{a}$. Note that forms pull back and vectors push forward even if the map is not invertible or bijective. It is more natural for forms to pull back and for vectors to push forward under a map.
However, if the map is smooth and invertible it is a [[diffeomorphism]] and its inverse partial derivative, $\fr{\pa x^i}{\pa y^j} = \pa_j \ph^-i(y)$, is also well defined. For this kind of map, forms also push forward and vectors also pull back,
\begin{eqnarray}
\f{\phi_*a} &=& \lb \f{\phi_*a} \rl_y = \phi_* \lb \f{a} \rl_x = \phi_* \f{a} = \phi_* \f{dx^i} a_i(x) = \f{dy^j} \lb \fr{\pa x^i}{\pa y^j} \rl_y a_i(x) = \lp \f{\pa} \ve{\ph^-} \rp \f{a} \\
\ve{\phi^*v} &=& \lb \ve{\phi^*v} \rl_x = \phi^* \lb \ve{v} \rl_y = \phi^* \ve{v} = \phi^* \ve{\fr{\pa}{\pa y^j}} v_j(y) = \ve{\fr{\pa}{\pa x^i}} \lb \fr{\pa x^i}{\pa y^j} \rl_y v_j(y) = \lp \ve{v} \f{\pa} \rp \ve{\ph^-}
\end{eqnarray}
This also allows the pushforward and pullback to be defined for [[vector valued form]]s,
\begin{eqnarray}
\f{\ve{\phi_*A}} &=& \lb \f{\ve{\phi_*A}} \rl_y = \phi_* \lb \f{\ve{A}} \rl_x = \phi_* \f{\ve{A}} = \phi_* \lp \f{dx^i} A_i{}^j(x) \ve{\fr{\pa}{\pa x^j}} \rp =
\f{dy^j} \lb \fr{\pa x^i}{\pa y^j} \rl_y A_i{}^k(x) \lb \fr{\pa y^m}{\pa x^k} \rl_x \ve{\fr{\pa}{\pa y^m}}
= \lp \f{\pa} \ve{\ph^-} \rp \lp \f{\ve{A}} \f{\pa} \rp \ve{\ph} \\
\f{\ve{\phi^*A}} &=& \lb \f{\ve{\phi^*A}} \rl_x = \phi^* \lb \f{\ve{A}} \rl_y = \phi^* \f{\ve{A}} = \phi^* \lp \f{dy^i} A_i{}^j(y) \ve{\fr{\pa}{\pa y^j}} \rp =
\f{dx^i} \lb \fr{\pa y^j}{\pa x^i} \rl_x A_j{}^k(y) \lb \fr{\pa x^m}{\pa y^k} \rl_y \ve{\fr{\pa}{\pa x^m}}
= \lp \f{\pa} \ve{\ph} \rp \lp \f{\ve{A}} \f{\pa} \rp \ve{\ph^-}
\end{eqnarray}
Pullbacks and/or pushforwards can also be done for vector and form fields over manifolds by extending the operation over every manifold point.
Using the basis of normalized [[Dirac solutions]], the ''quantum Dirac spinor'' (or ''Dirac spinor operator'') can be written as a [[quantum field]],
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp
$$
in which $\ud{\hat{b}}_p^{\wedge/\vee \, \da}$ and $\ud{\hat{a}}_p^{\wedge/\vee}$ are [[creation and annihilation operators]] for the corresponding anti-fermion and fermion of [[momentum]] $p$. The factor of $(2E)$ gives [[Lorentz invariance of the measure]]. There is also a [[massless quantum Dirac spinor]] for the case $m=0$.
The [[Dirac adjoint]] of a [[quantum Dirac spinor]] field is the ''quantum Dirac spinor adjoint'',
$$
\ud{\hat{\bar{\Psi}}} = \ud{\hat{\Psi}}^\da \ga^0
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee \, \da} {\bar{u}}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{\wedge/\vee} {\bar{v}}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp
$$
in which $\ud{\hat{a}}_p^{\wedge/\vee \, \da}$ and $\ud{\hat{b}}_p^{\wedge/\vee}$ are [[creation and annihilation operators]] for the corresponding fermion and anti-fermion, and ${\bar{u}}_p^{\wedge/\vee}$ and ${\bar{v}}_p^{\wedge/\vee}$ are the Dirac adjoints of [[Dirac solutions]].
Just as a [[Dirac spinor]] has an associated [[quantum Dirac spinor]] field operator, a [[Majorana spinor]] has a ''quantum Majorana spinor'' field operator,
$$
\ud{\hat{\Ps}}^M = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{a}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp
$$
This is simply a quantum Dirac spinor, with the same [[Dirac solutions]], but with $\ud{\hat{b}}_p^{\wedge/\vee} = \ud{\hat{a}}_p^{\wedge/\vee}$, so the quantum Majorana spinor equals its [[charge conjugate]] and has two particle degrees of freedom instead of four.
Alternatively, a massive quantum Majorana spinor can be described as a massive [[quantum Weyl spinor]] by taking the left or right chiral part of the above expression, so
$$
\ud{\hat{\ps}}_L^M = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_{Lp}^{\wedge/\vee} e^{-i p_\mu x^\mu} + \ud{\hat{a}}_p^{\wedge/\vee \, \da} {v}_{Lp}^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp
$$
in which
$$
{u}_{Lp}^{\wedge/\vee} = \fr{1}{\sqrt{2m}} \sqrt{\lp E - p^\va \si_\va \rp} \, \ch^{\wedge/\vee}
\s \s
{v}_{Lp}^{\wedge/\vee} = \fr{1}{\sqrt{2m}} \sqrt{\lp E - p^\va \si_\va \rp} \, \xi^{\wedge/\vee}
$$
which is in agreement with quantization using normalized [[Majorana solutions]].
If we consider that particles are fundamentally Weyl fermions, a ''quantum Majorana-Weyl spinor'' can be expressed using [[helicity state]]s as
$$
\ud{\hat{\ps}}_L = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^L \ch_- e^{-i p_\mu x^\mu} - \ud{\hat{a}}_p^{R \da} \ch_- e^{+i p_\mu x^\mu} \rp
$$
Using the basis of [[Weyl solutions]], which incorporate [[helicity state]]s, $\ch_\pm$, the left-handed ''quantum Weyl spinor'' (or ''Weyl spinor operator'') corresponding to a left-chiral fermion can be written as a [[quantum field]],
$$
\ud{\hat{\ps}}_L = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^L \ch_- e^{-i p_\mu x^\mu} - \ud{\hat{b}}_p^{R \da} \ch_- e^{+i p_\mu x^\mu} \rp
$$
in which $\ud{\hat{a}}_p^L$ and $\ud{\hat{b}}_p^{R \da}$ are annihilation and creation operators for the corresponding left-handed fermion and right-handed anti-fermion of [[momentum]] $p$.
A left-handed quantum Weyl spinor can be thought of as the left-chiral half of a [[massless quantum Dirac spinor]]. A left-handed quantum Weyl spinor does not have a [[charge conjugate]] or a [[parity conjugate]] if it does not have a right-handed partner -- if it does, the two partners are parts of a [[massless quantum Dirac spinor]] which does have charge, parity, and time [[conjugates|conjugates of a massless quantum Dirac spinor]]. However, the charge-parity conjugate of a left-handed quantum Weyl spinor fermion always exists and is the right-handed anti-fermion with opposite momentum. The [[time conjugate]] of annihilating a left-handed Weyl fermion is creating the left-handed fermion with opposite momentum. Combining the [[conjugates of a massless quantum Dirac spinor]], the CPT-conjugate of annihilating a left-handed Weyl fermion is creating the right-handed anti-fermion with the same momentum.
The right-handed quantum Weyl spinor CPT conjugate partner to a left-handed quantum Weyl spinor is
$$
\lp \ud{\hat{\ps}}_L \rp^{CPT} = \ud{\hat{\ps}}_R = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp - i \ud{\hat{b}}_{p}^R \ch_+ e^{-i p_\mu x^\mu} + i \ud{\hat{a}}_{p}^{L \da} \ch_+ e^{+i p_\mu x^\mu} \rp
$$
// check that //
If [[spin(11,3) GraviGUT fermions]], with the $\om_t$ quantum number switched for anti-particle annihilation operators, and magically converted from real to imaginary, are fit in the compact real form of [[e8]] then they are in the $8^+ \times 8^+$ representation space. This requires the introduction of another root coordinate, $p$, with the fermions and anti-fermions having $p = \mp 1$; and so the following e8 root coordinates:
$$
\begin{array}{|l|cccccc|}
\hline
& \om_t & \om_s & u & v & p & x & y & z \\
\hline
\nu_{eL}^{\wedge/\vee} & \mp & \pm & - & + & - & - & - & - \\
\nu_{eR}^{\wedge/\vee} & \pm & \pm & + & + & - & - & - & - \\
e_L^{\wedge/\vee} & \mp & \pm & + & - & - & - & - & - \\
e_R^{\wedge/\vee} & \pm & \pm & - & - & - & - & - & - \\
u^{r\wedge/\vee}_L & \mp & \pm & - & + & - & - & + & + \\
u^{r\wedge/\vee}_R & \pm & \pm & + & + & - & - & + & + \\
d^{r\wedge/\vee}_L & \mp & \pm & + & - & - & - & + & + \\
d^{r\wedge/\vee}_R & \pm & \pm & - & - & - & - & + & + \\
u^{g\wedge/\vee}_L & \mp & \pm & - & + & - & + & - & + \\
u^{g\wedge/\vee}_R & \pm & \pm & + & + & - & + & - & + \\
d^{g\wedge/\vee}_L & \mp & \pm & + & - & - & + & - & + \\
d^{g\wedge/\vee}_R & \pm & \pm & - & - & - & + & - & + \\
u^{b\wedge/\vee}_L & \mp & \pm & - & + & - & + & + & - \\
u^{b\wedge/\vee}_R & \pm & \pm & + & + & - & + & + & - \\
d^{b\wedge/\vee}_L & \mp & \pm & + & - & - & + & + & - \\
d^{b\wedge/\vee}_R & \pm & \pm & - & - & - & + & + & - \\
\hline
\end{array}
\s\s\;
\begin{array}{|l|cccccc|}
\hline
& \om_t & \om_s & u & v & p & x & y & z \\
\hline
\bar{\nu}_{eR}^{\wedge/\vee} & \mp & \pm & + & - & + & + & + & + \\
\bar{\nu}_{eL}^{\wedge/\vee} & \pm & \pm & - & - & + & + & + & + \\
\bar{e}_R^{\wedge/\vee} & \mp & \pm & - & + & + & + & + & + \\
\bar{e}_L^{\wedge/\vee} & \pm & \pm & + & + & + & + & + & + \\
\bar{u}^{r\wedge/\vee}_R & \mp & \pm & + & - & + & + & - & - \\
\bar{u}^{r\wedge/\vee}_L & \pm & \pm & - & - & + & + & - & - \\
\bar{d}^{r\wedge/\vee}_R & \mp & \pm & - & + & + & + & - & - \\
\bar{d}^{r\wedge/\vee}_L & \pm & \pm & + & + & + & + & - & - \\
\bar{u}^{g\wedge/\vee}_R & \mp & \pm & + & - & + & - & + & - \\
\bar{u}^{g\wedge/\vee}_L & \pm & \pm & - & - & + & - & + & - \\
\bar{d}^{g\wedge/\vee}_R & \mp & \pm & - & + & + & - & + & - \\
\bar{d}^{g\wedge/\vee}_L & \pm & \pm & + & + & + & - & + & - \\
\bar{u}^{b\wedge/\vee}_R & \mp & \pm & + & - & + & - & - & + \\
\bar{u}^{b\wedge/\vee}_L & \pm & \pm & - & - & + & - & - & + \\
\bar{d}^{b\wedge/\vee}_R & \mp & \pm & - & + & + & - & - & + \\
\bar{d}^{b\wedge/\vee}_L & \pm & \pm & + & + & + & - & - & + \\
\hline
\end{array}
$$
with fermions having $p=-1$ and anti-fermions $p=+1$. If these are the quantum numbers of annihilation operators, then there will be a complementary set of quantum numbers for the corresponding creation operators, with all quantum numbers opposite, in agreement with the [[spin quantum numbers of a massless quantum Dirac spinor]]. The existence of [[e8 triality|e8 triality decomposition]] suggests that the second and third generation fermions could be in $8^- \times 8^-$ and $8^v \times 8^v$.
Using the basis of normalized [[Maxwell solutions]] in Weyl gauge, the ''quantum [[electromagnetic field]]'' in a [[rest frame]] can be written as
$$
\hat{A} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \hat{a}_{p}^{\pm} \, \ep'_{\pm} e^{-i p_\mu x^\mu} + \hat{a}_{p}^{\pm \, \da} \, \ep'_{\pm} e^{+i p_\mu x^\mu} \rp
$$
in which $\ep'_{\pm}(p)$ are momentum-dependent spatial [[Cl(1,3)]] circular [[polarization]] basis vectors, $p_\mu$ are the [[momentum]] components, and $\hat{a}_p^{\pm \, \da}$ and $\hat{a}_p^{\pm}$ are [[creation and annihilation operators]] for the corresponding right or left-handed [[quantum field]] -- the ''photon''.
Within the [[Dirac Lagrangian]] the interaction term between the [[electromagnetic field]] and a [[Dirac spinor]] is ${\cal L}_I = i \bar{\Ps} \ve{e} \f{A} \Ps = i \bar{\Ps} A \Ps$. In a [[rest frame]], a positive energy [[Maxwell solution|Maxwell solutions]] circularly [[polarized|polarization]] plane wave (a ''classical photon''), $A=\ep'_\pm e^{-i p' x}$, acts on a positive energy [[massless Dirac solution|massless quantum Dirac spinor]] plane wave (a ''classical fermion''), $\Ps = u_p^{L/R} e^{-i p x}$, to give
$$
A \Ps = \ep'_\pm e^{-i p' x} \, u_p^{L/R} e^{-i p x}
= \Ps''_{R/L} e^{-i (p' + p) x}
$$
in which $\ep'_\pm = -i\si_2 \otimes \ep_\pm(p'_u)$ are the circular [[polarization]]s, and $\Ps''_{R/L}$ consists of right or left [[chiral]] [[Weyl spinor]] parts, $\ps''_{R/L}$. Note that, from interacting with a photon of either polarization, a massless Dirac fermion changes chirality from left to right or right to left handed. The helicity eigenstate resulting from the interaction is
$$
\ps''_{R/L} = \mp \ep_\pm(p'_u) \, \ch_\mp(p_u) = \pm \ch_\pm(p'_u) \, \ch_\mp(p'_u)^\da \, \ch_\mp(p_u)
$$
using [[helicity state]]s, $\ch_\pm$.The polarization and helicity of the initial photon and massless fermion are either opposite or matching, giving
$$
\ps''_{R/L} = \mp \ep_\pm(p'_u) \, \ch_\mp(p_u)
= \pm \lp e^{\pm i \fr{\ph+\ph'}{2}} \cos \fr{\th'}{2} \cos \fr{\th}{2} + e^{\mp i \fr{\ph+\ph'}{2}} \sin \fr{\th'}{2} \sin \fr{\th}{2} \rp \ch_\pm(p'_u)
$$
$$
\ps''_{L/R} = \pm \ep_\pm(p'_u) \, \ch_\pm(p_u)
= - \lp e^{\pm i \fr{\ph+\ph'}{2}} \cos \fr{\th'}{2} \sin \fr{\th}{2} - e^{\mp i \fr{\ph+\ph'}{2}} \sin \fr{\th'}{2} \cos \fr{\th}{2} \rp \ch_\pm(p'_u)
$$
If the initial fermion has $p_u = \si_3$ and the photon an angle $\th'$ to this, $p'_u = \cos \th' \, \si_3 + \sin \th' \, \si_1$, which we can always [[Clifford rotate|Clifford rotation]] to any other configuration, then the resulting chiral fermion is
$$
\ps''_{R/L} = \mp \ep_\pm(p'_u) \, \ch_\mp(p_u)
= \pm \cos \fr{\th'}{2} \, \ch_\pm(p'_u)
$$
$$
\ps''_{L/R} = \pm \ep_\pm(p'_u) \, \ch_\pm(p_u)
= \sin \fr{\th'}{2} \, \ch_\pm(p'_u)
$$
Using [[helicity notation]] the possible interactions are
$$
a] = b] [b \, c\!\!>
\s
a] = b] [b \, c]
\s
a\!\!>\, = b\!\!> <\!\!b \, c\!\!>
\s
a\!\!>\, = b\!\!> <\!\!b \, c]
$$
Note that this calculation just gives a feel for field interactions, since a massless photon and massless fermion cannot combine to produce a massless fermion without violating conservation of [[momentum]] or angular momentum. What actually happens is described by a QFT interaction, in which two photons interact with an incoming and outgoing fermion.
We're not exactly sure how to do quantum field theory in curved spacetime, but we can try doing it in a [[rest frame]] as an approximation valid locally. Using a basis of normalized solutions, such as $\Ph^\pm_p(x) = e^{\pm i p_\mu x^\mu} \!$, to a relativistic field equation, such as the ''Klein-Gordon field equation'',
$$
0 = ( \pa_t^2 - d_s \cdot d_s ) \, \Ph(x) + m^2 \Ph
$$
a ''quantum field'' (or ''quantum field operator'') can be written as
$$
\hat{\Ph} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \hat{a}_p \, e^{-i p_\mu x^\mu} + \hat{a}_p^\da \, e^{+i p_\mu x^\mu} \rp
$$
in which $\hat{a}_p^\da$ and $\hat{a}_p$ are [[creation and annihilation operators]] for the corresponding boson of that [[momentum]]. In essence, the [[Fourier transform]] pairs for the field get replaced with quantum operators. The factor of $(2E)$ ensures the measure is Lorentz invariant.
A quantum field operator is a [[unitary operator|unitary representation]] on an infinite-dimensional [[Fock space]], similar to the quantum position operator of the [[quantum harmonic oscillator]].
A ''quantum harmonic oscillator'' can be described by starting with a classical simple harmonic oscillator, described by the equation of motion,
$$
m \fr{d^2}{dt^2} x = - k \, x
$$
and following the prescription of ''quantization''. The position is converted to the [[quantum position operator|wavefunction]], $\hat{x}$, on a [[Hilbert space]] of quantum states. The Hamiltonian is also converted to an operator,
$$
\hat{H} = \fr{1}{2m} \hat{p}^2 + \fr{k}{2} \hat{x}^2 = \hbar \om \lp \hat{a}^\da \hat{a} + \ha \rp = \hbar \om \lp \hat{N} + \ha \rp
$$
in which $\hbar$ is Planck's constant, $\om = \sqrt{k/m}$ is the ''oscillator frequency'', and the position and momentum operators can be written in terms of [[creation and annihilation operators]] as
$$
\begin{array}{rcl}
\hat{x} &\!\!=\!\!& \sqrt{\fr{\hbar}{2m\om}} \lp \hat{a}^\da + \hat{a} \rp \\
\hat{p} &\!\!=\!\!& i \sqrt{\fr{\hbar m \om}{2}} \lp \hat{a}^\da - \hat{a} \rp \\
\end{array}
$$
which act on the harmonic energy states, $| n \rangle$. These [[Hermitian]] operators satisfy the canonical commutation relation, $\lb \hat{x}, \hat{p} \rb = i \hbar$. The ground state, $| 0 \rangle$, has energy $E_0 = \ha \hbar \om$, and non-zero ground state expectation values such as
$$
\langle 0 | \hat{x}^2 | 0 \rangle = \fr{\hbar}{2m\om}
$$
which can be calculated from the explicit form of the ground state [[wavefunction]] in the position basis,
$$
\ps_0(x) = \langle x | 0 \rangle = \lp \fr{m \om}{\pi \hbar} \rp^{\fr{1}{4}} e^{-\fr{m \om x^2}{2 \hbar}}
$$
From applying the creation operator, the first excited state is
\begin{eqnarray}
\ps_1(x) &=& \langle x | 1 \rangle = \langle x | a^\da | 0 \rangle
= \langle x | \sqrt{\fr{m \om}{2 \hbar}} \hat{x} - i \sqrt{\fr{1}{2 \hbar m \om}} \hat{p} | 0 \rangle \\
&=& \lp \sqrt{\fr{m \om}{2 \hbar}} x - \sqrt{\fr{\hbar}{2 m \om}} \fr{d}{dx}\rp \ps_0(x)
= \sqrt{2} \lp \fr{m \om}{\pi \hbar} \rp^{\fr{3}{4}} x e^{-\fr{m \om x^2}{2 \hbar}}
\end{eqnarray}
Alternatively, the framework of [[quantum mechanics]], with a Hilbert space spanned by basis states, $| n \rangle$, along with the above Hamiltonian and creation and annihilation operators, can be considered as a fundamental structure.
The time evolution of the harmonic energy states is, by the [[Schrödinger equation|quantum mechanics]],
$$
\fr{d}{dt} | n(t) \rangle = -i \om \lp n + \ha \rp \, | n(t) \rangle
$$
and therefore
$$
| n(t) \rangle = e^{-i \om \lp n + \ha \rp t} | n(0) \rangle
$$
Or, in the [[Heisenberg picture]], the time evolution of the creation and annihillation operators is
$$
\fr{d}{dt} \hat{a}_t = \fr{i}{\hbar} \lb \hat{H}, \hat{a}_t \rb = - i \om \hat{a}_t
\s\;\;
\fr{d}{dt} \hat{a}^\da_t = \fr{i}{\hbar} \lb \hat{H}, \hat{a}^\da_t \rb = + i \om \hat{a}^\da_t
$$
and therefore
$$
\hat{a}_t = e^{-i \om t} \hat{a}
\s\;\;
\hat{a}^\da_t = e^{+i \om t} \hat{a}^\da
$$
In ''quantum mechanics'', a pure ''quantum state'', $\ps$, corresponds to a unit vector called a [[ket]], $| \ps \rangle \in \mathcal{H}$ satisfying $\langle \ps | \ps \rangle = 1$, in a complex [[Hilbert space]]. The state actually corresponds to the equivalence class of kets related by arbitrary complex phase (or a ''ray'' in ''projective Hilbert space''), but we usually simply describe a state by a ket and understand there to be an arbitrary global phase, $| \ps \rangle \sim e^{i \ph} | \ps \rangle$.
An ''observable'' corresponds to a [[Hermitian]] operator, $\hat{O}$, on the Hilbert space, which returns a real [[eigen]]value, $\la$, as a ''measurement'' result, with the state then a (normalized) eigenvector,
$$
\hat{O} \, | \ps_\la \rangle = \la \, | \ps_\la \rangle
$$
According to the ''Born rule'', the complex ''amplitude'' (or ''quantum amplitude'' or ''probability amplitude'') of observing a value $\la$ while in a state $\ps$ is $\mathcal{A} = \langle \ps_\la | \ps \rangle$, and the ''probability'' (between $0$ and $1$) of observing a value $\la$ is $\mathcal{P} = \mathcal{A}^* \mathcal{A} = | \langle \ps_\la | \ps \rangle |^2$. The ''expectation value'' of an observable in some state is $\langle \ps | \hat{O} | \ps \rangle$.
The ''energy'', $E$, of a quantum state is an eigenvalue of the ''Hamiltonian'' operator, $\hat{H}$; so $\hat{H} \, | \ps_E \rangle = E \, | \ps_E \rangle$ for energy eigenstates. A state changes under ''time evolution'' according to the ''Schrödinger equation'',
$$
\fr{d}{dt} | \ps(t) \rangle = \fr{1}{i \hbar} \hat{H} \, | \ps(t) \rangle
$$
so $| \ps_E(t) \rangle = e^{-i\fr{E}{\hbar}t} \, | \ps_E(0) \rangle$ for energy eigenstates, in which $\hbar$ is ''Planck's constant''. Integrating the Schrödinger equation, an arbitrary pure quantum state evolves by a [[unitary]] ''time evolution operator'', $| \ps(t) \rangle = \hat{U}_t | \ps(0) \rangle$.
Any ''quantum symmetry'' of a quantum state necessarily corresponds to some [[unitary]] transformation, $| \ps \rangle \mapsto | \ps' \rangle = \hat{U} \, | \ps \rangle$, in order to maintain the unit norm,
$$
1 = \langle \ps' | \ps' \rangle = \langle \ps | \hat{U}^\da \hat{U} | \ps \rangle
$$
The ''quaternions'', $\mathbb{H}$, are a four-dimensional [[division algebra]], spanned by four basis elements, $\{ e_0 = 1, e_1 = i, e_2 = j, e_3 = k \}$. The quaternion identity element (spanning the ''real part'') is $e_0=1$, and the other three (spanning the ''vector part'' or //''imaginary part''//) can be thought of as different imaginary directions, squaring to $-1$. Quaternionic multiplication is non-commutative and associative. The multiplication table for the basis elements is
| $\; e_0 \;$ | $e_1$ | $e_2$ | $e_3$ |
| $e_1$ | $-e_0$ | $e_3$ | $-e_2$ |
| $e_2$ | $-e_3$ | $-e_0$ | $e_1$ |
| $e_3$ | $e_2$ | $-e_1$ | $-e_0$ |
which can be written using a ''quaternion multiplication coefficient matrix'' as
$$
e_a e_b = M_{ab}{}^c e_c
$$
so, for example, $e_1 e_2 = e_3$ and $M_{12}{}^3 = 1$. ''Quaternionic conjugation'' is given by
$$
\os{e}_0 = e_{\os{0}} = e_0 \;\;\;\;\; \os{e}_1 = e_{\os{1}} = -e_1 \;\;\;\; \os{e}_2 = e_{\os{2}} = -e_2 \;\;\;\; \os{e}_3 = e_{\os{3}} = -e_3
$$
and satisfies $\widetilde{(e_a e_b)} = e_{\os{b}} e_{\os{a}}$, so we have $M_{ab}{}^c = M_{\os{b} \os{a}}{}^{\os{c}}$.
Multiplying a quaternion by its conjugate gives a real, its norm,
$$
\os{q} \, q = q^a q^b \os{e}_a e_b =
q^0 q^0 + q^1 q^1 + q^2 q^2 + q^3 q^3
= q \tilde{q} = | q |^2
$$
allowing us to calculate the inverse of any nonzero quaternion, $q^- = \frac{\tilde{q}}{|q|^2}$. Note that $q q$ is real only if $q$'s real part or vector part is zero. The ''quaternion [[metric]]'' is defined as
$$
(u,v) = \ha \lp \os{u} v + \os{v} u \rp = u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3
$$
so
$$
(e_a, e_b) = \ha ( \os{e}_a e_b + \os{e}_b e_a ) = n_{ab}
$$
with $n_{ab} = \de_{ab}$ for the quaternions. This metric can be used to raise or lower quaternion indices (which has no effect for the quaternions, but matters for the [[split-quaternion]]s, for which $n'_{ab} = \text{diag}(+1,+1,-1,-1)$). The combination of metric and quaternionic conjugates, $\os{e}^a = n^{ab} \os{e}_b \in \tilde{\mathbb{H}}$, are the [[duals|dual space]] to the quaternions, $\os{e}^a e_b = \de^a_b$. The matrix $\de^\os{b}_a = \text{diag}(+1,-1,-1,-1)$ can be used to twiddle or untwiddle indices, such as $M^\os{b}{}_{ac} = \de^\os{b}_d M^d{}_{ac} = n^{\os{b}d} M_{dac}$.
The quaternions have a matrix representation relating to generalized [[Pauli matrices]], with
$$
e_0 = \si_0 \;\;\;\;\; \s e_A = - i \, \si_A
$$
which we can see reproduces the multiplication table, thanks to
$$
e_A e_B = - \de_{AB} \, e_0 + \ep_{ABC} \, e_C = M_{ABc} \, e_c
$$
and allows a quaternion to be expressed as an element of $\mathbb{C}(2)$,
$$
q = q^a e_a =
\lb \begin{array}{cc}
v^0 - i v^3 & - i v^1 - v^2 \\
- i v^1 + v^2 & v^0 + i v^3 \\
\end{array} \rb
$$
Since the Pauli matrices are [[Hermitian]], quaternionic conjugation corresponds to Hermitian conjugation of the representative matrix, $\tilde{q} = q^\da = - \ep \, q^T \ep$, or of the [[2D matrix conjugate|determinant]]. Also, the norm is the [[determinant]], $|q|^2 = {\rm Det}(q)$.
The quaternions also have a real matrix representation that, similar to the [[adjoint representation]], comes from their multiplication table,
$$
(e_c)^a{}_b = M^\os{a}{}_{bc}
$$
which relates to a [[realification|realify]] of the Pauli matrices. This representation is faithful, satisfying
$$
(e_a)^d{}_e (e_b)^e{}_f = M_{ab}{}^c (e_c)^d{}_f
$$
but $\tilde{q} \ne q^\da$.
There are two [[similar|Dirac matrices]] real, $8 \times 8$, [[chiral]] [[representation|Clifford matrix representation]]s of the [[Cl(4)]] [[Clifford algebra]] related to [[quaternion]]s. Via [[Clifford division algebra representation]] there are the ''direct quaternionic representation'', and the ''cyclic quaternionic representation''. Explicitly, from the quaternion multiplication coefficient matrix, the cyclic quaternionic representation of the chiral Dirac matrix is given by
$$
v^c \Ga_c =
\lb \begin{array}{cccc}
v^0 & -v^1 & -v^2 & -v^3 \\
-v^1 & -v^0 & v^3 & -v^2 \\
-v^2 & -v^3 & -v^0 & v^1 \\
-v^3 & v^2 & -v^1 & -v^0 \\
\end{array} \rb
$$
The resulting [[pseudoscalar]] is $\ga' = \ga'_0 \ga'_1 \ga'_2 \ga'_3 = \si_3 \otimes 1$.
When describing the [[Lie algebra structure]] of a [[real form]] of a [[Lie algebra]], we find that the Cartan-Weyl basis, $\{ T^\mathfrak{C}_I, V_{\pm i} \}$, consists of complex generators. If the Cartan subalgebra is compact (or split), these generators can be broken into real and imaginary (or positive and negative) parts, or if the Cartan subalgebra is mixed they can be broken into four parts, with complex and real [[structures on a real representation space]], $J^C$ and $J^S$, acting on these. The ''real Cartan-Weyl basis'' is then $\{T^\mathfrak{C}_I, V^{{\mathbb R}p}_{j}, V^{{\mathbb I}p}_{k}, V^{{\mathbb R}n}_{l},V^{{\mathbb I}n}_{m} \}$, with indices $\{j,k,l,m\}$ spanning different subsets of the range of $i$, which was half the dimension of the Lie algebra minus its Cartan subalgebra. The root vectors can be systematically decomposed into these orthogonal real spaces.
Each positive root basis vector, $V_{+i}$, corresponds to a positive root, $\al_i$. If $\al_l = \al_l^{\mathbb R} \in {\mathbb R}$ then $V_{\pm l} = V^{{\mathbb R}p}_{l} \pm V^{{\mathbb R}n}_{l}$. If $\al_k = i \al_k^{\mathbb I} \in {\mathbb I}$ then $V_{\pm k} = V^{{\mathbb R}p}_{k} \mp i \, V^{{\mathbb I}p}_{k}$. Otherwise, with $\al_m = \al_m^{\mathbb R} + i \al_m^{\mathbb I} \in {\mathbb C}$, we have
$$
V_{\pm m} = V^{{\mathbb R}p}_{m} \mp i \, V^{{\mathbb I}p}_{m} \pm V^{{\mathbb R}n}_{m} + i \, V^{{\mathbb I}n}_{m}
$$
The four parts of each complex root vector are thus
$$
\begin{array}{rcl}
V^{{\mathbb R}p}_{m} \!\!&\!\!=\!\!&\!\! \fr{1}{4} \lp V_{+m} + V_{-m} + V_{+m}^* + V_{-m}^* \rp \\
V^{{\mathbb I}p}_{m} \!\!&\!\!=\!\!&\!\! \fr{i}{4} \lp V_{+m} - V_{-m} - V_{+m}^* + V_{-m}^* \rp \\
V^{{\mathbb R}n}_{m} \!\!&\!\!=\!\!&\!\! \fr{1}{4} \lp V_{+m} - V_{-m} + V_{+m}^* - V_{-m}^* \rp \\
V^{{\mathbb I}n}_{m} \!\!&\!\!=\!\!&\!\! \fr{1}{4i} \lp V_{+m} + V_{-m} - V_{+m}^* - V_{-m}^* \rp \\
\end{array}
$$
and the (real) Lie brackets with these and the Cartan basis elements are
$$
\begin{array}{rcl}
\lb T^\mathfrak{C}_I, V^{{\mathbb R}p}_{m} \rb \!\!&\!\!=\!\!&\!\! \al_m^{\mathbb R} V^{{\mathbb R}n}_{m} + \al_m^{\mathbb I} V^{{\mathbb I}p}_{m} \\
\lb T^\mathfrak{C}_I, V^{{\mathbb I}p}_{m} \rb \!\!&\!\!=\!\!&\!\! - \al_m^{\mathbb R} V^{{\mathbb I}n}_{m} - \al_m^{\mathbb I} V^{{\mathbb R}p}_{m} \\
\lb T^\mathfrak{C}_I, V^{{\mathbb R}n}_{m} \rb \!\!&\!\!=\!\!&\!\! \al_m^{\mathbb R} V^{{\mathbb R}p}_{m} - \al_m^{\mathbb I} V^{{\mathbb I}n}_{m} \\
\lb T^\mathfrak{C}_I, V^{{\mathbb I}n}_{m} \rb \!\!&\!\!=\!\!&\!\! - \al_m^{\mathbb R} V^{{\mathbb I}p}_{m} + \al_m^{\mathbb I} V^{{\mathbb R}n}_{m} \\
\end{array}
$$
in which we can see the $J^C$ and $J^S$ structures in play, such as
$$
J^C V^{{\mathbb R}p}_{m} = V^{{\mathbb I}p}_{m} \s J^C V^{{\mathbb I}p}_{m} = - V^{{\mathbb R}p}_{m}
\s
J^S V^{{\mathbb R}p}_{m} = V^{{\mathbb R}n}_{m} \s J^S V^{{\mathbb R}n}_{m} = V^{{\mathbb R}p}_{m}
$$
Converting a real Lie algebra (or representation space) to a complex Cartan-Weyl basis and then to a real Cartan-Weyl basis is a way of decomposing a Lie algebra into a preferred real presentation consistent with its root system and a complex structure.
A [[Lie algebra]], $\mathfrak{g}$, has a ''real form'', $\mathfrak{g}_{\mathbb R}$, if its complex form has a [[complex structure]] giving a $\mathfrak{g}_{\mathbb R}$ [[subalgebra]] as a real half, allowing the description:
$$
\mathfrak{g}_{\mathbb C} = \mathfrak{g}_{\mathbb R} + J \, \mathfrak{g}'_{\mathbb R}
$$
A ''real Lie algebra'' can have all real [[structure constants|Lie algebra structure]].
A real Lie algebra has a [[Cartan decomposition]] into an even part (with negative definite [[Killing form]]) and an odd part (with positive definite Killing form),
$$
\mathfrak{g}_{\mathbb R} = \mathfrak{h}_+ + \mathfrak{h}_-
$$
The ''compact real form'' of the Lie algebra has negative definite Killing form, and can be constructed from any real form as $\mathfrak{h}_+ + i \, \mathfrak{h}_-$.
The core of a real algebra consists of interlinked [[sl(2)]]'s and [[su(2)]]'s.
Real and complex forms of a Lie algebra share [[representation space]]s.
We can ''realify'' a [[Clifford matrix representation]] by replacing the [[complex structure]] with a [[skew]] matrix, doubling the matrix size. For example, the [[Dirac matrices]],
\begin{eqnarray}
\ga_0 &=& \;\;\;\, \si_1 \otimes \si_0 \\
\ga_1 &=& -i \si_2 \otimes \si_1 \\
\ga_2 &=& -i \si_2 \otimes \si_2 \\
\ga_3 &=& -i \si_2 \otimes \si_3
\end{eqnarray}
let a [[Cl(1,3)]] Clifford vector be written as
$$
v = v^\mu \ga_\mu =
\lb \ba{cccc}
0 & 0 & v^0-v^3 & -v^1+iv^2 \\
0 & 0 & -v^1-iv^2 & v^0+v^3 \\
v^0+v^3 & v^1-iv^2 & 0 & 0 \\
v^1+iv^2 & v^0-v^3 & 0 & 0
\ea \rb
$$
Their ''realification'' is
\begin{eqnarray}
\ga_0 &=& \;\;\;\, \si_1 \otimes \si_0 \otimes \si_0 \\
\ga_1 &=& -i \si_2 \otimes \si_1 \otimes \si_0 \\
\ga_2 &=& \;\;\, i \si_2 \otimes \si_2 \otimes \si_2 \\
\ga_3 &=& -i \si_2 \otimes \si_3 \otimes \si_0
\end{eqnarray}
with a vector written as
$$
v = v^\mu \ga_\mu =
\lb \ba{cccccccc}
0 & 0 & 0 & 0 & v^0-v^3 & 0 & -v^1 & -v^2 \\
0 & 0 & 0 & 0 & 0 & v^0-v^3 & v^2 & -v^1 \\
0 & 0 & 0 & 0 & -v^1 & v^2 & v^0+v^3 & 0 \\
0 & 0 & 0 & 0 & -v^2 & -v^1 & 0 & v^0+v^3 \\
v^0+v^3 & 0 & v^1 & v^2 & 0 & 0 & 0 & 0 \\
0 & v^0+v^3 & -v^2 & v^1 & 0 & 0 & 0 & 0 \\
v^1 & -v^2 & v^0-v^3 & 0 & 0 & 0 & 0 & 0 \\
v^2 & v^1 & 0 & v^0-v^3 & 0 & 0 & 0 & 0 \\
\ea \rb
$$
which can act on a ''realified [[Dirac spinor]]'',
$$
\Ps^r = \lb
\ps_{L \mathbb{R}}^\wedge \;
\ps_{L \mathbb{I}}^\wedge \;
\ps_{L \mathbb{R}}^\vee \;
\ps_{L \mathbb{I}}^\vee \;
\ps_{R \mathbb{R}}^\wedge \;
\ps_{R \mathbb{I}}^\wedge \;
\ps_{R \mathbb{R}}^\vee \;
\ps_{R \mathbb{I}}^\vee
\rb^T
$$
One can also ''complexify'' real things by identifying a complex structure and converting it to $i$.
Sometimes for a [[principal bundle]], with [[entire space|fiber bundle]], $E$, and structure group, $G$, there is a [[subgroup]], $H \subset G$, such that the [[fiber bundle]] $E/H$ (with fibers homeomorphic to the [[coset]] space $G/H$) admits a global section. Such a section is then a ''reduction of the structure group'' of the bundle from $G$ to $H$. The (fiberwise) inverse image of the values of this section form a subbundle of $E$ that is a principal $H$-bundle.
Ref:
http://en.wikipedia.org/wiki/Principal_bundle#Reduction_of_the_structure_group
A [[Lie group]], $G$, has a [[Lie algebra]], $\mathfrak{g}$, spanned by $n$ generators, $T_I$. A [[subgroup]], $H \subset G$, has its own set of $n_H$ generators, $H_P \in \mathfrak{h}$, which can be chosen from some of $G$'s, $H_P = T_P$ (with $P$-series indices running from $1$ to $n_H$). The $n_S = n_G - n_H$ remaining generators, $K_A = T_A$, are the ''coset generators'' -- so $\{T_I\} = \{H_P\} \oplus \{K_A\}$. Let the [[vector space]] spanned by the coset generators be labeled $\mathfrak{g}/\mathfrak{h}$ (even though the [[homogeneous space]], $S=G/H$, isn't necessarily a group). The subgroup $H$ is ''reductive'' in $G$ (or subalgebra $\mathfrak{h}$ is ''reductive'' in $\mathfrak{g}$, or the decomposition of $\mathfrak{g}$ is ''reductive'') iff the Lie algebra decomposition,
$$
\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{g}/\mathfrak{h}
$$
is invariant under the adjoint action of $\mathfrak{h}$,
$$
\begin{eqnarray}
{\rm Ad}_{\mathfrak{h}} \mathfrak{h} &=& \lb \mathfrak{h}, \mathfrak{h} \rb = \mathfrak{h} \\
{\rm Ad}_{\mathfrak{h}} \mathfrak{g}/\mathfrak{h} &=& \lb \mathfrak{h}, \mathfrak{g}/\mathfrak{h} \rb = \mathfrak{g}/\mathfrak{h}
\end{eqnarray}
$$
Specifically, iff $\mathfrak{h}$ is reductive in $\mathfrak{g}$ the commutation relations are:
$$
\begin{eqnarray}
\lb H_P, H_Q \rb &=& C_{PQ}{}^R H_R \\
\lb H_P, K_A \rb &=& C_{PA}{}^B K_B \\
\lb K_A, K_B \rb &=& C_{AB}{}^C K_C + C_{AB}{}^R H_R
\end{eqnarray}
$$
If, in addition, we have $[\mathfrak{g}/\mathfrak{h},\mathfrak{g}/\mathfrak{h}] \subset \mathfrak{h}$ (equivalently $\lb K_A, K_B \rb = C_{AB}{}^R H_R)$), then we say that $\mathfrak{h}$ is a ''symmetric'' subalgebra of $\mathfrak{g}$, and $\mathfrak{g}$ has a $\mathbb{Z}_2$ ''grading''.
Here's another way of looking at what reductive means: By choosing a particular set of generators, $T_I$, for $G$, the [[Killing form]] for $\mathfrak{g}$,
$$
g_{IJ} = C_{IK}{}^L C_{JL}{}^K
$$
can be diagonalized, with the structure constants then satisfying $C_{IJK} = -C_{IKJ}$ after lowering indices with $g$. A subalgebra, $\mathfrak{h}$, is ''reductive'' in $\mathfrak{g}$ iff its generators can be chosen from these, $H_P = T_P$, that diagonalized the Killing form.
Note that this terminology comes from the study of [[symmetric space]]s, and a "reductive Lie algebra" conventionally means something different.
When a [[subgroup]], $H \subset G$, is [[reductive]] in $G$ the [[Lie group tangent bundle geometry]], represented by the [[frame]] 1-forms,
$$
\f{E^J} = \f{\xi_R^J} = \f{{\cal I}^J} = \lp T^J , \f{\cal I} \rp = \lp T^J , \f{\cal I} \rp = \lp T^J , g^-(z) \f{d} g(z) \rp
$$
splits into two parts. The coordinates, $z^i$, over the [[Lie group manifold|Lie group geometry]] split into two sets of coordinates: the coordinates, $y^p$, over $H$, and the "leftover" coordinates, $x^a$, over a base manifold, $M$, of dimension $n_S = (n_G - n_H)$. So a point (element) of $G$ is specified by
$$
g(z) = g(x,y) = r(x) \, h(y)
$$
with $h(y) \in H$ acting on $r(x) \in G$ via the [[right action|group]]. The arbitrarily chosen ''reference section'', $r : M \to G$, corresponds to the [[submanifold]] in $G$ corresponding to $y=0$. The base manifold may be thought of as the [[homogeneous space]], $M=S=G/H$. With this choice of coordinates, and reductivity assumed, the frame 1-forms over $G$ split as:
$$
\begin{eqnarray}
\f{E^A}(z) &=& \lp K^A , h^- \f{e_S} h(y) \rp = \f{e_S^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(z) &=& \lp H^P , h^- \f{A_S} h(y) + h^- \f{d} h(y) \rp = \f{A_S^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
$$
in which $\f{e_S}(x) = \f{e_S^B} K_B$ is the [[homogeneous space frame|homogeneous space geometry]], $\f{A_S}(x) = \f{A_S^Q} H_Q$, is the [[homogeneous H-connection|homogeneous space geometry]], $\f{{e'}_H^P}(y)$ are the frame 1-forms over $H$. In this way, a reductive Lie group geometry is equivalent to an [[Ehresmann homogeneous space geometry]]. The above [[left-right rotator]] for $H$ in $G$ is,
$$
\lp L^h \rp^I{}_J = \lp T^I , h^- T_J h(y) \rp
$$
and the left-right rotator for $G$ splits as
$$
L^I{}_J = \lp T^I , g^- T_J g(z) \rp = \lp T^I , h^- r^- T_J r(x) h(y) \rp = \lp L^h \rp^I{}_K \, \lp L^r \rp^K{}_J
$$
in which the left-right rotator for $r$ is
$$
\lp L^r \rp^K{}_J = \lp T^I , r^- T_J r(x) \rp
$$
Rotating the frame 1-forms gives the frame of a particular [[Kaluza-Klein]] spacetime,
$$
\begin{eqnarray}
\f{E'^A}(z) &=& \lp L^h \rp_B{}^A \, \f{e^B} = \f{e_S^B}(x) \\
\f{E'^P}(z) &=& \lp L^h \rp_Q{}^P \, \f{e^Q} = \f{A_S^P}(x) + \f{e_H^Q} \lp L^h \rp_Q{}^P
\end{eqnarray}
$$
with [[spacetime]] frame 1-forms, $\f{e_S^B}(x)$, over $M$, and $\f{e'_H}(y) = \f{e_H^Q} \lp L^h \rp_Q{}^P$ identified as the frame 1-forms over the small compact Kaluza-Klein manifold, $H$.
It is straightforward to calculate the inverse to the matrix of frame 1-form components, and get the [[orthornormal basis vector fields|frame]] over $G$,
$$
\begin{eqnarray}
\ve{E_A}(z) &=& \lp L^h \rp_A{}^B \, \ve{e^S_B}(x) - \lp L^h \rp_A{}^C \lp \ve{e^S_C} \f{A_S^Q} \rp \lp L^h \rp^P{}_Q \, \ve{e^H_P} \\
\ve{E_P}(z) &=& \ve{e^H_P}(y)
\end{eqnarray}
$$
corresponding to the [[left invariant Killing vector fields|Lie group geometry]], $\ve{\xi^R_J} = \ve{E_J}$, and satisfying $\ve{E_I} \f{E^J} = \de_I^J$. Note that, in the coordinates we have chosen, the Killing vector fields over $G$ corresponding to the [[Lie algebra]] generators of $H$ equal the Killing vectors over $H$,
$$
\ve{\xi^R_P}(z) = \ve{\xi^{HR}_P}(y)
$$
a fact that is true iff $H$ is reductive in $G$.
The [[reductive Lie group tangent bundle geometry]], including the connection and curvature, also splits in an interesting way.
A ''reductive Lie group tangent bundle geometry'' is a [[Lie group tangent bundle geometry]] for a [[reductive Lie group geometry]]. The [[frame]] 1-forms, $\f{E^J}=\f{{\cal I}^J}$, over the Lie group manifold, $G$, split in adapted coordinates as
\begin{eqnarray}
\f{E^A}(x,y) &=& \f{e_S^B}(x) \, \lp L^h\rp^A{}_B(y) \\
\f{E^P}(x,y) &=& \f{A_S^Q}(x) \, \lp L^h \rp^P{}_Q(y) + \f{e_H^P}(y)
\end{eqnarray}
in which $\f{e_S^B}$ and $\f{A_S^Q}$ are the [[homogeneous space frame|homogeneous space geometry]] forms and [[homogeneous H-connection|homogeneous space geometry]] forms, $\lp L^h \rp^A{}_B = \lp H^A, h^- H_B h(y) \rp$ is the [[left-right rotator]] over $H$, and $\f{e_H^P}$ are the frame 1-forms over $H$. Using the [[Maurer-Cartan equation|Maurer-Cartan form]] over $G$,
$$
0 = \f{d} \f{E^J} + \ha \f{E^I} \f{E^K} C_{IK}{}^J
$$
and insisting that the [[torsion]] vanish over $G$,
$$
\ff{T^J} = 0 = \f{d} \f{E^J} + \f{W}^J{}_K \f{E^K}
$$
gives the same [[tangent bundle spin connection|tangent bundle connection]],
$$
\f{W}^J{}_K = \ha \f{E^I} C_{IK}{}^J
$$
over $G$, as for a Lie group tangent bundle geometry. These split to:
\begin{eqnarray}
\f{W}^B{}_C &=& \ha \f{e_S^D} \, \lp L^h\rp^A{}_D C_{AC}{}^B + \ha \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp C_{PC}{}^B \\
\f{W}^B{}_R &=& \ha \f{e_S^D} \, \lp L^h\rp^A{}_D C_{AR}{}^B \\
\f{W}^Q{}_R &=& \ha \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp C_{PR}{}^Q
\end{eqnarray}
Note that these are the same values taken by the [[Cartan tangent bundle spin connection]] when $\f{e^A}=\f{e_S^A}$ and $\f{A^P}=\f{A_S^P}$, with
\begin{eqnarray}
F^{HS}_{DEQ} &=& \ve{e^S_E} \ve{e^S_D} \lp \f{d} \f{A^S_Q} + \ha \f{A_S^P} \f{A_S^R} C_{PRQ} \rp = -C_{DEQ} \\
\f{\nu^S}_{EF} &=& - \ha \f{e_S^D} C_{DEF} - \f{A_S^Q} C_{QEF}
\end{eqnarray}
The [[Riemann curvature]] is also the same as for the Lie group tangent bundle geometry, $\ff{R}{}^J{}_K = - \fr{1}{4} \f{E^I} \f{E^M} C_{KLI} C_M{}^{JL}$, which splits as:
\begin{eqnarray}
\ff{R}{}^B{}_C &=& - \fr{1}{4} \f{e_S^E} \, \lp L^h\rp^A{}_E \, \f{e_S^F} \, \lp L^h\rp^D{}_F \, C_{CLA} C_D{}^{BL} \\
\ff{R}{}^B{}_R &=& - \fr{1}{4} \lp \f{A_S^Q} \, \lp L^h \rp^P{}_Q + \f{e_H^P} \rp \f{e_S^F} \, \lp L^h\rp^D{}_F \, C_{RLP} C_D{}^{BL} \\
\ff{R}{}^Q{}_R &=& - \fr{1}{4} \lp \f{A_S^U} \, \lp L^h \rp^P{}_U + \f{e_H^P} \rp \lp \f{A_S^V} \, \lp L^h \rp^T{}_V + \f{e_H^T} \rp C_{RLP} C_T{}^{QL}
\end{eqnarray}
The [[Ricci curvature]] is $\f{R}{}_J = - \fr{1}{4} \f{E}{}_J$ and the [[curvature scalar]] is $R = -\fr{1}{4} n_G$.
[[Carlo Rovelli]] has a nice paper out on a local interpretation of EPR setup:
[[Relational EPR|http://arxiv.org/abs/quant-ph/0604064]]
Conventionally, an observer, $A$, at $\al$ measures a state,
$$\ll \ps \ri = \fr{1}{\sqrt{2}} \lp \ll + \ri^z_\al \ll - \ri^z_\be - \ll - \ri^z_\al \ll + \ri^z_\be \rp$$
and thus collapses the wave function for the spin partner at spatially distant $\be$. Rovelli says $A$ only measures and determines the new information locally, with an observation $S^z_{A, \al} = \pm$, adding to the known square of total spin, $S^2_{A, \al + \be} = 0$, and thus allowing $A$ to infer the state that will be measured at $\be$ once it is back in causal contact.
Suggests the wave function, $\psi$, should be static and represent state of information, while observables (operators) should evolve in time -- i.e. the Heisenberg picture.
Perhaps the wavefunction, and its collapse, is a poor descriptor? Since what's really going on with "collapse" is just new information being acquired by an observer.
Hmm, this seems similar to Von Neuman's treatment of QM using a density matrix.
A [[Lie group]] ''representation'', $\Pi$, is a map (homomorphism) from Lie group elements to general linear operators on a [[representation space]],
$$
\Pi : G \mapsto GL(V)
$$
Via [[exponentiation]], each Lie group representation gives rise to a [[Lie algebra]] ''representation'' (usually also just called the ''representation''),
$$
\pi(X) = \fr{d}{dt} \Pi(e^{tX}) |_{t=0} \in GL(V)
$$
for all $X \in {\frak g}$. We also have $\Pi(e^X) = e^{\pi(X)}$.
The Lie group product and Lie algebra bracket in a representation corresponds to operator composition,
$$
\Pi(g_1 g_2) = \Pi(g_1) \Pi(g_2) \s\;\;\; \pi([X,Y]) = \pi(X) \pi(Y) - \pi(Y) \pi(X)
$$
A representation is ''faithful'' if it is an injective homomorphism, with different group elements mapped to different linear operators.
Ref:
*Clara Loeh
**[[Representation Theory of Lie Algebras|papers/Loeh - Representation Theory of Lie Algebras.pdf]]
A ''represenation space'' (sometimes lazily called the ''representation'', or ''G-module''), $V$, is a real or complex [[vector space]] upon which a [[Lie group]], $G$, [[representation]], $\Pi(G)$, or Lie algebra representation, $\pi(\mathfrak{g})$, acts.
A [[unitary representation]] space must be ''real'', ''complex'', xor ''quaternionic'' (aka ''pseudo-real''), characterized by $c \in \{1,0,-1\}$. Somewhat confusingly, a representation space is real iff it has an [[antiunitary]] operator, $J \ni J^2=1$, commuting with $\pi(\mathfrak{g})$, squaring to the identity, that operates on it, quaternionic iff it has an antiunitary, $J \ni J^2=-1$, commuting with $\pi(\mathfrak{g})$, squaring to minus the identity, that operates on it, and complex iff it has neither.
A real representation space is the [[complexification|realify]] of a real vector space, $V = V_\mathbb{R} \otimes \mathbb{C}$, with a [[real structure|complex structure]], $J$, and has a symmetric bilinear form from the [[Hermitian form]], $g(u,v) = \li J u | v \ri = \li J v | u \ri = g(v,u)$ -- an ''orthogonal structure'' on $V$.
A [[quaternion]]ic representation space (for example, [[Pauli spinor]]s), with a [[complex structure]], and an antiunitary operator corresponding to a quaternion, $j = J$, necessarily also has unitary and antiunitary operators, $i \ni i^2 = -1$ and $k \ni k^2=-1$, satisfying $i j = k = -i j$, and has an antisymmetric bilinear form, $g(u,v) = \li J u | v \ri = - \li J v | u \ri = -g(v,u)$ -- a ''symplectic structure'' on $V$.
Both real and quaternionic representations spaces are isomorphic to their complex conjugates -- with the duals determined by either $J$ -- while complex representation spaces are not. The direct product of two representation spaces is characterized by $c = c_1 c_2$, so, for example, the direct produce of a complex representation space, $c_1=0$, with any other representation space is complex.
A ''rest frame'', or //''Minkowski spacetime''//, $M'$, is an extended inertial reference frame. It is flat, having no curvature, and has cartesian coordinates, $x^\al$, which multiply orthonormal vectors, $\ga_\al$, to designate points, $x=x^\al \ga_\al$, in the spacetime. Since the spacetime is flat and cartesian, the basis vectors, $\ga_\al$, [[Clifford basis vectors]], serve as both unit rulers on the spacetime (the ''coordinate axes'') and as unit vectors at every spacetime point. The ''temporal coordinate'' (often $x^0=t$) has [[units]] of time, $[x^0]=T$, such as seconds (or $T_n=E^{-1}$ in natural units or $T_g = L$ in geometric units). We convert the ''spatial spacetime coordinates'', $[x^\pi]=T$ (with spatial [[indices]]), to ''spatial coordinates'' , $x_s^\pi = c \, x^\pi$ (having units of length, $[x_s^\pi]=L$, such as meters or light-seconds (or $L_n=E^{-1}$ in natural units)), using the speed of light, $c$.
A rest frame is a local, flat approximation to a curved [[manifold]] at a point, with the Clifford basis vectors associated to [[frame]] vectors at that point, $\ga_\al \leftrightarrow \ve{e}_\al(x)$. If, within some patch, the manifold point coordinates near $x^i$ are $x'^i$, then rest frame coordinates corresponding to these nearby points are
$$
x^\al(x'^i) \simeq \left. \fr{\pa x^\al}{\pa x'^i} \right|_{x^i} (x^i - x'^i) = (e_i)^\al \, (x^i - x'^i)
$$
in which $(e_i)^\al$ are the frame components at $x$. Unitless [[tangent vector]]s and [[differential form]]s at a point on a manifold [[Cliffordize|Cliffordization]] back and forth to temporal ($[\f{e}]=T$) Clifford vectors and multivectors in a rest frame via the [[frame]], such as $v = \ve{v} \f{e}$ and $a = \ve{e} \f{a}$, using [[vector-form algebra]]. Since all physics can be described by local interactions, physics described locally in a rest frame can be mapped back and forth to physics on a curved manifold near a point, via the frame.
If we associate a rest frame with each manifold point, we have the [[Clifford vector bundle]]. Vector and form fields on a manifold, such as $\ve{v}(x)$ or $\f{a}(x)$, Cliffordize, via the frame, to and from Clifford fields -- sections of the associated Clifford vector bundle. However, points in a rest frame only map approximately to points on a manifold near the point for which the rest frame is defined, $x'^i(x^\al)$, via the inverse of the formula above.
In four dimensional [[spacetime]] the [[Cl(1,3)]] Clifford basis vectors in the [[Weyl representation|Dirac matrices]] are $\ga_0 = \si_1 \otimes \si_0$ and $\ga_\va = -i \si_2 \otimes \si_\va$, in which $\si_\va$ are the [[Pauli matrices]]. So rest frame points are
$$
x = x^\mu \ga_\mu =
\lb
\begin{array}{cc}
0 & x_L \\
x_R & 0
\end{array}
\rb
=
\lb
\begin{array}{cc}
0 & x^\mu \bar{\si}_\mu \\
x^\mu \si_\mu & 0
\end{array}
\rb
=
\lb
\begin{array}{cc}
0 & t - \fr{1}{c} x_s \\
t + \fr{1}{c} x_s & 0
\end{array}
\rb
$$
in which $x_s = x_s^\va \si_\va$ are ''spatial points'', and $x_{L/R}$ are ''chiral position''s. Rest frame points can be described using two [[Weyl spinor]]s in a [[twistor]].
The [[right action|group]] of one [[Lie group]] element on all others induces a [[diffeomorphism]], $\ph_h(x)$ on the [[Lie group manifold|Lie group geometry]],
$$
R_h g(x) = g(x) h = g(\ph_h(x))
$$
A vector field on the Lie group manifold is ''right invariant'' iff it is invariant under the [[pushforward|pullback]] of this diffeomorphism for arbitrary $h$,
$$
R_{h*} \ve{v}(x) = \ve{v} \f{\pa} \ph_h(x) = \ve{v}(\ph_h(x))
$$
The partial derivative of the diffeomorphism in the above expression is computed explicitly by using the [[chain rule|partial derivative]],
$$
\f{\pa} g(\ph_h(x)) = \lp \f{\pa} \ve{\ph_h}(x) \rp \f{\pa} g(\ph_h) = \f{\pa} g(x) h
$$
and the defining equation for the [[left action vector fields and 1-forms|Lie group geometry]],
$$
\f{\pa} g = \f{\xi_L^B} T_B g
$$
to write
$$
\lp \f{\pa} \ve{\ph_h}(x) \rp \f{\xi_L^B}(\ph_h) T_B g h = \f{\xi_L^B}(x) T_B g h
$$
and get
$$
\f{\pa} \ve{\ph_h}(x) = \f{\xi_L^B}(x) \ve{\xi^L_B}(\ph_h(x))
$$
This implies the left action vector fields are right invariant,
$$
R_{h*} \ve{\xi^L_C}(x) = \ve{\xi^L_C}(x) \f{\pa} \ph_h(x) = \ve{\xi^L_C}(x) \f{\xi_L^B}(x) \ve{\xi^L_B}(\ph_h) = \ve{\xi^L_C}(\ph_h(x))
$$
A [[differential form]] is right invariant iff it is invariant under the [[pullback]], $R_h^* \nf{F}(\phi_h(x)) = \nf{F}(x)$. The 1-form duals to right invariant vector fields, such as the duals to the left action vector fields, are right invariant. Since autodiffeomorphisms are invertible, these statements may be summarized by defining any form or [[vector valued form]] to be right invariant iff it is invariant under the pushforward, $R_{h*} \nf{\ve{K}}(x) = \nf{\ve{K}}(\phi_h(x))$, or pullback.
A root system is a collection of points (vectors) in an $R$ dimensional Euclidean (or pseudo-Euclidean) [[vector space]], $V$, with inner product $<\! \al, \be \!>$. To be a ''root system'', $\Ph$, the vectors must satisfy the following:
1. They must span $V$.
2. Closure under reflections. Every $\al, \be \in \Ph$ satisfy $\be - 2 \fr{<\al,\be>}{<\al,\al>} \al \in \Ph$.
3. Projections of roots along other roots are integers or half integers. Every $\al, \be \in \Ph$ satisfy $2 \fr{<\al,\be>}{<\al,\al>} \in \mathbb{Z}$.
With these restrictions any two roots can only have angular separations and length ratios of $\{ 30^\circ \textrm{ or } 150^\circ, \sqrt{3} \}$, $\{ 45^\circ \textrm{ or } 135^\circ, \sqrt{2} \}$, $\{ 60^\circ \textrm{ or } 120^\circ, 1 \}$, or $\{ 90^\circ , \textrm{any} \}$.
One can choose a set of $R$ non-orthogonal ''simple roots'', $\al_a \in \De$, such that all roots can be written as positive integral or negative integral combinations of simple roots,
$$
\be = \sum_{a=1}^R k^a \al_a \textrm{ with } k^a \textrm{ all in } \mathbb{Z}^+ \textrm{ or all in } \mathbb{Z}^-
$$
The ''negative roots'', $\Ph^-$, are negatives of the ''positive roots'', $\Ph^+$. The ''height'' (or sometimes ''level'') of a root is the sum of its coefficients, $n=\sum_{a=1}^R k^a$. The ''level'' of a root with respect to any root is its coefficient with respect to that root. One can thus decompose a root system, or Lie algebra, into grade levels.
The $R \times R$ ''Cartan matrix'', $A$, is the matrix of projections of simple roots along other simple roots,
$$
A_{ab} = 2 \fr{<\! \al_a , \al_b \!>}{<\! \al_a , \al_a \!>} = \lp \fr{2}{<\! \al_a , \al_a \!>} \rp \lp <\! \al_a , \al_b \!> \rp = D_a S_{ab}
$$
and consists of only $0$'s, $2$'s, and negative integers. As above, it can be written as a diagonal matrix times the ''symmetric Cartan matrix'', $S_{ab} = \, <\! \al_a , \al_b \!> = g_{ab}$, which is the [[metric]] on the space of simple roots. The Cartan matrix describes the root system and can be summarized by a [[Dynkin diagram]].
[[Lie algebra structure]] can be described and classified by root systems, and therefore by their Cartan matrix or Dynkin diagrams.
The roots in a root system can all be determined algorithmically from the Cartan matrix. By using the [[Chevalley-Serre relations|Lie algebra structure]], an algorithm for determining roots, level-by-level is:
1. Start with the simple roots, $\Ph_1^+=\De$.
2. For each $\ga = \sum k^a \al_a \in \Ph_n^+$ determine the largest $r \ge 0$ such that $\ga - r \al_b \in \Ph_{n-r}^+$ for some $\al_b$.
3. Let $q=r-\sum k^a A_{ab}$. If $q>0$ then $\ga + \al_b$ is a root in $\Ph_{n+1}^+$.
This algorithm terminates at the highest level for a finite dimensional Lie algebra, and proceeds indefinitely for an infinite dimensional Lie algebra. For a ''simply laced'' root system, having simple roots of all the same length, only $\ga$'s with $r=1 \textrm{ or } 0$ can produce $q>0$, which simplifies the algorithm. (Root chains can't be longer than $3$.)
Root systems can be quite pretty. (See, for example, the [[e8 root system]].)
Ref:
*de Graaf, Lie Algebras: Theory and Algorithms
The [[Clifford grade]] $0$ operator, $\li A\ri = \li A\ri_0 = A^0 = A^s$, is the ''scalar part'' operator, which gives the scalar part of a [[Clifford element]], $A$. For a Clifford element with real coefficients, this will be a real number. The scalar part is equal to the [[trace]] of the element in a [[Clifford matrix representation]] divided by the dimension of the representation.
Ref:
*Jeffrey D. Olson
**[[Instantons and Self-Dual Gauge Fields|papers/selfdual.ps]]
The real, [[chiral]] "conformal" representation of [[Cl(4,4)]], as the realification of [[Cl(2,4)]], may be related to the (real, chiral) [[split-octonionic representation of Cl(4,4)]] by a [[similarity transformation|Dirac matrices]]. If the ordering of the conformal generators is $\Ga'_\al = \{ \Ga'_1, \Ga'_2, \Ga'_3, \Ga'_4, \Ga'_5, \Ga'_6, \Ga'_7, \Ga'_8 \}$ and we re-order the split-octonions so the signature matches and the unit split-octonion matches the time direction, $\ga'_\al = \{ \ga'_5, \ga'_6, \ga'_7, \ga'_0, \ga'_4, \ga'_2, \ga'_1, \ga'_3 \}$, then the similarity transformation between them is
$$
\lb \begin{array}{cc}
& \overline{\Ga}{}'_\al \\
\Ga'_\al &
\end{array} \rb
=
\ga'_\al = U \, \Ga'_\al \, U^T
=
\lb \ba{cc}
U_- & \\
& U_+
\ea \rb
\lb \ba{cc}
& \Ga'_-{}_\al \\
\Ga'_+{}_\al &
\ea \rb
\lb \ba{cc}
U^T_- & \\
& U^T_+
\ea \rb
=
\lb \ba{cc}
& U_- \Ga'_-{}_\al U_+^T \\
U_+ \Ga'_+{}_\al U_-^T &
\ea \rb
$$
in which the positive and negative chiral similarity matrices are
$$
U_- = \ha \lb \ba{cccccccc}
- & & & + & + & & & - \\
& + & - & & & - & + & \\
& + & + & & & - & - & \\
& + & - & & & + & - & \\
+ & & & + & - & & & - \\
- & & & - & - & & & - \\
& - & - & & & - & - & \\
- & & & + & - & & & + \\
\ea \rb
\s
U_+ = \ha \lb \ba{cccccccc}
- & & & + & + & & & - \\
& + & - & & & - & + & \\
& + & + & & & - & - & \\
& + & - & & & + & - & \\
+ & & & + & - & & & - \\
+ & & & + & + & & & + \\
& + & + & & & + & + & \\
+ & & & - & + & & & - \\
\ea \rb
$$
These are related to [[triality matrices|triality matrix]].
A connected [[Lie group]], $G$, is ''simple'' iff it has no [[normal subgroup]]s. In this sense, other Lie groups can have simple Lie groups as "prime factors." A Lie algebra, $\mathfrak{g}$, is ''simple'' iff it's only [[ideal|spinor]] is itself -- i.e. there is no other $\mathfrak{h} \in \mathfrak{g}$ such that $\mathfrak{g} \, \mathfrak{h} = \mathfrak{h}$. A Lie group is simple iff its Lie algebra is simple.
A Lie algebra is ''semi-simple'' iff it is the direct sum of simple Lie algebras. A connected Lie group is ''semi-simple'' iff its Lie algebra is semi-simple.
The spinor ''skew'',
$$
\ep = \lb \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \rb = -i \si_2 = J
$$
is an operator on [[Weyl spinor]]s, arising from [[charge conjugation|charge conjugate]], and acts as a [[metric on Weyl spinors|flipped spin]]. It is also a [[complex structure]] and [[sl(2)]] operator, and relates to a two dimensional [[permutation symbol]], $\ep = - \ep_{AB}$, and to the second [[Pauli matrix|Pauli matrices]]. As well as the [[permutation identity|permutation identities]], $\ep_{ab} \ep_{bc} = \de_{ac}$, its components satisfy the ''Schouten identity'',
$$
0 = \ep_{ab} \ep_{cd} + \ep_{ac} \ep_{db} + \ep_{ad} \ep_{bc}
$$
The three [[Lie algebra]] generators, $\{L,J,K\}$, for the ''three dimensional special linear group Lie algebra'', $sl(2)$, corresponding to the $SL(2)$ Lie group may be [[represented|representation]] by $2 \times 2$ traceless real matrices related to the [[Pauli matrices]],
$$
\begin{array}{ccc}
L = T_1 = \sigma_{1} = \left[\begin{array}{cc}
0 & 1\\
1 & 0\end{array}\right] &
J = T_2 = - i \sigma_{2} = \left[\begin{array}{cc}
0 & -1 \\
1 & 0\end{array}\right] &
K = T_3 = \sigma_{3} = \left[\begin{array}{cc}
1 & 0\\
0 & -1 \end{array}\right]\end{array}
$$
The commutation relations, $\lb T_A, T_B \rb = T_A T_B - T_B T_A = C_{AB}^{C} T_C$, are
$$
\lb T_1, T_2 \rb = 2 \, T_3 \s \lb T_2, T_3 \rb = 2 \, T_1 \s \lb T_1, T_3 \rb = 2 \, T_2
$$
the structure coefficients for this Lie algebra differ from those of [[su(2)]] by a sign. The generators are orthogonal -- the [[Killing form]] is
$$
\lp T_A, T_B \rp = g_{AB} = C_{AC}{}^D C_{BD}{}^C = 4 \, {\rm Tr}\lp T_A T_B \rp = 8 \, \mathrm{diag}(+1,-1,+1)
$$
which is equal to four times the [[trace]] of two multiplied $sl(2)$ generators. As a real Lie algebra. $sl(2)$ is the split [[real form]] of $A_1 = sl(2,C)$, and related to the compact real form, [[su(2)]]. It is also isomorphic to [[so(2,1)|spin Lie algebra]], with the [[bivector|Clifford basis elements]] equivalents $\{ T_1 = \ga_{02}, \, T_2 = \ga_{12}, \, T_3 = \ga_{01} \}$, in which $\ga_0$ has the different (time-like) signature. And it is isomorphic to $su(1,1)$ and $sp(2)$.
If we re-define the basis generators as
$$
E_+ = \ha (T_1 - T_2) = \left[\begin{array}{cc}
0 & 1\\
0 & 0\end{array}\right]
\s \;\;\;
E_- = \ha (T_1 + T_2) = \left[\begin{array}{cc}
0 & 0 \\
1 & 0\end{array}\right]
\s \;\;\;
H = T_3 = \left[\begin{array}{cc}
1 & 0\\
0 & -1 \end{array}\right]
$$
The commutation relations are
$$
\lb H, E_+ \rb = +2 \, E_+ \s \lb H, E_- \rb = -2 \, E_- \s \lb E_+, E_- \rb = H
$$
consistent with the [[Chevalley basis|Lie algebra structure]]. Or we could choose $T_2$ instead of $T_3$ to generate our Cartan subalgebra, which is anti-[[Hermitian]] instead of [[Hermitian]], thus producing a compact torus (circle) within sl(2), and giving
$$
E_+ = \ha (T_1 + i \, T_3)
\s \;\;\;
E_- = \ha (-T_1 + i \, T_3)
\s \;\;\;
H = T_2
$$
$$
\lb H, E_+ \rb = +2 \, i \, E_+ \s \lb H, E_- \rb = -2 \, i \, E_- \s \lb E_+, E_- \rb = i \, H
$$
The algebra of [[complex structure]] is $sl(2)$, with $J=i$ interpreted as a [[complex number]], $K$ complex conjugation, and $L=JK$ their composition, acting on the 2D real representation space, $2$, of
$$
V = V_{\mathbb R} + V_{\mathbb I} = V_{\mathbb R} + J \, V'_{\mathbb R}
= \lb \begin{array}{c} V_{\mathbb R} \\ V'_{\mathbb R} \end{array} \rb
$$
The $sl(2)$ Lie algebra is isomorphic to ''su(1,1)'', ''so(2,1)'', and ''sp(2)''.
The $sl(3)$ [[Lie algebra]] is represented by traceless real $3 \times 3$ matrices. It is the split-[[real form]] of $sl(3,\mathbb{C}) = A_2$, of which [[su(3)]] is the compact real form. (Another real form is $su(2,1)$.) As our basis generator matrices, we choose a real version of the [[Gell-Mann matrices]],
$$
T_1 = \la_1 \s T_2 = -i \, \la_2 \s T_3 = \la_3 \s T_4 = \la_4 \s T_5 = -i \, \la_5 \s T_6 = \la_6 \s T_7 = -i \, \la_7 \s T_8 = \la_8
$$
The structure constants resulting from the commutation relations, $\lb T_A, T_B \rb = C_{AB}{}^C T_C$, are not anti-symmetric in their indices, but the lowered ones, $C_{ABC} = C_{AB}{}^D g_{DC}$, are, with nonzero values:
$$
\begin{array}{ccccc}
C_{123} = 2 & C_{147} = 1 & C_{156} = -1 & C_{246} = 1 & C_{257} = -1 \\
C_{345} = 1 & C_{367} = -1 & C_{458} = \sqrt{3} & C_{678} = \sqrt{3}
\end{array}
$$
The [[Killing form]] is proportional to the [[trace]] of two multiplied $su(3)$ generators,
$$
\lp T_A, T_B \rp = g_{AB} = C_{AC}{}^D C_{BD}{}^C = 6 \, {\rm Tr}(T_A T_B) = 12 \, {\rm Diag}(+1,-1,+1,+1,-1,+1,-1,+1)
$$
Using the ''real Gell-Mann matrices'', any Lie algebra element may be represented as a $3 \times 3$ real matrix,
$$
\ba{rcl}
B^A T_A \!\!&\!\!=\!\!&\!\!
\lb\ba{ccc}
B^3 + \fr{1}{\sqrt{3}} B^8 & B^1 - B^2 & B^4 - B^5 \\
B^1 + B^2 & - B^3 + \fr{1}{\sqrt{3}} B^8 & B^6 - B^7 \\
B^4 + B^5 & B^6 + B^7 & -\fr{2}{\sqrt{3}} B^8
\ea\rb
=
\lb\ba{ccc}
V-M & v & m' \\
v' & P-V & p & \\
m & p' & M-P
\ea\!\!\!\!\!\rb \\
\!\!&\!\!=\!\!&\!\! V \, H_v + M \, H_m + P \, H_p + (v \, E^+_v - v^* E^-_v) + (m \, E^+_m - m^* E^-_m) + (p \, E^+_p - p^* E^-_p) \\
\!\!&\!\!=\!\!&\!\! u(1)_3 + u(1)_8 + 2_v + 2_m + 2_p
\ea
$$
with $v$, $m$, $p$, $v'$, $m'$, $p'$, real numbers (or half as many [[split-complex number]]s), and $M$, $P$, $V$ real numbers. Note that the same Lie algebra element is specified by $(M+c, P+c, V+c)$, so there are only two degrees of freedom in those three parameters, $B^3$ and $B^8$. With orthogonal [[Cartan subalgebra|Lie algebra structure]] basis generators $(T_3,T_8)$, or non-orthogonal $(\mathcal{H}_m, \mathcal{H}_p, \mathcal{H}_v)$, the root vectors, their root coordinates, and their Lie brackets, are
$$
\begin{array}{|rcl|cc|ccc|ccc|}
\hline
& & & g^3 & g^8 & v & m & p & r & g & b\\
\hline
E^+_v = g^{r\bar{g}} \!\!&\!\!=\!\!&\!\! \ha ( -T_2 + T_1 ) & +2 & 0 & +2 & -1 & -1 & +1 & -1 & 0\\
E^-_v = g^{\bar{r}g} \!\!&\!\!=\!\!&\!\! \ha ( T_2 + T_1 ) & -2 & 0 & -2 & +1 & +1 & -1 & +1 & 0\\
E^+_m = g^{\bar{r}b} \!\!&\!\!=\!\!&\!\! \ha ( T_5 + T_4 ) & -1 & -\sqrt{3} & -1 & +2 & -1 & -1 & 0 & +1\\
E^-_m = g^{r\bar{b}} \!\!&\!\!=\!\!&\!\! \ha (- T_5 + T_4 ) & +1 & +\sqrt{3} & +1 & -2 & +1 & +1 & 0 & -1\\
E^+_p = g^{g\bar{b}} \!\!&\!\!=\!\!&\!\! \ha ( - T_7 + T_6 ) & -1 & +\sqrt{3} & -1 & -1 & +2 & 0 & +1 & -1\\
E^-_p = g^{\bar{g}b} \!\!&\!\!=\!\!&\!\! \ha ( T_7 + T_6 ) & +1 & -\sqrt{3} & +1 & +1 & -2 & 0 & -1 & +1\\
\hline
\end{array}
\s \s
\ba{lcrcl}
\!\!&\!\! \!\!&\!\! [ T_3, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm g_3 \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ T_8, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm g_8 \, E^\pm_\al \\
H_v \! \!\!&\!\!=\!\!&\!\! [ E^+_v, E^-_v ] \!\!&\!\!=\!\!&\!\! T_3 \\
H_m \! \!\!&\!\!=\!\!&\!\! [ E^+_m, E^-_m ] \!\!&\!\!=\!\!&\!\! (- \ha T_3 - \fr{\sqrt{3}}{2} T_8) \\
H_p \! \!\!&\!\!=\!\!&\!\! [ E^+_p, E^-_p ] \!\!&\!\!=\!\!&\!\! (- \ha T_3 + \fr{\sqrt{3}}{2} T_8) \\
\!\!&\!\! \!\!&\!\! [ H_v, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm v \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ H_m, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm m \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ H_p, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm p \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ E^+_v, E^+_m ] \!\!&\!\!=\!\!&\!\! -E^-_p \\
\ea
$$
This description of [[Lie algebra structure]] is consistent with a Cartan-Weyl basis and Chevalley-Serre basis, with [[simple root|root system]] vectors chosen as any two of $\{ E^+_v, E^+_m, E^+_p \}$. The $sl(3)$ [[root system]] is a hexagon, which we can understand as a 2D projection of the roots of [[sp(6,R)]].
The $sl(3)$ Lie algebra is related to the $su(3)$ associated with the strong force. It is useful to consider yet a different orthogonal set of ''psuedo-color'' basis elements, $( C_r, C_g, C_b )$, that are in the Cartan subalgebra of $gl(3)$ but are not in $sl(3)$, such that [>img[images/png/g2rs2.png]]
$$
\s \s \s H_v = C_r - C_g \s \s H_m = - C_r + C_b \s \s H_p = C_g - C_b \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
$$
The corresponding ''pseudo-quark'' weight vectors and weights are
$$
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\begin{array}{|rcl|cc|ccc|ccc|}
\hline
& & & g^3 & g^8 & v & m & p & r & g & b \\
\hline
q^r \!\!&\!\!=\!\!&\!\! [ 1, 0, 0 ]^T & +1 & +\fr{1}{\sqrt{3}} & +1 & -1 & 0 & +1 & 0 & 0\\
\bar{q}^r \!\!&\!\!=\!\!&\!\! [ 1, 0, 0 ] & -1 & -\fr{1}{\sqrt{3}} & -1 & +1 & 0 & -1 & 0 & 0\\
q^g \!\!&\!\!=\!\!&\!\! [ 0, 1, 0 ]^T & -1 & +\fr{1}{\sqrt{3}} & -1 & 0 & +1 & 0 & +1 & 0\\
\bar{q}^g \!\!&\!\!=\!\!&\!\! [ 0, 1, 0 ] & +1 & -\fr{1}{\sqrt{3}} & +1 & 0 & -1 & 0 & -1 & 0\\
q^b \!\!&\!\!=\!\!&\!\! [ 0, 0, 1 ]^T & 0 & -\fr{2}{\sqrt{3}} & 0 & +1 & -1 & 0 & 0 & +1\\
\bar{q}^b \!\!&\!\!=\!\!&\!\! [ 0, 0, 1 ] & 0 & + \fr{2}{\sqrt{3}} & 0 & -1 & +1 & 0 & 0 & -1\\
\hline
\end{array}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
$$
This structure of $sl(3)$ is consistent with a [[triality decomposition]], in which each of the three triples,
$$
\{ H_{v/m/p}, E^+_{v/m/p}, E^-_{v/m/p} \} \sim \{ H , E^+ , E^- \} = \{ K , \ha (L-J) , \ha (L+J) \}
$$
corresponds to a different [[sl(2)]] related to each other by complex triality, with the triality-related $sl(2)$ Cartan generators in the $su(3)$ Cartan subalgebra necessarily overlapping. A [[triality automorphism of sl(3)]] preserves the structure.
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<<ListTagged sm>>
The compact real form of the rank 3 [[symplectic Lie algebra|sp(n)]], ''sp(3)'', has an interesting structure related to [[quaternion]]ic [[triality]] and the [[exceptional magic square]]. The 21 dimensional Lie algebra decomposes as
$$
sp(3) = su(2)^M + su(2)^P + su(2)^V + 2^m \! \times 2^p + 2^v \! \times 2^p + 2^v \! \times 2^m
$$
using representation spaces of [[su(2)]], and the labels, $\{ V, M, P, v, m, p \}$, suggestive of quaternionic [[confusion|division algebra confusion]]. The Lie algebra basis elements can be [[represented|representation]] as $3 \times 3$ matrices of [[quaternion]]s,
$$
\{ T^M_A, T^P_B, T^V_C, T^v_c, T^m_a, T^p_b \} =
\lb
\begin{array}{ccc}
e^M_A & e^v_c & -\os{e}^m_a \\
-\os{e}^v_c & e^P_B & e^p_b & \\
e^m_a & -\os{e}^p_b & e^V_C \\
\end{array}
\!\!\!\!\! \rb
\;\; \in \; \pi \! \lp sp(3) \rp
$$
with $A,B,C \in \{1,2,3\}$ and $a,b,c \in \{0,1,2,3\}$. The Lie algebra structure can then be obtained from the [[commutator]] of these matrices, such as
\begin{eqnarray}
\lb T^v_a, T^m_b \rb & = & -M_{ab}{}^\os{c} T^p_c \s \s \lb v , m \rb = v \, m = - \os{p} \\
\lb T^v_a, T^p_b \rb & = & M_{ab}{}^\os{c} T^m_c \\
\lb T^v_a, T^v_b \rb & = & (-M_{a\os{b}}{}^A + M_{b\os{a}}{}^A ) T^M_A + (-M_{\os{a}b}{}^A + M_{\os{b}a}{}^A ) T^P_A
= - \de_{0a} (T^M_b-T^P_b) + \de_{0b} (T^M_a-T^P_a) - \ep_{abc} (T^M_c+T^P_c) \\
\lb T^m_a, T^m_b \rb & = & (-M_{a\os{b}}{}^A + M_{b\os{a}}{}^A ) T^P_A + (-M_{\os{a}b}{}^A + M_{\os{b}a}{}^A ) T^V_A \\
\lb T^p_a, T^p_b \rb & = & (-M_{a\os{b}}{}^A + M_{b\os{a}}{}^A ) T^V_A + (-M_{\os{a}b}{}^A + M_{\os{b}a}{}^A ) T^M_A \\
\end{eqnarray}
from matrix multiplication and the quaternion product. There is a [[triality automorphism of sp(3)]].
The split real form, [[sp(6,R)]], has a similar description using [[split-quaternion]]s.
The split real form of the rank 2 [[symplectic Lie algebra|sp(n)]], $sp(4,\mathbb{R})$, has an interesting structure. The 10 dimensional Lie algebra decomposes as
$$
sp(4,\mathbb{R}) = sl(2)^- + sl(2)^+ + 2^- \! \times 2^+ = spin(3,2)
$$
using representation spaces of [[sl(2)]]. The Lie algebra basis elements can be [[represented|representation]] as $2 \times 2$ matrices of [[split-quaternion]]s,
$$
\{ T^-_A, T^+_B, T^v_a \} =
\lb
\begin{array}{cc}
e'{}^-_A & e'{}^v_a \\
-\os{e}'{}^v_a & e'{}^+_B \\
\end{array}
\rb
\;\; \in \; \pi \! \lp sp(4,\mathbb{R}) \rp
$$
with $A,B \in \{1,2,3\}$ and $a \in \{0,1,2,3\}$, so as $4 \times 4$ real matrices. The four basis "vector" generators, $T^v_a$, and the "psuedoscalar vector",
$$
T^p_4 = T^v_0 T^v_1 T^v_2 T^v_3 =
\lb
\begin{array}{cc}
e'_0 & 0 \\
0 & -e'_0 \\
\end{array}
\rb
$$
are a [[Clifford matrix representation]] of basis vectors for $Cl(3,2)$.
For understanding the [[Lie algebra structure]] we can choose a split Cartan subalgebra, such as $C_s = C_s^1 T^-_3 + C_s^2 T^+_3$, producing real roots and root vectors, or a compact Cartan subalgebra, such as $C_c = C_c^1 T^-_2 + C_c^2 T^+_2$, producing imiginary roots and complex root vectors. These root coordinates and root vectors are:
$$
\begin{array}{|c|cc|c|}
\hline
\al' & \al_s^- & \al_s^+ & V'_\al \\
\hline
T^-_3, T^+_3 & 0 & 0 & T^-_3,T^+_3 \\
\om'{}^-_+ & +2 & 0 & - T^-_1 + T^-_2 \\
\om'{}^-_- & -2 & 0 & + T^-_1 + T^-_2 \\
\om'{}^+_+ & 0 & +2 & - T^+_1 + T^+_2 \\
\om'{}^+_- & 0 & -2 & + T^+_1 + T^+_2 \\
v'_{+-} & +1 & -1 & + T^v_0 + T^v_3 \\
v'_{-+} & -1 & +1 & - T^v_0 + T^v_3 \\
v'_{++} & +1 & +1 & - T^v_1 + T^v_2 \\
v'_{--} & -1 & -1 & + T^v_1 + T^v_2 \\
\hline
\end{array}
\s\;\;\;
\s\;\;\;
\s\;\;\;
\begin{array}{|c|cc|c|}
\hline
\al & \al_c^- & \al_c^+ & V_\al \\
\hline
T^-_1, T^+_1 & 0 & 0 & T^-_2,T^+_2 \\
\om^-_+ & +2i & 0 & -i T^-_1 + T^-_3 \\
\om^-_- & -2i & 0 & +i T^-_1 + T^-_3 \\
\om^+_+ & 0 & +2i & -i T^+_1 + T^+_3 \\
\om^+_- & 0 & -2i & +i T^+_1 + T^+_3 \\
v_{+-} & +i & -i & +i T^v_0 + T^v_2 \\
v_{-+} & -i & +i & -i T^v_0 + T^v_2 \\
v_{++} & +i & +i & -i T^v_1 + T^v_3 \\
v_{--} & -i & -i & +i T^v_1 + T^v_3 \\
\hline
\end{array}
$$
Either of these [[different Cartans]] can be used to form a different [[Cartan-Weyl basis|Lie algebra structure]] for sp(2). We can also transform from the (real) split-Cartan basis to the (complex) compact-Cartan basis,
$$
\begin{array}{l}
\lb \begin{array}{c}
T^\pm_1 \\ V^\pm_+ \\ V^\pm_- \\
\end{array} \rb
= \ha
\lb \begin{array}{ccc}
-i & \sqrt{2} & i \\
i & \sqrt{2} & -i \\
\sqrt{2} & 0 & \sqrt{2}
\end{array} \rb
\lb \begin{array}{c}
V'{}^\pm_+ \\ T{}^\pm_2 \\ V'{}^\pm_-
\end{array} \rb
\\
\\
\lb \begin{array}{c}
V^v_{+-} \\ V^v_{-+} \\ V^v_{++} \\ V^v_{--}
\end{array} \rb
= \ha
\lb \begin{array}{cccc}
i & -i & 1 & 1 \\
-i & i & 1 & 1 \\
1 & 1 & -i & i \\
1 & 1 & i & -i
\end{array} \rb
\lb \begin{array}{c}
V'{}^v_{+-} \\ V'{}^v_{-+} \\ V'{}^v_{++} \\ V'{}^v_{--}
\end{array} \rb
\end{array}
$$
The split and compact Cartans acting on the four dimensional space of vectors are compatible with a [[split structure|structures on a real representation space]], $\lb J^s, C_s \rb = 0$, and a [[complex structure]], $\lb J^c, C_c \rb = 0$,
$$
J^s
\lb \begin{array}{c}
T^v_0 \\ T^v_1 \\ T^v_2 \\ T^v_3
\end{array} \rb
=
\lb \begin{array}{cccc}
& & & 1 \\
& & -1 & \\
& -1 & & \\
1 & & & \\
\end{array} \rb
\lb \begin{array}{c}
T^v_0 \\ T^v_1 \\ T^v_2 \\ T^v_3
\end{array} \rb
\s\;\;\;
\s\;\;\;
\s\;\;\;
J^c
\lb \begin{array}{c}
T^v_0 \\ T^v_1 \\ T^v_2 \\ T^v_3
\end{array} \rb
=
\lb \begin{array}{cccc}
& &-1 & \\
& & & 1 \\
1 & & & \\
& -1 & & \\
\end{array} \rb
\lb \begin{array}{c}
T^v_0 \\ T^v_1 \\ T^v_2 \\ T^v_3
\end{array} \rb
$$
So, $J^s = -\si_2 \otimes \si_2$ and $J^c = - i \si_2 \otimes \si_3$.Their commutator gives $J^{sc} = -\ha \lb J^s, J^c \rb = \si_0 \otimes \si_1$, which is compatible with another split Cartan, $C_{sc} = C_{sc}^1 T^-_1 + C_{sc}^2 T^+_1$. Together with the identity these two split and one compact structure operators are a ''split-quaternionic structure'' on the vector space (as are their three Cartans and the identity).
The split real form of the rank 3 [[symplectic Lie algebra|sp(n)]], ''sp(6,R)'', has an interesting structure related to [[quaternion]]ic [[triality]] and the [[exceptional magic square]]. It is also interesting to describe it using two [[different Cartans]], such as in the description of [[sp(4,R)]]. The triality decomposition is
$$
sp(6,\mathbb{R}) = sl(2)^M + sl(2)^P + sl(2)^V + 2^m \! \times 2^p + 2^v \! \times 2^p + 2^v \! \times 2^m
$$
using representation spaces of [[sl(2)]], and can be represented as a $3 \times 3$ matrix of [[split-quaternion]]s, or as $6 \times 6$ real matrices,
$$
\{ T^M_A, T^P_B, T^V_C, T^v_c, T^m_a, T^p_b \} =
\lb
\begin{array}{ccc}
e'{}^M_A & e'{}^v_c & -\os{e}'{}^m_a \\
-\os{e}'{}^v_c & e'{}^P_B & e'{}^p_b \\
e'{}^m_a & -\os{e}'{}^p_b & e'{}^V_C \\
\end{array}
\rb
\;\; \in \; \pi \! \lp sp(6,\mathbb{R}) \rp
$$
with $A,B,C \in \{1,2,3\}$ and $a,b,c \in \{0,1,2,3\}$, which is the split real version of compact [[sp(3)]].
From split-quaternionic triality we can see that the ''triality automorphism of sp(6,R)'' matches the [[triality automorphism of sp(3)]]. We can describe the [[Lie algebra structure]] by choosing [[different Cartans]]; either compact, split, or mixed. A triality automorphism matches to root maps under a compact or split Cartan, but it is more complicated to describe the structure and [[triality automorphism of sp(6,R) under a mixed Cartan]].
The ''symplectic Lie algebra'', ''sp(n)'' (also labeled $C_n$), is the [[Lie algebra]] of the symplectic Lie group of rank $n$ and dimension $n(2n+1)$. In general, the ''symplectic group'', $Sp(n)$, is the compact [[real form]] of the complex Lie group $Sp(2n,\mathbb{C})$, which has as split real form, $Sp(2n,\mathbb{R})$. (//Yes, this naming convention is pretty messed up!//)
The noncompact real symplectic Lie algebra of rank $n$, $Sp(2n,\mathbb{R})$, corresponds to $2n \times 2n$ complex matrices, $A$, satisfying
$$
\Om \, A^T \, \Om = A
$$
in which
$$
\Om=\left[\begin{array}{cc}
0 & I_n\\
-I_n & 0\end{array}\right]
$$
which may be rearranged by row and column.
For sp(n), the compact real form of $C_n$, the matrix $A$ may be written as a $n \times n$ special [[unitary]] matrix of [[quaternion]]s. A $n \times n$ matrix of [[split-quaternion]]s with this property corresponds to $sp(2n,\mathbb{R})$.
''Spacetime'' (//''Lorentzian spacetime''//) is a four dimensional [[manifold]], $M$, with coordinates (possibly on patches), $x^a$, and a [[metric]], $g_{ab}$. This metric may be derived from four ''spacetime [[orthonormal basis vectors|frame]]'', $\ve{e_\mu} = \lp e_\mu \rp^a \ve{\pa_a}$ (spanning the ''spacetime [[tangent bundle]]''), with appropriate [[indices]], along with the [[Minkowski metric]] (chosen to have positive time signature, unless stated otherwise). The ''spacetime [[torsion]]'', usually taken to be zero, determines the ''spacetime [[tangent bundle spin connection|tangent bundle connection]]'', $\f{w}^\mu{}_\nu = \f{dx^a} w_a{}^\mu{}_\nu$, which in turn determines the ''spacetime [[Riemann curvature]]''. The spacetime orthonormal basis vectors have an inverse, the ''spacetime [[frame]] 1-forms'', $\f{e^\mu} = \f{dx^a} \lp e_a \rp^\mu$.
This structure matches that of a [[Clifford vector bundle]] with the spacetime manifold as its base, and an identification between orthonormal basis vectors and [[Clifford basis vectors]], $\ve{e_\mu} \leftrightarrow \ga_\mu$. The Clifford algebra fiber of this bundle is the ''spacetime [[Clifford algebra]]'', [[Cl(1,3)]], or with the other choice of signature, Cl(3,1). The [[spacetime frame]] is then defined as $\f{e} = \f{dx^a} \lp e_a \rp^\mu \ga_\mu$, and the [[spacetime spin connection]] is $\f{\om} = \f{dx^a} \ha w_a{}^{\mu \nu} \ga_{\mu \nu}$, which is determined by the (usually vanishing) spacetime [[torsion]],
$$
\ff{T} = \f{d} \f{e} + \f{\om} \times \f{e}
$$
The spacetime spin connection determines the ''spacetime [[Clifford-Riemann curvature]]'',
$$
\ff{R} = \f{d} \f{\om} + \ha \f{\om} \f{\om}
$$
which has coefficients equal to the spacetime Riemann curvature tensor, $R_{ab}{}^{\mu \nu}$. This, along with the spacetime frame, determines the ''spacetime [[Clifford-Ricci curvature]]'' and ''spacetime [[Clifford curvature scalar]]''.
Sometimes //''spacetime''// is used to refer to manifolds of dimension higher than four, along with a metric. In these cases the word should be used with qualifiers such as "any" or "generalized". A ''Riemannian spacetime'' is like a Lorentzian spacetime, but with a positive definite metric.
A [[frame]] over [[spacetime]] is a [[Cl(1,3)]] vector valued [[1-form]] field (a [[Clifform]]),
$$
\f{e} = \f{dx^a} \lp e_a \rp^\mu \ga_\mu
$$
which may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as
\begin{eqnarray}
\f{e} &=& \f{e^\mu} \ga_\mu =
\lb \begin{array}{cc}
0 & \f{e_L} \\
\f{e_R} & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \f{e^\mu} \bar{\si}_\mu \\
\f{e^\mu} \si_\mu & 0
\end{array} \rb
\\
&=&
\lb \begin{array}{cccc}
0 & 0 & \f{e^0}-\f{e^3} & -\f{e^1}+i\f{e^2} \\
0 & 0 & -\f{e^1}-i\f{e^2} & \f{e^0}+\f{e^3} \\
\f{e^0}+\f{e^3} & \f{e^1}-i\f{e^2} & 0 & 0 \\
\f{e^1}+i\f{e^2} & \f{e^0}-\f{e^3} & 0 & 0
\end{array} \rb
\end{eqnarray}
with left and right [[chiral]] parts, $\f{e_{L/R}}$, represented by $2\times2$ complex matrix (or [[quaternion]]) valued 1-forms. The inverse of the frame is Cl(1,3) vector valued [[tangent vector]] field,
$$
\ve{e} = \ga^\mu \lp e_\mu \rp^a \ve{\pa_a}
$$
which may be written as a matrix valued tangent vector as
\begin{eqnarray}
\ve{e} &=& \ga^\mu \ve{e_\mu} =
\lb \begin{array}{cc}
0 & \ve{e_R} \\
\ve{e_L} & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \ve{e_0} + \si^\va \ve{e_\va} \\
\ve{e_0} - \si^\va \ve{e_\va} & 0
\end{array} \rb
\\
&=&
\lb \begin{array}{cccc}
0 & 0 & \ve{e_0}+\ve{e_3} & \ve{e_1}-i\ve{e_2} \\
0 & 0 & \ve{e_1}+i\ve{e_2} & \ve{e_0}-\ve{e_3} \\
\ve{e_0}-\ve{e_3} & -\ve{e_1}+i\ve{e_2} & 0 & 0 \\
-\ve{e_1}-i\ve{e_2} & \ve{e_0}+\ve{e_3} & 0 & 0
\end{array} \rb
\end{eqnarray}
A [[spin connection]] over [[spacetime]] is a [[Cl(1,3) bivector]] valued 1-form (a [[Clifform]]),
$$
\f{\om} = \f{dx^a} \ha \om_a{}^{\mu \nu} \ga_{\mu \nu}
$$
which may be written as a matrix valued 1-form, using the [[Weyl representation|Dirac matrices]], as
\begin{eqnarray}
\f{\om} &=& \ha \f{\om}{}^{\mu \nu} \ga_{\mu \nu} =
\lb \begin{array}{cc}
\f{\om}{}_L & 0 \\
0 & \f{\om}{}_R
\end{array} \rb
=
\lb \begin{array}{cc}
\f{\om}{}^{0 \va} \si_\va - i \ha \f{\om}{}^{\va \ze} \ep_{\va \ze \ta} \si_\ta & 0 \\
0 & -\f{\om}{}^{0 \va} \si_\va - i \ha \f{\om}{}^{\va \ze} \ep_{\va \ze \ta} \si_\ta
\end{array} \rb \\
&=&
\lb \begin{array}{cccc}
\f{\om}{}^{03}- i \f{\om}{}^{12} & \f{\om}{}^{01}+\f{\om}{}^{13}-i \f{\om}{}^{02}- i \f{\om}{}^{23} & 0 & 0 \\
\f{\om}{}^{01}- \f{\om}{}^{13}+i \f{\om}{}^{02}-i \f{\om}{}^{23} & -\f{\om}{}^{03}+i \f{\om}{}^{12} & 0 & 0 \\
0 & 0 & -\f{\om}{}^{03}- i \f{\om}{}^{12} & -\f{\om}{}^{01}+\f{\om}{}^{13}+i \f{\om}{}^{02}- i \f{\om}{}^{23} \\
0 & 0 & -\f{\om}{}^{01}- \f{\om}{}^{13}-i \f{\om}{}^{02}-i \f{\om}{}^{23} & -\f{\om}{}^{03}+i \f{\om}{}^{12}
\end{array} \rb
\end{eqnarray}
with left and right [[chiral]] parts, $\f{\om}{}_{L/R}$, projected out by the [[left/right chirality projector]]. These $2\times2$ complex matrix (or [[quaternion]]) valued 1-forms satisfy $\f{\om}{}_L^\da = - \f{\om}{}_R$, using the [[Hermitian]] conjugate. The spacetime spin connection is the [[connection]] of the spacetime [[Lorentz group]], valued in [[spin(1,3)]].
The ''spacetime spin group'', $Spin(1,3)$, equivalent to $Spin(3,1)$, with Lie algebra [[spin(1,3)]], is the double cover of the special [[Lorentz group]], $SO(1,3)$. The spacetime spin group is a subgroup of the ''spacetime pin group'', $Pin(1,3)$, not equivalent to $Pin(3,1)$, which is a double cover of the orthogonal group, $O(1,3)$, and a subgroup of the [[Clifford group]]. We have
$$
\mbox{Spin}{}^+(1,3) \rtimes \{1,P,T,PT \} = \mbox{Spin}(1,3) \rtimes \{1,P \} = \mbox{Pin}(1,3) \subset Cl^*
$$
In which the identity component of the spacetime spin group is $Spin^+(1,3)$, the ''spacetime [[orthochronous spin group|Lorentz group]]'' -- the the group of [[Clifford rotation]]s obtained by [[exponentiating|exponentiation]] [[Cl(1,3) bivector]]s -- equivalent to $SL(2,\mathbb{C})$, and the double cover of $SO^+(1,3)$.
The $\mbox{Pin}(1,3)$ group (and thus its subgroups, including the spacetime spin group) elements, $U$, act as [[spacetime]] [[Cl(1,3)]] [[Clifford algebra]] elements on [[Dirac spinor]]s, $\Ps \to U \Ps$, and via the [[Clifford adjoint]] on [[Clifford vector]]s and other Clifford algebra elements. $A \to U A \, U^-$. Dirac spinors are not irreducible representation space of $Spin(1,3)$, since they decompose into left and right [[chiral]] parts, via the [[left/right chirality projector]], acted on by $SL(2,\mathbb{C})$. Dirac spinors (technically ''Dirac pinor''s) are a irreducible representation space of $Pin(1,3)$. The $\mbox{Spin}{}^+(1,3)$ subgroup elements of $\mbox{Pin}(1,3)$ consist of combinations of [[spatial rotation]]s and [[Lorentz boost]]s,
$$
U_\th = e^{-\ha \ga \ga_0 n_1 \th} = \cos{\fr{\th}{2}} - \ga \, \ga_0 n_1 \sin{\fr{\th}{2}}
\s \;\;\;\;\;
U_\ze = e^{-\ha \ga_0 n_2 \ze} = \cosh{\fr{\ze}{2}} - \ga_0 n_2 \sinh{\fr{\ze}{2}}
$$
which can combine to make any scalar plus bivector plus [[pseudoscalar]] element, $U \in \mbox{Spin}{}^+(1,3)$. These Clifford rotation elements can alternatively be built from successive [[Clifford reflection]]s. Other elements of $\mbox{Spin}(1,3)$, and of $\mbox{Pin}(1,3)$, corresponding to [[CPT symmetry]], can be built from other reflections, such as ''space reflection'', $U_P = \ga_0$, directly related to [[parity conjugation|parity conjugate]], and ''time reflection'', $U_T = \ga_0 \ga = \ga_1 \ga_2 \ga_3$, directly related to [[time conjugation|time conjugate]], as well as their combination, $U_{PT} = \ga$. These reflections anticommute, $P T = - T P$, and close to form a subgroup of the [[CPT group]]. For $\mbox{Pin}(1,3)$ these satisfy $\{ U_P^2 = 1, U_T^2 = 1, U_{PT}^2 = -1 \}$, while for $\mbox{Pin}(3,1)$ we have $\{ U'{}_P^2 = -1, U'{}_T^2 = -1, U'{}_{PT}^2 = -1 \}$, so the groups are not isomorphic. Also note that $\mbox{Spin}(1,3) = \mbox{Spin}{}^+(1,3) \rtimes \{ 1, PT \}$. Since we like working with $Spin{}^+(1,3)$ because of the isomorphism to $SL(2,\mathbb{C})$, but we would also like our spinors to change sign under $P^2$, corresponding to a $2 \pi$ rotation, as in $\mbox{Pin}(3,1)$, we can fudge a bit by defining $U'_P = i \ga_0$, which lets us get the best of both worlds.
In physics, $P$ symmetry is violated by the weak interaction, and $CP$ is violated by the Yukawa interaction with the Higgs field, but $PT$ holds (with $T=T_U$ here, the unitary $T$, not the $T' = C T_U$), so $\mbox{Spin}(1,3) = \mbox{Spin}^+(1,3) \rtimes \{1, PT \}$ appears to be a good symmetry of nature. With $PT \sim \ga$, the [[center]] of the spacetime spin group is $\mathbb{Z}_2 \times \mathbb{Z}_2 = \{ \pm1, \pm \ga \}$.
A ''spatial rotation'' is a [[Lorentz rotation]] of a [[spacetime]] vector or [[Dirac spinor]] along a spatial ''rotation vector'', $a = \ga_\pi a^\pi$, and corresponds to a [[Clifford rotation]] by the [[Cl(1,3) bivector]] corresponding to the perpendicular spatial plane,
$$
B = - \ga \ga_0 a = \ga_{123} \ga_\pi a^\pi = - \ha \ga_\si \ga_\rh \ep^{\si \rh \pi} a^\pi
$$
The rotation angle is the magnitude of the rotation vector, $\th = |a|$, and the ''rotation axis'' is $n = \fr{a}{\left| a \right|}$, so $a = \th n$. The ''spatial rotor'' is then
$$
U_a = e^{\ha B} = e^{-\ha \ga \ga_0 n \th} = \cos{\fr{\th}{2}} - \ga \ga_0 n \sin{\fr{\th}{2}}
$$
Any [[spacetime]] Clifford vector, $u$, decomposes into its temporal and spatial parts, and its spatial part splits into parts parallel and perpendicular to the rotation axis,
$$
u = u^\mu \ga_\mu = u^0 \ga_0 + u^\pi \ga_\pi
= u_t + u_s = u_t + u_\parallel + u_\perp
$$
with $u_\parallel = n u^n$ and $u_\perp = u_s - u_\parallel$, in which $u^n = n \cdot u$ is the magnitude of $u$ along the rotation axis. Working through some trig and Clifford algebra, we see that the ''rotated vector'' is
$$
\begin{array}{rcl}
u' \!\!&\!\!=\!\!&\!\! u^\mu \ga'_\mu = u^\mu L^\nu{}_\mu \ga_\nu = U_a \, u \, U_a^- \\
\!\!&\!\!=\!\!&\!\! \lp \cos{\fr{\th}{2}} - \ga \ga_0 n \sin{\fr{\th}{2}} \rp \lp u_t + u_\parallel + u_\perp \rp \lp \cos{\fr{\th}{2}} + \ga \ga_0 n \sin{\fr{\th}{2}} \rp \\
\!\!&\!\!=\!\!&\!\! u_t + u_\parallel + u_\perp \cos{\th} + u_\top \sin{\th}
\end{array}
$$
in which $u_\top = u_\perp n \, \ga_{123} $ is a spatial vector perpendicular to both $n$ and $u$.
A ''rotated Dirac spinor'', using the Weyl rep of the [[Dirac matrices]] and the corresponding Cl(1,3) bivectors, is
$$
\ps' = U_a \, \ps =
\lb \begin{array}{cc}
\cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}} & 0 \\
0 & \cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}}
\end{array} \rb
\lb \begin{array}{c}
\ps_L \\
\ps_R
\end {array} \rb
$$
in which the ''left and right chiral rotation rotor''s, $U^{L/R}_a = \cos{\fr{\th}{2}} + i n^\pi \si_\pi \sin{\fr{\th}{2}}$, are [[unitary]], $U^{L/R \, \da}_a U^{L/R}_a = 1$, and thus so is $U_a$.
It is sometimes useful to factor the co[[frame]] matrix, $\lp e_i\rp^\al$, into a ''conformal scalar'', $s$, times the ''special coframe matrix'',
\begin{eqnarray}
\lp e_i\rp^\al &=& s \lp e^s_i\rp^\al\\
\f{e} = s \f{e^s}
\end{eqnarray}
such that the frame [[determinant]] depends only on the conformal scalar
\[ \ll e \rl = \det \lp e_i\rp^\al = s^n \]
and the special coframe matrix is restricted to satisfy
\[ \ll e^s \rl = \det \lp e^s_i\rp^\al = 1 \]
This factorization allows separate consideration of this conformal factor, $s$, which carries [[units]] of time, $T$.
Ref:
*L. Smolin
**[[The quantization of unimodular gravity and the cosmological constant problem|http://arxiv.org/abs/0904.4841v1]]
*Philip D. Mannheim
**[[Making the Case for Conformal Gravity|http://arxiv.org/abs/1101.2186]]
The ''special orthogonal group'' of order $n$, $G=SO(n)$, is the [[simple]], compact, connected, $\ha n (n-1)$ dimensional [[Lie group]] of [[orthogonal]] $n \times n$ real matrices, $R$, with unit [[determinant]]. The Lie algebra, $so(n)$, with $n \times n$ basis generators that are antisymmetric and traceless, is isomorphic to the corresponding [[spin Lie algebra]].
The ''generalized special orthogonal group'', $SO(p,q)$, noncompact for $q>0$, are $n \times n$ real matrices, $R$, with unit determinant and signature $(p,q)$, satisfying
$$
R^T \lb
\ba{cccccc}
+ 1 & & & & & \\
& \ddots & & & & \\
& & +1 & & & \\
& & & -1 & & \\
& & & & \ddots & \\
& & & & & -1 \\
\ea
\rb R =
\lb \ba{cccccc}
+ 1 & & & & & \\
& \ddots & & & & \\
& & +1 & & & \\
& & & -1 & & \\
& & & & \ddots & \\
& & & & & -1 \\
\ea \rb
$$
for $p$ positive and $q$ negative on the diagonal. Elements of the corresponding Lie algabra, $A \in so(p,q)$, are traceless and satisfy $A^T D + D A = 0$, with $D$ the diagonal matrix above. The fundamental [[representation space]] of the generalized special orthogonal group inherits this [[metric]], $v^T D v \in \mathbb{R}$. The matrix $D$ can be any [[similar|Dirac matrices]] matrix, as long as its spectral decomposition produces the diagonal [[eigen]]value matrix above.
The ''special unitary group'' of order $n$, $G=SU(n)$, is the [[simple]], compact, connected, $(n^2-1)$ dimensional [[Lie group]] of [[unitary]] $n \times n$ complex matrices, $U$, with unit [[determinant]]. The [[Lie algebra]], $\mathfrak{g} = su(n)$, has $n \times n$ basis generators that are anti-[[Hermitian]] and [[trace]]less.
The ''generalized special unitary group'', $SU(p,q)$, noncompact for $q>0$, are $n \times n$ complex matrices, $U$, with unit determinant and signature $(p,q)$, satisfying
$$
U^\da \lb
\ba{cccccc}
+ 1 & & & & & \\
& \ddots & & & & \\
& & +1 & & & \\
& & & -1 & & \\
& & & & \ddots & \\
& & & & & -1 \\
\ea
\rb U =
\lb \ba{cccccc}
+ 1 & & & & & \\
& \ddots & & & & \\
& & +1 & & & \\
& & & -1 & & \\
& & & & \ddots & \\
& & & & & -1 \\
\ea \rb
$$
for $p$ positive and $q$ negative on the diagonal. Elements of the corresponding Lie algabra, $A \in su(p,q)$, are traceless and satisfy $A^\da D + D A = 0$, with $D$ the diagonal matrix above. The fundamental [[representation space]] of the generalized special unitary group inherits this [[Hermitian form]], $\Ps^\da D \Ps \in \mathbb{C}$. The matrix $D$ can be any [[similar|Dirac matrices]] matrix, as long as its spectral decomposition produces the diagonal [[eigen]]value matrix above.
Ref:
*Ali H. Chamseddine and Alain Connes
**[[Quantum Gravity Boundary Terms from Spectral Action|http://arxiv.org/abs/0705.1786]]
**[[Gravity and the standard model with neutrino mixing|http://arxiv.org/abs/hep-th/0610241]]
**[[The Spectral Action Principle|http://arxiv.org/abs/hep-th/9606001]]
An [[Ehresmann principal bundle connection]] is a [[vector valued 1-form|vector valued form]],
$$
\f{\ve{\cal A}} = \f{A^B} \ve{\xi^L_B} + \f{\ve{\cal I}}
$$
with an interesting [[spectral decomposition|eigen]]. The the eigenequations for the eigenvectors and eigenforms are solved by:
\begin{eqnarray}
\ve{\xi^L_B} \f{\ve{\cal A}} &=& \ve{\xi^L_B}\\
\ve{h_i} \f{\ve{\cal A}} &=& 0\\
\f{\ve{\cal A}} \f{dx^i} &=& 0\\
\f{\ve{\cal A}} \lp \f{\xi_L^B} + \f{A^B} \rp &=& \lp \f{\xi_L^B} + \f{A^B} \rp\\
\end{eqnarray}
with the horizontal eigenvectors, $\ve{h_i} = \ve{\pa_i} - A_i{}^B \ve{\xi^L_B} \in \ve{\De_H}$. These eigenvectors and eigenforms are chosen to satisfy the normality conditions,
\begin{eqnarray}
\ve{h_i} \f{dx^j} &=& \de_i^j\\
\ve{h_i} \lp \f{\xi_L^B} + \f{A^B} \rp &=& 0\\
\ve{\xi^L_B} \f{dx^j} &=& 0\\
\ve{\xi^L_B} \lp \f{\xi_L^C} + \f{A^C} \rp &=& \de_B^C
\end{eqnarray}
and the resulting ''spectral decomposition of the Ehresmann principal bundle connection'' is
$$
\f{\ve{\cal A}} = \lp \f{\xi_L^B} + \f{A^B} \rp \ve{\xi^L_B}
$$
<<ListTagged speculative>>
In flat three dimensional space, the most common coordinates are the ''cartesian coordinates'', $(x^1 = x, x^2 = y, x^3 = z)$. The second most common coordinates are ''spherical coordinates'', $(r,\th,\ph)$, describing the distance, $r$, from some "center" point, the angle, $\th$, from the z-axis, and the angle, $\ph$, from the x-axis. In terms of these ''angular'' variables, $(a^1 = r, a^2 = \th, a^3 = \ph)$, the equivalent cartesian coordinates are
\begin{eqnarray}
x^1 &=& r \sin(\th) \cos(\ph) \\
x^2 &=& r \sin(\th) \sin(\ph) \\
x^3 &=& r \cos(\th)
\end{eqnarray}
Elements, $T_A$, of the ''spin [[Lie algebra]]'', $spin(n)$, equivalent to the [[special orthogonal group]] Lie algebra, $so(n)$, may be represented by $\ha n (n-1)$ antisymmetric $n \times n$ matrices, $T_A \sim J_{ij}$, antisymmetric in the $ij$ labels, or alternatively by [[bivectors|Clifford basis elements]], $T_A \sim \ga_{\al \be}$, of the $Cl(n,0)$ [[Clifford algebra]]. The bivectors satisfy the [[Clifford basis identities]]:
$$
\lb \ga_{\al \be}, \ga_{\ga \de} \rb = 2 \left\{ - \et_{\al \ga} \ga_{\be \de} + \et_{\al \de} \ga_{\be \ga} + \et_{\be \ga} \ga_{\al \de} - \et_{\be \de} \ga_{\al \ga} \right\} = C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb} \ga_{\ep \up}
$$
giving the $spin(n)$ structure constants,
$$
C_{\lb{\al \be}\rb \lb \ga \de \rb}{}^{\lb \ep \up \rb}
= 2 \left\{ - \et_{\al \ga} \de^{\lb\ep \up\rb}_{\be \de} + \et_{\al \de} \de^{\lb\ep \up\rb}_{\be \ga} + \et_{\be \ga} \de^{\lb\ep \up\rb}_{\al \de} - \et_{\be \de} \de^{\lb\ep \up\rb}_{\al \ga} \right\}
$$
with $\et_{\al \ga} = \de_{\al \ga}$ for $Cl(n,0)$. The [[Killing form]] for $spin(n)$ is
$$
g_{\lb \al \be \rb \lb \ga \de \rb} = C_{\lb \al \be \rb \lb \ep \up \rb}{}^{\lb \ze \et \rb} C_{\lb \ga \de \rb \lb \ze \et \rb}{}^{\lb \ep \up \rb}
= 8 \lp n-2 \rp \lp \et_{\al \de} \et_{\be \ga} - \et_{\al \ga} \et_{\be \de} \rp
$$
The Lie algebra, $spin(p,q)=so(p,q)$, of the generalized [[spin group]], $Spin(p,q)$, is similarly described by the bivectors of $Cl(p,q)$. The spin Lie algebra with only one positive dimension, $spin(1,n-1)$, is the [[Lorentz algebra]].
The ''spin connection'', a Clifford bivector valued 1-form,
$$
\f{\om} = \f{dx^i} \ha \om_i{}^{\al \be} \ga_{\al \be} \in \f{Cl^2}
$$
serves as the [[Clifford connection]] for the [[Clifford vector bundle]] or for the [[graded Clifford bundle|Clifford vector bundle]], since the structure group for both of these is the group of [[Clifford rotation]]s. The [[vector bundle covariant derivative|vector bundle connection]] acting on the [[Clifford basis vectors]] for the fiber of either bundle is
$$
\f{\na} \ga_\al = \f{\om} \times \ga_\al = \f{\om}^\be{}_\al \ga_\be
$$
This gives the covariant derivative acting on the [[Clifford basis elements]] of the graded Clifford bundle as the [[cross product|antisymmetric bracket]] of the connection with the basis element,
$$
\f{\na} \ga_{\al \dots \be} = \f{\om} \times \ga_{\al \dots \be}
$$
and the covariant derivative for a section of any Clifford bundle,
$$
\f{\na} C = \f{d} C + \f{\om} \times C
$$
This covariant derivative necessarily preserves [[Clifford grade]]. The ''Clifford vector bundle covariant derivative'' acting on any [[Clifform]] is
$$
\f{\na} \nf{C} = \f{d} \nf{C} + \f{\om} \times \nf{C}
$$
We can also [[Cliffordize|Cliffordization]] the covariant derivative to get the [[Clifford covariant derivative]], $\na = \ve{e} \f{\na}$, which acts on any Clifford bundle section as
$$
\na C = d C + \ve{e} (\f{\om} \times C)
$$
in which $d = \ve{e} \f{d}$. Expanding the product leads us to define the ''Cliffordized spin connection'',
$$
\om = \ve{e} \f{\om} = \ha \ga^\al \om_\al{}^{\be\de} \ga_{\be\de}
= \ha \om^{\al\be\de} \ga_{\al\be\de} + \om_\al{}^{\al\de} \ga_\de
$$
a Clifford trivector plus vector.
Through the identification of orthonormal basis vectors with Clifford basis vectors, $\ve{e_\al} \leftrightarrow \ga_\al$, via the [[frame]], $\ve{e_\al} \f{e} = \ga_\al$, the spin connection coefficients equal the [[tangent bundle spin connection|tangent bundle connection]] coefficients, $\om_i{}^\be{}_\al = w_i{}^\be{}_\al$, since the defining equation,
$$
\f{\na} \ve{e_\al} = \f{w}^\be{}_\al \ve{e_\be}
$$
must agree with the similar equation for the covariant derivative of Clifford basis elements.
refs:
*[[Conrady and Freidel - Path integral representation of spin foam models of 4d gravity|http://arxiv.org/abs/0806.4640]]
**good recent review
*Thiemann, p464
*Rovelli, p249
*[[Freidel and Krasnov - Spin Foam Models and the Classical Action Principle|http://arxiv.org/abs/hep-th/9807092v2]]
*[[Freidel and Starodubtsev - Quantum gravity in terms of topological observables|http://arxiv.org/abs/hep-th/0501191v2]]
*[[Freidel and Krasnov - A New Spin Foam Model for 4d Gravity|http://arxiv.org/abs/0708.1595v1]]
*[[Baez - An Introduction to Spin Foam Models of Quantum Gravity and BF Theory|http://arxiv.org/abs/gr-qc/9905087]]
**good review
*[[Baez - Spin Foam Models|http://xxx.lanl.gov/abs/gr-qc/9709052]]
**one of the first introductions
*[[Baez - Spin Networks, Spin Foams, and Quantum Gravity|http://math.ucr.edu/home/baez/foam/]]
**collection of lecture tiddlers
*[[Baez - Towards a Spin Foam Model of Quantum Gravity|http://www.youtube.com/watch?v=cVfE6aK57S8]]
**youtube and pdf tiddlers from Loops 05
*[[Perez - Spin Foam Models for Quantum Gravity|http://arxiv.org/abs/gr-qc/0301113]]
**review paper
*[[Conrady - Geometric spin foams, Yang-Mills theory and background-independent models|http://arxiv.org/abs/gr-qc/0504059]]
**recent review, related to lattice gauge theory
A ''spin group'', $G=Spin(n)$, is the double cover of the [[special orthogonal group]]. The two groups have the same [[spin Lie algebra]], $spin(n) = so(n)$. The ''generalized spin group'', $Spin(p,q)$, is the double cover of the corresponding generalized special orthogonal group. The generalized spin group is a [[subgroup]] of the corresponding [[pin group]] and [[Clifford group]],
$$
Spin(p,q) = \{ U \in Cl^{+}(p,q) \ni U \tilde{U} = \pm 1 \}
$$
consisting of even grade Clifford elements of the pin group. The ''orthochronous spin group'' is a subgroup of the spin group,
$$
Spin^+(p,q) = \{ U \in Cl^{+}(p,q) \ni U \tilde{U} = + 1 \}
$$
in which $\tilde{U}$ is the [[Clifford reverse|Clifford conjugate]]. [check these] Spin group elements act on Clifford elements and spinors as [[Clifford rotation]]s.
The [[center]] of $Spin(p,q)$ is $\mathbb{Z}_2$ if $p$ or $q$ are odd, $\mathbb{Z}_4$ if $p$ and $q$ are even and $p+q = 4 k + 2$, or $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ if $p$ and $q$ are even and $p+q = 4 k$.
good new paper of tiddler, by Levine and Terno:
[[Reconstructing Quantum Geometry from Quantum Information: Area Renormalisation, Coarse-Graining and Entanglement on Spin Networks|http://arxiv.org/abs/gr-qc/0603008]]
The ''spin operator'', $\ve{S} = S^\va \ve{e}_\va$, is a specific [[Cl(1,3) bivector]] valued spatial [[tangent vector]] that can act on a [[Dirac spinor]], $\ve{S} \Ps$, or on a Cliffordized [[electromagnetic field]], $\ve{S} \times A$. In the [[Weyl representation|Dirac matrices]] it has Clifford algebra components
$$
S^\va = \fr{i}{2} \ga \ga^0 \ga^\va = \ha \si_0 \otimes \si_\va \;\;\;\;\; \text{and so} \;\;\;\;\; \ve{S} = \ha \si_0 \otimes \si_\va \, \ve{e}_\va = \ha \si_0 \otimes \ve{S}_s
$$
in which $\ve{S_s} = \si_\va \ve{e}_\va$ is the ''spatial spin operator'' that acts on [[Weyl spinor]]s.
By abuse of language, the "spin operator", $S_z$, is sometimes also defined as the spin operator component along the z axis,
$$
S_z = \ve{S} \f{e}^3 = \fr{i}{2} \ga \ga^0 \ga^3 = - \fr{i}{2} \ga_1 \ga_2 = \ha \si_0 \otimes \si_3
$$
which is simply $\si_3$ acting on Weyl spinors.
[[Eigen|eigen]]values of the spin operator, multiplied by ''Planck's constant'', $\hbar$ (which is 1 in natural [[units]]), correspond to ''spin angular momentum''.
//spin ½ vs spin 1 and... spin 2?//
A [[massless quantum Dirac spinor]],
$$
\ud{\hat{\Ps}} = \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{\, p}^{L/R} u_p^{L/R} e^{-i p_\mu x^\mu} + \ud{\hat{b}}_{\, p}^{R/L \, \da} v_p^{L/R} e^{+i p_\mu x^\mu} \rp
$$
and its [[adjoint|massless quantum Dirac spinor adjoint]],
$$
\ud{\hat{\bar{\Psi}}} = \ud{\hat{\Psi}}^\da \ga^0
= \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{L/R \, \da} {\bar{u}}_p^{L/R} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_p^{R/L} {\bar{v}}_p^{L/R} e^{-i p_\mu x^\mu} \rp
$$
include [[creation and annihilation operators|creation and annihilation operators of a massless quantum Dirac spinor]] for antiparticles and particles of left and right chirality, for all possible momenta. The particle momentum direction (a 3D unit vector) determines particle [[spin|spin operator]], and vice versa, via [[massless Dirac solutions]]. A right-handed particle with spin up, $\ch_+=\lb \begin{array}{c} 1 \\ 0 \end{array} \rb$, or a left-handed particle with spin down, $\ch_-=\lb \begin{array}{c} 0 \\ -1 \end{array} \rb$, are both traveling at the speed of light in the $\hat{z}$ direction. A right-handed particle with spin down, $\ch_+=\lb \begin{array}{c} 0 \\ 1 \end{array} \rb$, or a left-handed particle with spin up, $\ch_-=\lb \begin{array}{c} 1 \\ 0 \end{array} \rb$, are both traveling at the speed of light in the $-\hat{z}$ direction. In this way, the integrand of the massless quantum Dirac spinor, for momenta in the positive or negative $z$ directions, can be written as
$$
\ud{\hat{a}}_{+E\hat{z}}^{L/R} u_{z}^{L/R} e^{-i E (t - z)}
+ \ud{\hat{a}}_{-E\hat{z}}^{L/R} u_{-z}^{L/R} e^{-i E (t + z)}
+ \ud{\hat{b}}_{+E\hat{z}}^{R/L \, \da} v_z^{L/R} e^{+i E (t - z)}
+ \ud{\hat{b}}_{-E\hat{z}}^{R/L \, \da} v_{-z}^{L/R} e^{+i E (t + z)}
=
\lb
\begin{array}{c}
\ud{\hat{a}}_{-}^{L} e^{-i E (t + z)} \!-\! \ud{\hat{b}}_{-}^{R \, \da} e^{+i E (t + z)} \\
- \ud{\hat{a}}_{+}^{L} e^{-i E (t - z)} \!+\! \ud{\hat{b}}_{+}^{R \, \da} e^{+i E (t - z)} \\
\ud{\hat{a}}_{+}^{R} e^{-i E (t - z)} \!-\! \ud{\hat{b}}_{+}^{L \, \da} e^{+i E (t - z)} \\
\ud{\hat{a}}_{-}^{R} e^{-i E (t + z)} \!-\! \ud{\hat{b}}_{-}^{L \, \da} e^{+i E (t + z)} \\
\end{array}
\rb
$$
Similarly, the integrand of the massless quantum Dirac spinor adjoint can be written as
$$
\begin{array}{c}
\ud{\hat{a}}_{+E\hat{z}}^{L/R \, \da} {\bar{u}}_z^{L/R} e^{+i E (t - z)}
+ \ud{\hat{a}}_{-E\hat{z}}^{L/R \, \da} {\bar{u}}_{-z}^{L/R} e^{+i E (t + z)}
+ \ud{\hat{b}}_{+E\hat{z}}^{R/L} {\bar{u}}_z^{L/R} e^{-i E (t - z)}
+ \ud{\hat{b}}_{-E\hat{z}}^{R/L} {\bar{u}}_{-z}^{L/R} e^{-i E (t + z)}
= \\
\lb
\begin{array}{cccc}
\ud{\hat{a}}_{+}^{R \, \da} e^{+i E (t - z)} \!-\! \ud{\hat{b}}_{+}^{L} e^{-i E (t - z)} &
\ud{\hat{a}}_{-}^{R \, \da} e^{+i E (t + z)} \!-\! \ud{\hat{b}}_{-}^{L} e^{-i E (t + z)} &
\ud{\hat{a}}_{-}^{L \, \da} e^{+i E (t + z)} \!-\! \ud{\hat{b}}_{-}^{R} e^{-i E (t + z)} &
- \ud{\hat{a}}_{+}^{L \, \da} e^{+i E (t - z)} \!+\! \ud{\hat{b}}_{+}^{R} e^{-i E (t - z)}
\end{array}
\rb
\end{array}
$$
These are acted on by the [[spin(1,3)]] [[Lorentz algebra]], with [[Cartan subalgebra|Lie algebra structure]] basis elements chosen to be the (anti-[[Hermitian]]) rotation, $J_3 = \ha \ga_{12} = -\fr{i}{2} \si_0 \otimes \si_3 = - i \, S_z$, and (Hermitian) boost, $K_3 = \ha \ga_{03} = \ha \si_3 \otimes \si_3$, [[Cl(1,3) bivector]]s. Typically, such as for the anti-Hermitian rotation operator, $O = J_3$, there is a corresponding anti-Hermitian operator on the [[infinite-dimensional unitary representation]] space operators, such as $\hat{O} = \hat{J}_{\! 3}$, satisfying
$$
\lb \hat{O}, \ud{\hat{\Ps}} \rb = O \, \ud{\hat{\Ps}} \s
\lb \hat{J}_{\! 3}, \ud{\hat{\Ps}} \rb = J_3 \, \ud{\hat{\Ps}}
$$
and, for the adjoint,
$$
\lb \hat{O}^\da, \ud{\hat{\bar{\Psi}}} \rb = - \ud{\hat{\bar{\Psi}}} \, \lp \ga_0 O^\da \ga^0 \rp \s
\lb \hat{J}_{\! 3}, \ud{\hat{\bar{\Psi}}} \rb = - \ud{\hat{\bar{\Psi}}} \, J_3
$$
For the Hermitian boost operator, $K_3$, we take the corresponding operator, $\hat{K}_3$ on the infinite-dimensional unitary representation space to be anti-Hermitian to preserve quantum unitarity, and so we have
$$
\lb \hat{K}_3, \ud{\hat{\Ps}} \rb = K_3 \, \ud{\hat{\Ps}} \s
\lb \hat{K}_3, \ud{\hat{\bar{\Psi}}} \rb = \ud{\hat{\bar{\Psi}}} \, \lp \ga_0 K_3^\da \ga^0 \rp = - \ud{\hat{\bar{\Psi}}} \, K_3
$$
These formulas allow us to find the rotation and boost quantum numbers, $j_3$ and $k_3$, of the annihilation and creation operators,
$$
\begin{array}{rclcrcl}
\lb \hat{J}_{\! 3}, \ud{\hat{a}}_{\mp}^{L} \rb \!\!&\!\!=\!\!&\!\! \mp \fr{i}{2} \ud{\hat{a}}_{\mp}^{L} & \s &
\lb \hat{J}_{\! 3}, \ud{\hat{b}}_{\mp}^{R \, \da} \rb \!\!&\!\!=\!\!&\!\! \mp \fr{i}{2} \ud{\hat{b}}_{\mp}^{R \, \da} \\
\lb \hat{J}_{\! 3}, \ud{\hat{a}}_{\pm}^{R} \rb \!\!&\!\!=\!\!&\!\! \mp \fr{i}{2} \ud{\hat{a}}_{\pm}^{R} & \s &
\lb \hat{J}_{\! 3}, \ud{\hat{b}}_{\pm}^{L \, \da} \rb \!\!&\!\!=\!\!&\!\! \mp \fr{i}{2} \ud{\hat{b}}_{\pm}^{L \, \da} \\
\lb \hat{K}_3, \ud{\hat{a}}_{\mp}^{L} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{a}}_{\mp}^{L} & \s &
\lb \hat{K}_3, \ud{\hat{b}}_{\mp}^{R \, \da} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{b}}_{\mp}^{R \, \da} \\
\lb \hat{K}_3, \ud{\hat{a}}_{\pm}^{R} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{a}}_{\pm}^{R} & \s &
\lb \hat{K}_3, \ud{\hat{b}}_{\pm}^{L \, \da} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{b}}_{\pm}^{L \, \da} \\
\\
\lb \hat{J}_{\! 3}, \ud{\hat{a}}_{\pm}^{R \, \da} \rb \!\!&\!\!=\!\!&\!\! \pm \fr{i}{2} \ud{\hat{a}}_{\pm}^{R \, \da} & \s &
\lb \hat{J}_{\! 3}, \ud{\hat{b}}_{\pm}^{L} \rb \!\!&\!\!=\!\!&\!\! \pm \fr{i}{2} \ud{\hat{b}}_{\pm}^{L} \\
\lb \hat{J}_{\! 3}, \ud{\hat{a}}_{\mp}^{L \, \da} \rb \!\!&\!\!=\!\!&\!\! \pm \fr{i}{2} \ud{\hat{a}}_{\mp}^{L \, \da} & \s &
\lb \hat{J}_{\! 3}, \ud{\hat{b}}_{\mp}^{R} \rb \!\!&\!\!=\!\!&\!\! \pm \fr{i}{2} \ud{\hat{b}}_{\mp}^{R} \\
\lb \hat{K}_3, \ud{\hat{a}}_{\pm}^{R \, \da} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{a}}_{\pm}^{R \, \da} & \s &
\lb \hat{K}_3, \ud{\hat{b}}_{\pm}^{L} \rb \!\!&\!\!=\!\!&\!\! \mp \ha \ud{\hat{b}}_{\pm}^{L} \\
\lb \hat{K}_3, \ud{\hat{a}}_{\mp}^{L \, \da} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{a}}_{\mp}^{L \, \da} & \s &
\lb \hat{K}_3, \ud{\hat{b}}_{\mp}^{R} \rb \!\!&\!\!=\!\!&\!\! \pm \ha \ud{\hat{b}}_{\mp}^{R} \\
\end{array}
$$
Summarizing this structure, the table of spin and boost quantum numbers, $\om_S = s_z = - j_3^\mathbb{I} $ and $\om_T^\mathbb{R} = - k_3^\mathbb{R}$, of the annihilation and creation operators of a massless quantum Dirac spinor, with a relabeling for particle spin and helicity, is
$$
\begin{array}{ccc}
\begin{array}{|l|l|cc|c|c|}
\hline
& & \om_t^{\mathbb{R}} & \om_s^{\mathbb{I}} & h & q \\
\hline
a_L^\wedge & \ud{\hat{a}}_{-}^{L} & - & + & - & + \\
a_L^\vee & \ud{\hat{a}}_{+}^{L} & + & - & - & + \\
a_R^\wedge & \ud{\hat{a}}_{+}^{R} & + & + & + & + \\
a_R^\vee & \ud{\hat{a}}_{-}^{R} & - & - & + & + \\
\hline
\bar{a}_L^\wedge & \ud{\hat{b}}_{-}^{L} & - & + & - & - \\
\bar{a}_L^\vee & \ud{\hat{b}}_{+}^{L} & + & - & - & - \\
\bar{a}_R^\wedge & \ud{\hat{b}}_{+}^{R} & + & + & + & - \\
\bar{a}_R^\vee & \ud{\hat{b}}_{-}^{R} & - & - & + & - \\
\hline
\end{array}
& \s &
\begin{array}{|l|l|cc|c|c|}
\hline
& & \om_t^{\mathbb{R}} & \om_s^{\mathbb{I}} & h & q \\
\hline
a_L^{\wedge \, \da} & \ud{\hat{a}}_{-}^{L \, \da} & - & - & + & - \\
a_L^{\vee \, \da} & \ud{\hat{a}}_{+}^{L \, \da} & + & + & + & - \\
a_R^{\wedge \, \da} & \ud{\hat{a}}_{+}^{R \, \da} & + & - & - & - \\
a_R^{\vee \, \da} & \ud{\hat{a}}_{-}^{R \, \da} & - & + & - & - \\
\hline
\bar{a}_L^{\wedge \, \da} & \ud{\hat{b}}_{-}^{L \, \da} & - & - & + & + \\
\bar{a}_L^{\vee \, \da} & \ud{\hat{b}}_{+}^{L \, \da} & + & + & + & + \\
\bar{a}_R^{\wedge \, \da} & \ud{\hat{b}}_{+}^{R \, \da} & + & - & - & + \\
\bar{a}_R^{\vee \, \da} & \ud{\hat{b}}_{-}^{R \, \da} & - & + & - & + \\
\hline
\end{array}
\end{array}
$$
The [[helicity]] quantum number is $h \!=\! p_z s_z = 2 \, \om_T^\mathbb{R} \, \om_S$, and the $q = \pm 1$ quantum number is for whatever internal charge the particle has. Note that the helicity, spin, and charge, but not boost, quantum numbers of a creation operator are opposite that of the corresponding annihilation operator; and there is a weight match between annihilating particles and creating anti-particles, such as $\bar{a}_L^{\wedge \, \da} = a_R^{\vee}$. The spin and boost quantum numbers of the annihilation of a massless fermion match those of a Dirac spinor, as do those of the corresponding anti-fermion.
For the [[conjugates of a massless quantum Dirac spinor]], the weights change as
$$
\begin{array}{rclcrcl}
(a_L^\wedge)^C \!\!&\!\!=\!\!&\!\! \bar{a}_L^\wedge & \;\;\; & C : (\om_t, \om_s, q) \!\!&\!\!\mapsto\!\!&\!\! (-\om_t, \om_s, -q) \\
(a_L^\wedge)^P \!\!&\!\!=\!\!&\!\! - \, a_R^\wedge & \;\;\; & P : (\om_t, \om_s, q) \!\!&\!\!\mapsto\!\!&\!\! (-\om_t, \om_s, q) \\
(a_L^\wedge)^T \!\!&\!\!=\!\!&\!\! -i \, a_L^\vee & \;\;\; & T : (\om_t, \om_s, q) \!\!&\!\!\mapsto\!\!&\!\! (-\om_t, -\om_s, q) \\
\end{array}
$$
The [[Lorentz algebra]] in the four dimensional [[spacetime]] case (the ''spacetime Lorentz algebra'' or ''spacetime spin algebra'') is the $spin(1,3)$ [[spin Lie algebra]], with Lie algebra basis elements the six [[Cl(1,3) bivector]]s, $\ga_{\mu \nu}$. These can be scaled and relabelled as the three ''spatial rotation generators'', $J_\pi = \fr{1}{4} \ep_{\pi \rh \si} \ga_{\rh \si}$, such as $J_3 = \ha \ga_{12}$, and the three ''Lorentz boost generators'', $K_\pi = \ha \ga_{0 \pi}$, which satisfy the commutation relations,
$$
\left[ J_\pi, J_\rh \right] = \ep_{\pi \rh \si} J_\si \s
\left[ J_\pi, K_\rh \right] = \ep_{\pi \rh \si} K_\si \s
\left[ K_\pi, K_\rh \right] = - \ep_{\pi \rh \si} J_\si
$$
using the three dimensional [[permutation symbol]]. The Killing form for the three rotation generators is $-4$, and for the three boost generators is $+4$.
The spacetime spin algebra is the Lie algebra of the [[spacetime spin group]] -- the algebra of Cl(1,3) [[Clifford rotation]]s, including [[Lorentz boost]]s and [[spatial rotation]]s, which can act on vector and [[spinor]] [[representation space]]s. The spatial subalgebra of the spacetime Lorentz algebra is [[su(2)]], the Lie algebra of rotations in three dimensional space.
We can use these $spin(1,3)$ generators to define new, complex generators,
$$
N^L_\pi = \ha \lp J_\pi - i K_\pi \rp \s N^R_\pi = \ha \lp J_\pi + i K_\pi \rp
$$
satisfying
$$
\left[ N^L_\pi, N^L_\rh \right] = \ep_{\pi \rh \si} N^L_\si \s
\left[ N^R_\pi, N^R_\rh \right] = \ep_{\pi \rh \si} N^R_\si \s
\left[ N^L_\pi, N^R_\rh \right] = 0
$$
corresponding to representations of left or right [[chiral]]ity. These generators correspond to the decomposition
$$
spin(4,\mathbb{C}) = su(2,\mathbb{C})_L + su(2,\mathbb{C})_R = sl(2,\mathbb{C})_L + sl(2,\mathbb{C})_R
$$
using complex [[su(2)]] or equivalently complex [[sl(2)]]. A Lie algebra isomorphism, $spin(1,3) \sim sl(2,\mathbb{C})$, is established by decomposing either $N^L_\pi$ or $N^R_\pi$ into real and imaginary parts as above. The $su(2,\mathbb{C})_L$ generators are $T_A = N^L_A$, and the $sl(2,\mathbb{C})_L$ generators are
$$
L = -i N^L_1 \s J = N^L_2 \s K = -i N^L_3
$$
The [[Lie algebra structure]] of $spin(1,3)$ is found by identifying Cartan subalgebra basis elements, $\{ J_3, K_3 \}$, and then computing the roots (with rotation and boost quantum numbers, $\om_s$ and $\om_t$) and root vectors, with resulting nonzero Cartan-Weyl brackets,
$$
\ba{rclcrclcrcl}
[J_3,V_{+-}] \!\!&\!\!=\!\!&\!\! (+i) V_{+-} & \;\; & [K_3,V_{+-}] \!\!&\!\!=\!\!&\!\! (-1) V_{+-} & \;\; & V_{+-} \!\!&\!\!=\!\!&\!\! \ha \lp - J_1 + i J_2 + i K_1 + K_2 \rp \\
[J_3,V_{-+}] \!\!&\!\!=\!\!&\!\! (-i) V_{-+} & \;\; & [K_3,V_{-+}] \!\!&\!\!=\!\!&\!\! (+1) V_{-+} & \;\; & V_{-+} \!\!&\!\!=\!\!&\!\! \ha \lp J_1 + i J_2 - i K_1 + K_2 \rp \\
[J_3,V_{++}] \!\!&\!\!=\!\!&\!\! (+i) V_{++} & \;\; & [K_3,V_{++}] \!\!&\!\!=\!\!&\!\! (+1) V_{++} & \;\; & V_{++} \!\!&\!\!=\!\!&\!\! \ha \lp J_1 - i J_2 + i K_1 + K_2 \rp \\
[J_3,V_{--}] \!\!&\!\!=\!\!&\!\! (-i) V_{--} & \;\; & [K_3,V_{--}] \!\!&\!\!=\!\!&\!\! (-1) V_{--} & \;\; & V_{--} \!\!&\!\!=\!\!&\!\! \ha \lp - J_1 - i J_2 - i K_1 + K_2 \rp \\
[V_{+-},V_{-+}] \!\!&\!\!=\!\!&\!\! - i J_3 - K_3 & \;\; & [V_{++},V_{--}] \!\!&\!\!=\!\!&\!\! - i J_3 + K_3 & & & & \\
\ea
$$
These can be put into Chevalley-Serre form by defining complex Cartan basis elements, $H_{L/R} = - i J_3 \mp K_3$, which give brackets compatible with the structures of $su(2,\mathbb{C})_{L/R}$ or $sl(2,\mathbb{C})_{L/R}$,
$$
[H_L,V_{\pm \mp}] = \pm 2 i V_{\pm \mp} \s
[V_{+-},V_{-+}] = H_L \s
[H_R,V_{\pm \pm}] = \pm 2 i V_{\pm \pm} \s
[V_{+-},V_{-+}] = H_R
$$
Alternatively, the Cartan-Weyl basis can be transformed to a [[real Cartan-Weyl basis]], with real structure constants,
$$
\begin{array}{rclcrclcrcl}
[J_3,V^\mathbb{R}_{+}] \!\!&\!\!=\!\!&\!\! (+1) V^\mathbb{I}_{+} & \;\; & [K_3,V^\mathbb{R}_{+}] \!\!&\!\!=\!\!&\!\! (+1) V^\mathbb{R}_{+} & \;\; & V^\mathbb{R}_{+} \!\!&\!\!=\!\!&\!\! \ha \lp J_1 + K_2 \rp \\
[J_3,V^\mathbb{I}_{+}] \!\!&\!\!=\!\!&\!\! (-1) V^\mathbb{R}_{+} & \;\; & [K_3,V^\mathbb{I}_{+}] \!\!&\!\!=\!\!&\!\! (+1) V^\mathbb{I}_{+} & \;\; & V^\mathbb{I}_{+} \!\!&\!\!=\!\!&\!\! \ha \lp J_2 - K_1 \rp \\
[J_3,V^\mathbb{R}_{-}] \!\!&\!\!=\!\!&\!\! (+1) V^\mathbb{I}_{-} & \;\; & [K_3,V^\mathbb{R}_{-}] \!\!&\!\!=\!\!&\!\! (-1) V^\mathbb{R}_{-} & \;\; & V^\mathbb{R}_{-} \!\!&\!\!=\!\!&\!\! \ha \lp - J_1 + K_2 \rp \\
[J_3,V^\mathbb{I}_{-}] \!\!&\!\!=\!\!&\!\! (-1) V^\mathbb{R}_{-} & \;\; & [K_3,V^\mathbb{I}_{-}] \!\!&\!\!=\!\!&\!\! (-1) V^\mathbb{I}_{-} & \;\; & V^\mathbb{I}_{-} \!\!&\!\!=\!\!&\!\! \ha \lp - J_2 - K_1 \rp \\
[V^\mathbb{R}_{+},V^\mathbb{I}_{-}] \!\!&\!\!=\!\!&\!\! [V^\mathbb{R}_{-},V^\mathbb{I}_{+}] = - \ha J_3 & \;\; & [V^\mathbb{R}_{+},V^\mathbb{R}_{-}] \!\!&\!\!=\!\!&\!\! [V^\mathbb{I}_{+},V^\mathbb{I}_{-}] = \ha K_3 & & & & \\
\end{array}
$$
The [[standard model]] [[fermions|one generation of fermions]] of the [[SO(10)]] GUT live in a sixteen-dimensional complex negative [[chiral]] spinor representation space of the $spin(10) = so(10)$ Lie algebra. Using a [[chiral]] [[Clifford matrix representation]], a Clifford bivector element of so(10), $B = \ha B^{\al \be} \ga_{\al \be}$, acts on a negative [[spinor]], $\ps = \ps^a Q^-_a$, as
$$
B \, \ps = \ha B^{\al \be} (\ga^-_{\al \be})^b{}_a \Ps^a Q^-_b
$$
The Lie algebra has a five-dimensional Cartan subalgebra, so each fermion weight corresponds to five charges, $\{u, v, x, y, z\}$, of a $spin(10)$ [[spinor]]:
$$
\begin{array}{|l|ccccc|}
\hline
& u & v & x & y & z \\
\hline
\nu_e & - & + & - & - & - \\
\bar{\nu}_e & - & - & + & + & + \\
e & + & - & - & - & - \\
\bar{e} & + & + & + & + & + \\
u^r & - & + & - & + & + \\
\bar{u}^r & - & - & + & - & - \\
d^r & + & - & - & + & + \\
\bar{d}^r & + & + & + & - & - \\
u^g & - & + & + & - & + \\
\bar{u}^g & - & - & - & + & - \\
d^g & + & - & + & - & + \\
\bar{d}^g & + & + & - & + & - \\
u^b & - & + & + & + & - \\
\bar{u}^b & - & - & - & - & + \\
d^b & + & - & + & + & - \\
\bar{d}^b & + & + & - & - & + \\
\hline
\end{array}
$$
The $\{x, y, z\}$ charges combine to give two [[su(3)]] color charges, $\{g_3, g_8\}$, and ''Baryon minus Lepton number'', $B = (x+y+z)/3$. The $\{u, v\}$ charges combine to give weak charge, $W = (v-u)/2$, and ''weaker charge'', $W' = (v+u)/2$, which contributes to give hypercharge, $Y=W'+B$, or electric charge, $Q = v+B$, matching the weak, hyper, and color charges of [[one generation of fermions]]. This can be seen in a weight diagram of the [[Spin(10) GUT]]. Note that, being in a chiral representation space, all particles have an odd number of positive weight values. Also, each particle can be identified by its $\{v,x,y,z\}$ weights, without $u$, and these can be interpreted as all of the four weights of a $spin(8)$ spinor.
The [[standard model]] [[fermions|one generation of fermions]] of the [[SO(10)]] GUT live in a sixteen-dimensional complex negative [[chiral]] spinor representation space of the $spin(10) = so(10)$ Lie algebra, the [[spin(10) GUT fermions]]. All sixteen of these massless particle states have left [[helicity]], $h = -1$, while their [[charge-parity conjugates|conjugates of a massless quantum Dirac spinor]] have right helicity, $h=+1$. The full particle state spectrum of one generation of ''spin(10) GUT fermions with helicity'' thus has thirty-two states:
$$
\begin{array}{|l|cccccc|}
\hline
& h & u & v & x & y & z \\
\hline
\nu_{eL} & - & - & + & - & - & - \\
\bar{\nu}_{eL} & - & - & - & + & + & + \\
e_L & - & + & - & - & - & - \\
\bar{e}_L & - & + & + & + & + & + \\
u^r_L & - & - & + & - & + & + \\
\bar{u}^r_L & - & - & - & + & - & - \\
d^r_L & - & + & - & - & + & + \\
\bar{d}^r_L & - & + & + & + & - & - \\
u^g_L & - & - & + & + & - & + \\
\bar{u}^g_L & - & - & - & - & + & - \\
d^g_L & - & + & - & + & - & + \\
\bar{d}^g_L & - & + & + & - & + & - \\
u^b_L & - & - & + & + & + & - \\
\bar{u}^b_L & - & - & - & - & - & + \\
d^b_L & - & + & - & + & + & - \\
\bar{d}^b_L & - & + & + & - & - & + \\
\hline
\end{array}
\s\s\;
\begin{array}{|l|cccccc|}
\hline
& h & u & v & x & y & z \\
\hline
\bar{\nu}_{eR} & + & + & - & + & + & + \\
\nu_{eR} & + & + & + & - & - & - \\
\bar{e}_R & + & - & + & + & + & + \\
e_R & + & - & - & - & - & - \\
\bar{u}^r_R & + & + & - & + & - & - \\
u^r_R & + & + & + & - & + & + \\
\bar{d}^r_R & + & - & + & + & - & - \\
d^r_R & + & - & - & - & + & + \\
\bar{u}^g_R & + & + & - & - & + & - \\
u^g_R & + & + & + & + & - & + \\
\bar{d}^g_R & + & - & + & - & + & - \\
d^g_R & + & - & - & + & - & + \\
\bar{u}^b_R & + & + & - & - & - & + \\
u^b_R & + & + & + & + & + & - \\
\bar{d}^b_R & + & - & + & - & - & + \\
d^b_R & + & - & - & + & + & - \\
\hline
\end{array}
$$
If [[spin(10) GUT fermions]] with [[spin|spin operator]] are fit in the compact [[real form]] of [[e8]] they can all be in the $8^+ \times 8^+$ representation space. This requires the introduction of another root coordinate, $w$, with the fermions having $w = -1$ and anti-fermions having $w = +1$, and a coordinate, $\om'$, related to spin. Note that the [[helicity]] is determined by $\om'$, $\om_s$, and $w$ (or by $u$, $v$, and $w$), and not by $\om'$ and $\om_s$ alone.
$$
\begin{array}{|l|cccccc|}
\hline
& \om' & \om_s & u & v & w & x & y & z \\
\hline
\nu_{eL}^{\wedge/\vee} & \mp & \pm & - & + & - & - & - & - \\
\nu_{eR}^{\wedge/\vee} & \pm & \pm & + & + & - & - & - & - \\
e_L^{\wedge/\vee} & \mp & \pm & + & - & - & - & - & - \\
e_R^{\wedge/\vee} & \pm & \pm & - & - & - & - & - & - \\
u^{r\wedge/\vee}_L & \mp & \pm & - & + & - & - & + & + \\
u^{r\wedge/\vee}_R & \pm & \pm & + & + & - & - & + & + \\
d^{r\wedge/\vee}_L & \mp & \pm & + & - & - & - & + & + \\
d^{r\wedge/\vee}_R & \pm & \pm & - & - & - & - & + & + \\
u^{g\wedge/\vee}_L & \mp & \pm & - & + & - & + & - & + \\
u^{g\wedge/\vee}_R & \pm & \pm & + & + & - & + & - & + \\
d^{g\wedge/\vee}_L & \mp & \pm & + & - & - & + & - & + \\
d^{g\wedge/\vee}_R & \pm & \pm & - & - & - & + & - & + \\
u^{b\wedge/\vee}_L & \mp & \pm & - & + & - & + & + & - \\
u^{b\wedge/\vee}_R & \pm & \pm & + & + & - & + & + & - \\
d^{b\wedge/\vee}_L & \mp & \pm & + & - & - & + & + & - \\
d^{b\wedge/\vee}_R & \pm & \pm & - & - & - & + & + & - \\
\hline
\end{array}
\s\s\;
\begin{array}{|l|cccccc|}
\hline
& \om' & \om_s & u & v & w & x & y & z \\
\hline
\bar{\nu}_{eR}^{\wedge/\vee} & \mp & \pm & + & - & + & + & + & + \\
\bar{\nu}_{eL}^{\wedge/\vee} & \pm & \pm & - & - & + & + & + & + \\
\bar{e}_R^{\wedge/\vee} & \mp & \pm & - & + & + & + & + & + \\
\bar{e}_L^{\wedge/\vee} & \pm & \pm & + & + & + & + & + & + \\
\bar{u}^{r\wedge/\vee}_R & \mp & \pm & + & - & + & + & - & - \\
\bar{u}^{r\wedge/\vee}_L & \pm & \pm & - & - & + & + & - & - \\
\bar{d}^{r\wedge/\vee}_R & \mp & \pm & - & + & + & + & - & - \\
\bar{d}^{r\wedge/\vee}_L & \pm & \pm & + & + & + & + & - & - \\
\bar{u}^{g\wedge/\vee}_R & \mp & \pm & + & - & + & - & + & - \\
\bar{u}^{g\wedge/\vee}_L & \pm & \pm & - & - & + & - & + & - \\
\bar{d}^{g\wedge/\vee}_R & \mp & \pm & - & + & + & - & + & - \\
\bar{d}^{g\wedge/\vee}_L & \pm & \pm & + & + & + & - & + & - \\
\bar{u}^{b\wedge/\vee}_R & \mp & \pm & + & - & + & - & - & + \\
\bar{u}^{b\wedge/\vee}_L & \pm & \pm & - & - & + & - & - & + \\
\bar{d}^{b\wedge/\vee}_R & \mp & \pm & - & + & + & - & - & + \\
\bar{d}^{b\wedge/\vee}_L & \pm & \pm & + & + & + & - & - & + \\
\hline
\end{array}
$$
The [[standard model]] [[fermions|one generation of fermions]] of the [[SO(10)]] GUT live in a sixteen-dimensional complex negative [[chiral]] spinor representation space of the $spin(10) = so(10)$ Lie algebra, the [[spin(10) GUT fermions]]. All sixteen of these massless particle states have left [[helicity]], $h = -\ha$, while their [[charge-parity conjugates|conjugates of a massless quantum Dirac spinor]] have right helicity, $h=+\ha$. Under the [[spin(1,3)]] [[Lorentz algebra]], with a [[Cartan subalagebra|Lie algebra structure]] basis choice of $(\ga_{03},\ga_{12})$, with coordinate labels $(\om_t, \om_s)$, left-handed chirality [[Dirac spinor]]s (corresponding to [[Weyl spinor]]s) with spin up and down have weights $(\mp \ha, \pm \fr{i}{2})$ while right-handed chirality Dirac spinors have weights $(\pm \ha, \pm \fr{i}{2})$. The particle state spectrum of one generation of ''spin(11,3) GraviGUT fermions'' thus has thirty-two states:
$$
\begin{array}{|l|cccccc|}
\hline
& \om'_t & \om_s & u & v & x & y & z \\
\hline
\nu_{eL}^{\wedge/\vee} & \mp & \pm & - & + & - & - & - \\
\nu_{eR}^{\wedge/\vee} & \pm & \pm & + & + & - & - & - \\
e_L^{\wedge/\vee} & \mp & \pm & + & - & - & - & - \\
e_R^{\wedge/\vee} & \pm & \pm & - & - & - & - & - \\
u^{r\wedge/\vee}_L & \mp & \pm & - & + & - & + & + \\
u^{r\wedge/\vee}_R & \pm & \pm & + & + & - & + & + \\
d^{r\wedge/\vee}_L & \mp & \pm & + & - & - & + & + \\
d^{r\wedge/\vee}_R & \pm & \pm & - & - & - & + & + \\
u^{g\wedge/\vee}_L & \mp & \pm & - & + & + & - & + \\
u^{g\wedge/\vee}_R & \pm & \pm & + & + & + & - & + \\
d^{g\wedge/\vee}_L & \mp & \pm & + & - & + & - & + \\
d^{g\wedge/\vee}_R & \pm & \pm & - & - & + & - & + \\
u^{b\wedge/\vee}_L & \mp & \pm & - & + & + & + & - \\
u^{b\wedge/\vee}_R & \pm & \pm & + & + & + & + & - \\
d^{b\wedge/\vee}_L & \mp & \pm & + & - & + & + & - \\
d^{b\wedge/\vee}_R & \pm & \pm & - & - & + & + & - \\
\hline
\end{array}
\s\s\;
\begin{array}{|l|cccccc|}
\hline
& \om'_t & \om_s & u & v & x & y & z \\
\hline
\bar{\nu}_{eR}^{\wedge/\vee} & \pm & \pm & + & - & + & + & + \\
\bar{\nu}_{eL}^{\wedge/\vee} & \mp & \pm & - & - & + & + & + \\
\bar{e}_R^{\wedge/\vee} & \pm & \pm & - & + & + & + & + \\
\bar{e}_L^{\wedge/\vee} & \mp & \pm & + & + & + & + & + \\
\bar{u}^{r\wedge/\vee}_R & \pm & \pm & + & - & + & - & - \\
\bar{u}^{r\wedge/\vee}_L & \mp & \pm & - & - & + & - & - \\
\bar{d}^{r\wedge/\vee}_R & \pm & \pm & - & + & + & - & - \\
\bar{d}^{r\wedge/\vee}_L & \mp & \pm & + & + & + & - & - \\
\bar{u}^{g\wedge/\vee}_R & \pm & \pm & + & - & - & + & - \\
\bar{u}^{g\wedge/\vee}_L & \mp & \pm & - & - & - & + & - \\
\bar{d}^{g\wedge/\vee}_R & \pm & \pm & - & + & - & + & - \\
\bar{d}^{g\wedge/\vee}_L & \mp & \pm & + & + & - & + & - \\
\bar{u}^{b\wedge/\vee}_R & \pm & \pm & + & - & - & - & + \\
\bar{u}^{b\wedge/\vee}_L & \mp & \pm & - & - & - & - & + \\
\bar{d}^{b\wedge/\vee}_R & \pm & \pm & - & + & - & - & + \\
\bar{d}^{b\wedge/\vee}_L & \mp & \pm & + & + & - & - & + \\
\hline
\end{array}
$$
as well as their thirty-two [[charge-parity conjugate|conjugates of a massless quantum Dirac spinor]] anti-particles, making sixty-four states. This is a real chiral $64_+$ spinor of [[spin(11,3)|Cl(11,3)]]. Note that there are only thirty-two states for each momentum, but the spin of [[Dirac solutions]] is dependent on momentum, and an arbitrary spin state (for which we need a superposition of two spin states) corresponds to a momentum state.
Note that these are the quantum numbers for the classical degrees of freedom of one generation of fermions, but are in agreement with the [[spin quantum numbers of a massless quantum Dirac spinor]].
The ''spin(2,4)'' [[spin Lie algebra]], isomorphic to [[su(2,2)]] and to the [[conformal algebra]], is a subalgebra of [[Cl(2,4)]] spanned by its 15 basis bivectors,
\begin{eqnarray}
\ga'_{\mu \nu} &=& \;\;\;\, \, \si_0 \otimes \ga_{\mu\nu} \\
\ga'_{\mu 5 } &=& \; - \, \si_0 \otimes \ga_{\mu} \ga \\
\ga'_{\mu 6 } &=& \;\;\, i \, \si_3 \otimes \ga_{\mu} \\
\ga'_{5 6} &=& - i \, \si_3 \otimes \ga \\
\end{eqnarray}
The [[chiral]] parts of a $spin(2,4)$ bivector are thus
$$
B_\pm = B + P_\pm + K_\pm + dD_\pm
$$
including generators of spacetime [[Cl(1,3)]] [[Clifford rotation]]s (generated by [[Cl(1,3) bivector]]s), translations, special conformal transformations, and dilations,
\begin{eqnarray}
B &=& \ha B^{\mu\nu} \ga_{\mu\nu} \\
P_\pm &=& p^\mu P_{\mu\pm} = p^\mu ( \ga'_{\mu 5 \pm} + \ga'_{\mu 6 \pm} ) = p^\mu \ga_\mu ( - \ga \pm i) = \pm 2 i p P_{L/R} \\
K_\pm &=& k^\mu K_{\mu\pm} = k^\mu ( \ga'_{\mu 5 \pm} - \ga'_{\mu 6 \pm} ) = k^\mu \ga_\mu ( - \ga \mp i) = \mp 2 i k P_{R/L} \\
dD_\pm &=& \ha d \ga'_{5 6} = \pm \fr{i}{2} d \ga \\
\end{eqnarray}
in which $P_{L/R}$ is the [[left/right chirality projector]]. This complex [[Cl(1,3)]] bivector representative and all its parts satisfy $B_\pm^\da = -\ga_0 B_\pm \ga_0$. As matrices, these $Cl(2,4)$ bivectors of $spin(2,4)$, acting on a (internally represented) $Cl(2,4)$ vector, with both acting on a $spin(2,4)$ spinor, $\Ps'$, are
$$
\lb \ba{cccc}
B_L - \fr{d}{2} & -2 i p_L & i \, v_- & v_L \\
2 i k_R & B_R + \fr{d}{2} & v_R & - i \, v_+ \\
i \, v_+ & v_L & B_L + \fr{d}{2} & -2 i k_L \\
v_R & - i \, v_- & 2 i p_R & B_R - \fr{d}{2} \\
\ea \rb
\lb \ba{c} \ps_{-L} \\ \ps_{-R} \\ \ps_{+L} \\ \ps_{+R} \ea \rb
$$
A $spin(2,4) = su(2,2)$ element, as a chiral $Cl(2,4)$ bivector, $B_\pm$, acts on positive and negative [[chiral]] $spin(2,4)$ spinors as two $spin(1,3)$ [[Dirac spinor]]s or [[twistor]]s,
$$
B_\pm = \lb \ba{cc}
B_L \pm \fr{d}{2} & -2 i (k/p)_L \\
2 i (p/k)_R & B_R \mp \fr{d}{2}
\ea \rb
\s \s
\Ps_\pm = \lb \ba{c} \ps_{\pm L} \\ \ps_{\pm R} \ea \rb \in \mathbb{T} = \mathbb{C}^4
$$
The $spin(2,4)$ [[conjugate spinor]] is $\bar{\Ps'} = \Ps'{}^\da \ga'_{0 6}$, and resulting ''$spin(2,4)$ chiral spinor adjoint'' is
$$
\bar{\Ps}_\pm = \Ps_\pm^\da \ga'_{0 6}{}^{\!\! \pm} =
\pm i \lb \begin{array}{cc} \ps_{\pm R}^\da & \ps_{\pm L}^\da \\ \end{array} \rb
$$
with $\ga'_{0 6}{}^{\!\! \pm} = \pm i \, \si_1 \otimes \si_0 = \pm i \ga_0$. The chiral spinor contraction,
$$
\bar{\Ps}_\pm \Ps_\pm = \pm i \lp \ps_{\pm R}^\da \ps_{\pm L} + \ps_{\pm L}^\da \ps_{\pm R} \rp = \pm i \overline{\Ps_\pm} \Ps_\pm
$$
is the [[Hermitian form]], with signature $(2,2)$, invariant under $Spin(2,4)$ -- the [[conformal group]]. The isomorphism to [[su(2,2)]] is apparent from $B_\pm^\da \ga'_{0 6}{}^{\!\! \pm} + \ga'_{0 6}{}^{\!\! \pm} B_\pm = 0$.
Basis elements of two [[different Cartans]] for $spin(2,4)$, compact and non-compact, are
$$
\begin{array}{rcl}
\ga'_{0 5} \!\!&\!\!=\!\!&\!\! \; - \, \si_0 \otimes \si_2 \otimes \si_0 \\
\ga'_{1 2} \!\!&\!\!=\!\!&\!\! - i \, \si_0 \otimes \si_0 \otimes \si_3 \\
\ga'_{3 6} \!\!&\!\!=\!\!&\!\! \; - \, \si_3 \otimes \si_2 \otimes \si_3 \\
\end{array}
\s \s
\begin{array}{rcl}
\ga'_{0 3} \!\!&\!\!=\!\!&\!\! \;\;\;\, \, \si_0 \otimes \si_3 \otimes \si_3 \\
\ga'_{1 2} \!\!&\!\!=\!\!&\!\! - i \, \si_0 \otimes \si_0 \otimes \si_3 \\
\ga'_{5 6} \!\!&\!\!=\!\!&\!\! \; - \, \si_3 \otimes \si_3 \otimes \si_0 \\
\end{array}
$$
The 28-dimensional [[spin Lie algebra]] of split signature, $spin(4,4)$, has a nice representation as [[Cl(4,4)]] bivectors. In addition to the realified [[spin(2,4)]] [[chiral]] bivector parts,
$$
B_\pm = B \pm 2 p P_{L/R} J \mp 2 k P_{R/L} J \pm \ha d i \ga
$$
using the [[left/right chirality projector]] and a [[realified|realify]] [[complex structure]], $J$, there are thirteen new $spin(4,4)$ generators incorporated into chiral parts,
\begin{eqnarray}
b^{7+8}_\pm &=& b_\mathbb{I}^\mu \ga'_{\mu 7 \pm} \! + b_\mathbb{R}^\mu \ga'_{\mu 8 \pm}
= \pm \ga_\mu \ga \ga_2 (b_\mathbb{I}^\mu J + b_\mathbb{R}^\mu) K= \mp b P_{L/R} \ep K \\
b^{5}_\pm &=& b^5_\mathbb{I} \ga'_{5 7 \pm} \! + b^5_\mathbb{R} \ga'_{5 8 \pm}
= \pm \ga_2 b^5 K \\
b^{6}_\pm &=& b^6_\mathbb{I} \ga'_{6 7 \pm} \! + b^6_\mathbb{R} \ga'_{6 8 \pm}
= - \ga \ga_2 b^6 J K \\
\fr{\ph}{2} J_\pm &=& \fr{\ph}{2} \, \ga'^\pm_{7 8} = \fr{\ph}{2} J \\
\end{eqnarray}
with $K$ the realified complex conjugation operator, and a new complex spacetime vector defined as
$$
b = (b_\mathbb{I}^\mu J + b_\mathbb{R}^\mu) \ga_\mu = b_L P_R + b_R P_L
$$
and new complex scalars $b^{5/6} = b^{5/6}_\mathbb{I} J + b^{5/6}_\mathbb{R}$ (which we can use to define $b_\pm = \pm \, b^5 + b^6$), and the phase $\ph$. For expressing $Cl(4,4)$ vectors and bivectors, it's useful to remember
$$
\ga_2 = \lb \ba{cc} & - \ep J \\ \ep J & \ea \rb
\;\;\;\;
- \! \ga \ga_2 = \lb \ba{cc} & \ep \\ \ep & \ea \rb
\;\;\;\;
\text{so}
\;\;\;\;
b^5_\pm + b^6_\pm = \pm \ga_2 b^5 K - \ga \ga_2 b^6 J K
= \lb \ba{cc} & \mp b^5 + b^6 \\ \pm b^5 + b^6 & \ea \rb J \ep K
$$
A $Cl(4,4)$ bivector and vector (with $v_\pm = v^5 \pm v^6$ and $v_c = v^7 J + v^8$) acting on each other and a [[spinor]], represented as a $16 \times 16$ matrix times itself and a column, can then be expressed as
$$
\lb \ba{cccc}
B_L - b_L \ep K - \fr{d}{2} + \fr{\ph}{2} J & -2 p_L J + b_+ J \ep K & v_- J & v_L + v_c \ep K \\
2 k_R J + b_- J \ep K & B_R - b_R \ep K + \fr{d}{2} + \fr{\ph}{2} J & v_R + v_c \ep K & - v_+ J \\
v_+ J & v_L - v_c \ep K & B_L + b_L \ep K + \fr{d}{2} + \fr{\ph}{2} J & -2 k_L J + b_- J \ep K \\
v_R - v_c \ep K & - v_- J & 2 p_R J + b_+ J \ep K & B_R + b_R \ep K - \fr{d}{2} + \fr{\ph}{2} J \\
\ea \rb
\lb \ba{c} \ps_{-L} \\ \ps_{-R} \\ \ps_{+L} \\ \ps_{+R} \ea \rb
$$
in which all $i$'s are realified to $2 \times 2$ matrices, $i \to J$, and the skew, $\ep$, is acting on the "spin" level. If we close the Lie algebra of the above bivectors, vectors, and spinors with a bracket between the spinors, we have the Lie algebra [[f4(4)]].
The spinors are realified as
$$
\ps_{\pm L/R } = \lb \ba{c} \ps_{\pm L/R}^{\wedge} \\ \ps_{\pm L/R}^{\vee} \ea \rb
= \lb \ba{c} \ps_{\pm L/R}^{\wedge \mathbb{R}} \\ \ps_{\pm L/R}^{\wedge \mathbb{I}} \\ \ps_{\pm L/R}^{\vee \mathbb{R}} \\ \ps_{\pm L/R}^{\vee \mathbb{I}} \ea \rb
$$
so that, for example, $\ep K \ps_L = \ep \ps^*_L = (\ps_L)^C$ will give the [[Weyl spinor conjugate|charge conjugate]]. The $spin(4,4)$ [[spinor metric|conjugate spinor]] is
$$
n' = \Ga'_4 \Ga'_6 \Ga'_7 \Ga'_8 = - \si_3 \otimes \si_1 \otimes \si_0 \otimes \si_0
\s \s
n'^\pm = \pm \, \si_1 \otimes \si_0 \otimes \si_0 = \pm \, \ga_0 \otimes \si_0
$$
which gives the inverse of $spin(4,4)$ [[rotor|Clifford rotation]]s, $U = e^B$, as $U^- = e^{-B} = n' U^T n'$, and allows the ''spin(4,4) spinor adjoint'' to be defined as $\bar{\Ps}{}' = \Ps'^T n' $. Here, since our Clifford representation is real, the [[transpose]] is the same as the [[Hermitian]] conjugate. The $spin(4,4)$ invariant [[Hermitian form]] is
\begin{eqnarray}
\bar{\Ps}{}' \Ps' &=& -
\lb \ba{cccccccc}
\ps_{- R}^{\wedge \mathbb{R}} & \ps_{- R}^{\wedge \mathbb{I}} & \ps_{- R}^{\vee \mathbb{R}} & \ps_{- R}^{\vee \mathbb{I}} &
\ps_{- L}^{\wedge \mathbb{R}} & \ps_{- L}^{\wedge \mathbb{I}} & \ps_{- L}^{\vee \mathbb{R}} & \ps_{- L}^{\vee \mathbb{I}}
\ea \rb \Ps'_- + \Ps'^T_+ n'^+ \Ps'_+ \\
&=& - 2 \lp
\ps_{- R}^{\wedge \mathbb{R}} \ps_{- L}^{\wedge \mathbb{R}}
+ \ps_{- R}^{\wedge \mathbb{I}} \ps_{- L}^{\wedge \mathbb{I}}
+ \ps_{- R}^{\vee \mathbb{R}} \ps_{- L}^{\vee \mathbb{R}}
+ \ps_{- R}^{\vee \mathbb{I}} \ps_{- L}^{\vee \mathbb{I}}
\rp + 2 \lp \texttt{"}_+ \rp \\
&=& - \lp \ps_{- R}^\da \ps_{- L} + \ps_{- L}^\da \ps_{- R} \rp + \lp \texttt{"}_+ \rp \\
&=& - \overline{\Ps_-} \Ps_- + \overline{\Ps_+} \Ps_+
\end{eqnarray}
which relates directly to the Hermitian form for [[spin(2,4)]] complex chiral spinors. This Hermitian form is also invariant under $Spin(4,4)$ transformations that are not $Spin(2,4)$ transformations.
From [[division algebra confusion]] and the [[similarity transformation for Cl(4,4)]] to the [[Clifford division algebra representation]] of $Cl(4,4)$ there is a [[triality]], (//might need to add a term?//)
\begin{eqnarray}
T(v'_+, \Ps'_-, \Ps'_+) &=& \Ps'_+{}^T n'^+ v'_+ \, \Ps'_- = \Ps_+^\da \ga_0 \, v_+ \, \Ps_- \\
&=& \lb \ba{cc} \ps_{+R}^\da & \ps_{+L}^\da \ea \rb
\lb \ba{cc} v_+ i & v_L - v_c \ep K \\ v_R - v_c \ep K & - v_- i \ea \rb
\lb \ba{c} \ps_{-L} \\ \ps_{-R} \ea \rb \\
&=&
\ps_{+R}^\da \lp i \, v_+ \ps_{-L} + v_L \ps_{-R} - v_c \ep \, \ps_{-R}^* \rp
+ \ps_{+L}^\da \lp v_R \ps_{-L} - v_c \ep \, \ps_{-L}^* - i \, v_- \ps_{-R} \rp
\end{eqnarray}
in which we have re-complexified, and see terms emerging reminiscent of Dirac mass and Majorana mass.
The 28-dimensional [[spin Lie algebra]], ''spin(8)'' (also called ''d4''), has a nice representation as [[Cl(8)]] bivectors, $\Ga_{\al \be}$, which can be represented as $16 \times 16$ matrices. Alternatively, there is an [[octonionic representation of Cl(8)]], which facilitates the description of [[triality]] outer [[automorphism]]s of spin(8).
A good way to find an explicit description of the ''spin(8) triality'' automorphism is from the [[triality automorphism of f4]], from which we find the ''triality rotation matrix'', $\Ga'_{\al \be} = \Ga_{\ga \de} \, {\underset{bb}{T}}^{\ga \de}{}_{\al \be}$, with
$$
{\underset{bb}{T}}^{\ga \de}{}_{\al \be} = \ha ( \Ga^\ga \overline{\Ga}{}^\de )_{\al \be}
$$
This matrix is a 28-dimensional rotation, ${\underset{bb}{T}} \, {\underset{bb}{T}}^T = 1$, and is block-diagonalized by organizing the 28 spin(8) bivector basis generators into seven groups of four bivectors,
$$
\begin{array}{c}
\lb + \Ga_{12}, + \Ga_{34}, + \Ga_{56}, + \Ga_{78} \rb \\
\lb + \Ga_{13}, - \Ga_{24}, + \Ga_{57}, - \Ga_{68} \rb \\
\lb + \Ga_{14}, + \Ga_{23}, - \Ga_{58}, - \Ga_{67} \rb \\
\lb + \Ga_{15}, - \Ga_{26}, - \Ga_{37}, + \Ga_{48} \rb \\
\lb + \Ga_{16}, + \Ga_{25}, + \Ga_{38}, + \Ga_{47} \rb \\
\lb + \Ga_{17}, - \Ga_{28}, + \Ga_{35}, - \Ga_{46} \rb \\
\lb + \Ga_{18}, + \Ga_{27}, - \Ga_{36}, - \Ga_{45} \rb \\
\end{array}
$$
with each set of four a [[Cartan subalgebra|Lie algebra structure]] transforming by the same [[triality matrix]],
$$
T =
\lb \begin{array}{cccc}
- \ha & - \ha & - \ha & - \ha \\
\ha & \ha & - \ha & - \ha \\
\ha & - \ha & \ha & - \ha \\
\ha & - \ha & - \ha & \ha \\
\end{array} \rb
$$
For each of the seven sets of four bivectors, there is a two-dimensional space, $B2_0$, [[invariant|triality triplet]] under this triality, with the resulting 14 elements generating [[g2]]. Triality acts as a $\fr{2\pi}{3}$ rotation in the planes complementary to these, spanned by the orthogonal bivectors, $\Ga_{1x}$ and their [[T-complement|triality triplet]]s, $\Ga'_{1x}$, in $B2_\perp$,
$$
\begin{array}{ccccc}
B2_0 &\s& B2_\perp \\
\lb \Ga_{34} \!-\! \Ga_{56}, \Ga_{34} \!-\! \Ga_{78} \rb
&\s&
\lb \Ga_{12}, \Ga'_{12} \rb &\;\;\;& \Ga'_{12} = + \Ga_{34} \!+\! \Ga_{56} \!+\! \Ga_{78} \\
\lb \Ga_{24} \!+\! \Ga_{57}, \Ga_{24} \!-\! \Ga_{68} \rb
&\s&
\lb \Ga_{13}, \Ga'_{13} \rb &\;\;\;& \Ga'_{13} = - \Ga_{24} \!+\! \Ga_{57} \!-\! \Ga_{68} \\
\lb \Ga_{23} \!+\! \Ga_{58}, \Ga_{23} \!+\! \Ga_{67} \rb
&\s&
\lb \Ga_{14}, \Ga'_{14} \rb &\;\;\;& \Ga'_{14} = + \Ga_{23} \!-\! \Ga_{58} \!-\! \Ga_{67} \\
\lb \Ga_{26} \!-\! \Ga_{37}, \Ga_{26} \!+\! \Ga_{48} \rb
&\s&
\lb \Ga_{15}, \Ga'_{15} \rb &\;\;\;& \Ga'_{15} = - \Ga_{26} \!-\! \Ga_{37} \!+\! \Ga_{48} \\
\lb \Ga_{25} \!-\! \Ga_{38}, \Ga_{25} \!-\! \Ga_{47} \rb
&\s&
\lb \Ga_{16}, \Ga'_{16} \rb &\;\;\;& \Ga'_{16} = + \Ga_{25} \!+\! \Ga_{38} \!+\! \Ga_{47} \\
\lb \Ga_{28} \!+\! \Ga_{35}, \Ga_{28} \!-\! \Ga_{46} \rb
&\s&
\lb \Ga_{17}, \Ga'_{17} \rb &\;\;\;& \Ga'_{17} = - \Ga_{28} \!+\! \Ga_{35} \!-\! \Ga_{46} \\
\lb \Ga_{27} \!+\! \Ga_{36}, \Ga_{27} \!+\! \Ga_{45} \rb
&\s&
\lb \Ga_{18}, \Ga'_{18} \rb &\;\;\;& \Ga'_{18} = + \Ga_{27} \!-\! \Ga_{36} \!-\! \Ga_{45} \\
\end{array}
$$
So, for example,
$$
T \, \Ga_{12} = -\ha \Ga_{12} + \ha \Ga'_{12}
\s
T^2 \, \Ga_{12} = -\ha \Ga_{12} - \ha \Ga'_{12}
$$
This corresponds to the ''g2 triality decomposition of spin(8)'',
$$
spin(8) = g2_0 + 7 + \bar{7}
$$
in which the $7$ consists of the $\Ga_{1x}$ and the $\bar{7}$ their T-complements, with a [[T-compatible|triality triplet]] [[complex structure]] rotating between them.
Alternatively, there is a ''su(2) triality decomposition of spin(8)'',
$$
spin(8) = su(2)_0 + su(2)_1 + su(2)_2 + su(2)_3 + 2_0 \otimes 2_1 \otimes 2_2 \otimes 2_3
$$
with corresponding generators
$$
\begin{array}{ccccccc}
su(2)_0 &\s& su(2)_1 &\s& su(2)_2 &\s& su(2)_3 \\
\Ga_{34}-\Ga_{78} & & \Ga_{34}+\Ga_{78} & & -\Ga_{12}-\Ga_{56} & & \Ga_{12}-\Ga_{56} \\
\Ga_{37}+\Ga_{48} & & \Ga_{37}-\Ga_{48} & & \Ga_{15}-\Ga_{26} & & \Ga_{15}+\Ga_{26} \\
\Ga_{38}-\Ga_{47} & & \Ga_{38}+\Ga_{47} & & -\Ga_{16}-\Ga_{25} & & \Ga_{16}-\Ga_{25} \\
\hline
\Ga_{35}+\Ga_{46} & & \Ga_{35}-\Ga_{46} & & \Ga_{17}+\Ga_{28} & & \Ga_{17}-\Ga_{28} \\
\Ga_{36}-\Ga_{45} & & \Ga_{36}+\Ga_{45} & & \Ga_{18}-\Ga_{27} & & \Ga_{18}+\Ga_{27} \\
\hline
\Ga_{57}+\Ga_{68} & & \Ga_{57}-\Ga_{68} & & \Ga_{13}+\Ga_{24} & & \Ga_{13}-\Ga_{24}\\
\Ga_{58}-\Ga_{67} & & \Ga_{58}+\Ga_{67} & & \Ga_{14}-\Ga_{23} & & \Ga_{14}+\Ga_{23} \\
\end{array}
$$
//check signs above//
with rows related by $T \, su(2)_0 = su(2)_0$, $T \, su(2)_1 = su(2)_2$, and $T \, su(2)_2 = su(2)_3$. So, in a sense, spin(8) consists of three triality-invariant su(2)'s and three sets of three triality-related su(2)'s all sharing a 4-dimensional Cartan subalgebra.
If we form the [[coset]], $Spin(8)/\mathbb{Z}^T_3$ (technically $\mathbb{Z}^T_3$ is not a subgroup of $Spin(8)$, but it is an automorphism, and a subgroup of [[f4]], so we'll proceed as if it were), with the ''finite triality group'', $\mathbb{Z}^T_3 = \{ 1, T, T^2 \} $, then this gives a ''tessellation'' of $Spin(8)$ into four ''region''s: a region invariant under $T$ and three regions related by $T$.
A ''spinor'', $\Psi$, is an element of a minimal left ideal of a [[Clifford algebra]], $I_L \in Cl$. Each ''minimal left ideal'', $I_L$, is the smallest subset such that for all $\Psi \in I_L$ and $A \in Cl$ we have $A \Psi \in I_L$. Using a [[Clifford matrix representation]], each minimal left ideal corresponds to a collumn of the matrix, so a spinor, $\Ps = \Ps^a Q_a$, as a [[representation space]], is a ''matrix collumn'' on which the Clifford algebra elements, represented as matrices, such as the bivector $B = \ha B^{\al \be} \ga_{\al \be}$, act from the left.
$$
B \, \Ps = \ha B^{\al \be} \ga_{\al \be} \Ps^a Q_a = \ha B^{\al \be} \Ps^a Q_b (\ga_{\al \be})^b{}_c \de_a^c = \ha B^{\al \be} \Ps^a Q_b (\ga_{\al \be})^b{}_a
$$
In terms of matrix components, this gives
$$
\lp B \Psi \rp^b = \ha B^{\al \be} (\ga_{\al \be})^b{}_a \Ps^a
$$
A spinor, written as a collumn of numbers, may be real if the Clifford matrix representation is real, or [[chiral]] if the rep is chiral. A ''Grassmann spinor'' may be written as a collumn of real or complex [[Grassmann number]]s.
Geometrically, a ''spinor field'' is a section of a [[fiber bundle]] [[associated]] to a [[principal bundle]], with the elements of the structure group, in the appropriate representation, acting on the spinor from the left.
Ref:
http://en.wikipedia.org/wiki/Spinor
The ''spinor covariant derivative'' is the [[covariant derivative]] of a [[spinor]] field,
$$
\f{\nabla} \ps = \f{\pa} \ps + \ha \f{\om} \ps
$$
in which $\f{\pa}$ is the [[partial derivative]] and $\f{\om}$ is the [[spin connection]] acting on the spinor via [[Clifford algebra]] multiplication.
The ''split-complex numbers'', $\mathbb{C}'$, like the [[complex number]]s, are a two-dimensional [[derision algebra|division algebra]], spanned by two basis elements, $e'_0$ and $e'_1$. The split-complex identity element (spanning the ''real part'') is $e'_0=1$, and the other, $e'_1 = I$ (spanning the ''imaginary part''), with $I I = 1$, is a ''split-imaginary'' direction. So a split-complex number, $z = a + I \, b \in \mathbb{C}'$, has a ''real component'', $a \in \mathbb{R}$ (equal to its real part, $\text{Re}(z) = z_{\mathbb{R}} = a$), and a ''split-imaginary component'', $b = z_{\mathbb{I}} \in \mathbb{R}$ (the split-imaginary part being $\text{Im}(z) = I \, b$). Split-complex multiplication is commutative and associative. The multiplication table for the basis elements is
| | $1$ | $I$ |
| $1$ | $1$ | $I$ |
| $I$ | $I$ | $1$ |
which can be written using a ''complex multiplication coefficient matrix'' as $e'_a e'_b = M'_{ab}{}^c e'_c$, so, for example, $e'_1 e'_1 = e'_0$ and $M'_{11}{}^0 = 1$. ''Split-complex conjugation'' can be written synonymously as
$$
K e'_0 = e'^*_0 = \os{e}'_0 = \bar{e}'_0 = e'_{\os{0}} = e'_0 = 1 \;\;\;\;\; \;\;\;\;\; K e'_1 = e'^*_1 = \os{e}'_1 = \bar{e}'_1 = e'_{\os{1}} = - e'_1 = - I
$$
using the ''conjugation operator'', $K$, and satisfies $(z_1 z_2)^* = z_2^* z_1^* = z_1^* z_2^*$, so we have $M'_{ab}{}^c = M'_{\os{b} \os{a}}{}^{\os{c}}$.
Multiplying a split-complex number, $z = a + I \, b$, by its conjugate, $z^* = a - I \, b$, gives its norm,
$$
|z|^2 = z^* z = z^a z^b \os{e}_a e_b = a^2 - b^2 \in \mathbb{R}
$$
allowing us to calculate the inverse of any non null split-complex number, $z^- = \frac{z^*}{|z|^2}$. The [[metric]] on the split-complex numbers is then
$$
(e'_a, e'_b) = \ha ( \os{e}'_a e'_b + \os{e}'_b e'_a ) = n'_{ab} = \lb \ba{cc} 1 & \\ & -1 \ea \rb
\s \s
(z_1,z_2) = \ha \lp z^*_1 z_2 + z^*_2 z_1 \rp = a_1 a_2 - b_1 b_2
$$
The complex multiplication table and metric, $n'_{dc}$, can be used to define
$$
\Ga'_{cab} = M'_{\os{a} \os{b} c} = M'_{\os{a}\os{b}}{}^d n'_{dc} \s \text{and so} \s \Ga'_{000} = 1 \s \Ga'_{011} = \Ga'_{110} = \Ga'_{101} = 1
$$
Further defining $\overline{\Ga}{}'_{cab} = \Ga'_{cba}$ (so $\overline{\Ga}{}'_c = \Ga'_c{}^T$ in terms of a matrix [[transpose]]), these can be used to construct a $4 \times 4$ real [[chiral]] [[Clifford matrix representation]] of $Cl(1,1)$, the ''cyclic split-complex representation'', with the Clifford basis vectors represented as [[Hermitian]] matrices (symmetric matrices since they're real),
$$
\ga'_c =
\lb \begin{array}{cc}
0 & \overline{\Ga}{}'_c \\
\Ga'_c & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \overline{\Ga}{}'_c{}^b{}_a \\
\Ga'_c{}^b{}_a & 0
\end{array} \rb
=
\lb \begin{array}{cc}
0 & \Ga'_{ca}{}^b \\
\Ga'_c{}^b{}_a & 0
\end{array} \rb
$$
Explicity, from the complex multiplication coefficient matrix, the chiral Clifford vector and bivector representatives are
\begin{eqnarray}
\Ga'_0 &=& \lb \ba{cc} 1 & 0 \\ 0 & -1 \ea \rb = \si_3 = K \\
\Ga'_1 &=& \lb \ba{cc} 0 & 1 \\ -1 & 0 \ea \rb = i \si_2 = - \ep = - J \\
\Ga'_{01} &=& \lb \ba{cc} 0 & 1 \\ 1 & 0 \ea \rb = \si_1 = JK \\
\end{eqnarray}
which relate directly to the [[Pauli matrices]], the [[skew]], the matrix representation of [[sl(2)]], and the [[realification|realify]] of complex structure. These three operators, $\{J,K,L\}$, conflated with their matrix [[representation]], $\{\pi(J)=-i \si_2$, $\pi(K)=\si_3$, $\pi(L)=\si_1\}$, constitute a basis for the [[sl(2)]] Lie algebra, and generally a [[complex structure]], which acts on the representation space, $V = \pi(z) = \lb \ba{c} a \\ b \ea \rb$. A $spin(1,1)$ [[Clifford rotation]], with rotor $U = e^{\ha \ze \, \Ga'_{01}}$, similar to a [[Lorentz boost]], preserves the split-complex norm.
The split-complex numbers also have a different faithful real matrix representation that comes from their multiplication table,
$$
(e'_c)^b{}_a = M'^\os{b}{}_{ac} \s \s
e'_0 = \lb \ba{cc} 1 & \\ & 1 \ea \rb = 1 \s \s
e'_1 = \lb \ba{cc} & -1 \\ -1 & \ea \rb = -JK \s \s
\ga_c =
\lb \begin{array}{cc}
0 & e'_c \\
\os{e}'_c & 0
\end{array} \rb
$$
that can also be used to build the ''direct split-complex representation'' for $Cl(1,1)$, related to $\ga'_c$ by a [[similarity transformation|Dirac matrices]].
The ''split-octonions'', $v' = v'^a e'_a \in \mathbb{O}'$, are an eight-dimensional [[derision algebra|division algebra]], similar to the [[octonion]]s, spanned by eight basis elements, $e'_0,...,e'_7$. The split-octonion identity element is $e'_0=1$, and the other seven can be thought of as different imaginary directions, squaring to $\pm 1$. Split-octonionic multiplication is non-commutative and, unusually, non-associative. The multiplication table for the basis elements is
| $\;e'_0\;$ | $e'_1$ | $e'_2$ | $e'_3$ | $e'_4$ | $e'_5$ | $e'_6$ | $e'_7$ |
| $\;e'_1\;$ | $-e'_0$ | $e'_3$ | $-e'_2$ | $-e'_5$ | $e'_4$ | $-e'_7$ | $e'_6$ |
| $\;e'_2\;$ | $-e'_3$ | $-e'_0$ | $e'_1$ | $-e'_6$ | $e'_7$ | $e'_4$ | $-e'_5$ |
| $\;e'_3\;$ | $e'_2$ | $-e'_1$ | $-e'_0$ | $-e'_7$ | $-e'_6$ | $e'_5$ | $e'_4$ |
| $\;e'_4\;$ | $e'_5$ | $e'_6$ | $e'_7$ | $e'_0$ | $e'_1$ | $e'_2$ | $e'_3$ |
| $\;e'_5\;$ | $-e'_4$ | $-e'_7$ | $e'_6$ | $-e'_1$ | $e'_0$ | $e'_3$ | $-e'_2$ |
| $\;e'_6\;$ | $e'_7$ | $-e'_4$ | $-e'_5$ | $-e'_2$ | $-e'_3$ | $e'_0$ | $e'_1$ |
| $\;e'_7\;$ | $-e'_6$ | $e'_5$ | $-e'_4$ | $-e'_3$ | $e'_2$ | $-e'_1$ | $e'_0$ |
which can be written using a ''split-octonion multiplication coefficient matrix'' as
$$
e'_a e'_b = M'_{ab}{}^c e'_c
$$
so, for example, $e'_1 e'_2 = e'_3$ and $M'_{12}{}^4 = 1$. ''Split-octonionic conjugation'' is given by
$$
\os{e}'_0 = e'_{\os{0}} = e'_0 \;\;\;\;\; \os{e}'_1 = e'_{\os{1}} = -e'_1 \;\;\;\; ... \;\;\;\; \os{e}'_7 = e'_{\os{7}} = -e'_7
$$
like for the octonions, and satisfies $\widetilde{(e'_a e'_b)} = e'_{\os{b}} e'_{\os{a}}$, so we have $M'_{ab}{}^c = M'_{\os{b} \os{a}}{}^{\os{c}}$.
Multiplying a split-octonion by its conjugate gives a real, its norm,
$$
\os{v} v = v^a v^b \os{e}'_a e'_b =
v^0 v^0 + v^1 v^1 + v^2 v^2 + v^3 v^3 - v^4 v^4 - v^5 v^5 - v^6 v^6 - v^7 v^7
= v \, \tilde{v} = | v |^2
$$
allowing us to calculate the inverse of any non-null split-octonion, $v^- = \frac{\tilde{v}}{|v|^2}$. Note that $v v$ is real only if $v$'s real part or vector part is zero. The ''split-octonion [[metric]]'' is defined as
$$
(u,v) = \ha \lp \os{u} v + \os{v} u \rp = u^0 v^0 + u^1 v^1 + u^2 v^2 + u^3 v^3 - u^4 v^4 - u^5 v^5 - u^6 v^6 - u^7 v^7
$$
so
$$
(e'_a, e'_b) = \ha ( \os{e}'_a e'_b + \os{e}'_b e'_a ) = n'_{ab} = \text{diag}(+1,+1,+1,+1,-1,-1,-1,-1)
$$
"split" between four positive and four negative, unlike the octonion metric. This metric, $n'_{ab}$, can be used to raise or lower split-octonion indices. The combination of metric and split-octonionic conjugates, $\os{e}{}'^a = n'^{ab} \os{e}'_b \in \tilde{\mathbb{O}}'$, are the [[duals|dual space]] to the split-octonions, $\os{e}'^a e_b = \de^a_b$. The matrix $\de^\os{b}_a = \text{diag}(+1,-1,-1,-1,-1,-1,-1,-1)$ can be used to twiddle or untwiddle indices, such as $M'^\os{b}{}_{ac} = \de^\os{b}_d M'^d{}_{ac} = n^{\os{b}d} M'_{dac}$.
The split-octonions are not associative and so do not have a faithful matrix representation; however, a nonfaithful real $8 \times 8$ representation comes from their multiplication table,
$$
(e'_c)^b{}_a = M'^\os{b}{}_{ac}
$$
which satisfies the metric expression, $(e'_a, e'_b) = n'_{ab}$, but $e'_a e'_b \ne M'_{ab}{}^c e'_c$.
There are real, $16 \times 16$, [[chiral]] [[representation|Clifford matrix representation]]s of the [[Cl(4,4)]] [[Clifford algebra]] related to [[split-octonion]]s, via [[Clifford division algebra representation]]. Explicitly, from the split-octonion multiplication coefficient matrix, the cyclic octonionic representation chiral dirac matrix is given by
$$
v^c \Ga'_c =
\lb \begin{array}{cccccccc}
v^0 & -v^1 & -v^2 & -v^3 & v^4 & v^5 & v^6 & v^7 \\
-v^1 & -v^0 & v^3 & -v^2 & v^5 & -v^4 & v^7 & -v^6 \\
-v^2 & -v^3 & -v^0 & v^1 & v^6 & -v^7 & -v^4 & v^5 \\
-v^3 & v^2 & -v^1 & -v^0 & v^7 & v^6 & -v^5 & -v^4 \\
-v^4 & v^5 & v^6 & v^7 & -v^0 & -v^1 & -v^2 & -v^3 \\
-v^5 & -v^4 & -v^7 & v^6 & v^1 & -v^0 & -v^3 & v^2 \\
-v^6 & v^7 & -v^4 & -v^5& v^2 & v^3 & -v^0 & -v^1 \\
-v^7 & -v^6 & v^5 & -v^4 &v^3 & -v^2 & v^1 & -v^0 \\
\end{array} \rb
$$
The resulting [[pseudoscalar]] is $\ga' = \ga'_0 \ga'_1 \ga'_2 \ga'_3 \ga'_4 \ga'_5 \ga'_6 \ga'_7 = - \si_3 \otimes 1$.
The ''split quaternions'', $\mathbb{H}'$, are a four-dimensional [[derision algebra|division algebra]], spanned by four basis elements, $e'_0,e'_1,e'_2,e'_3$, related to the [[quaternion]]s. The split-quaternion identity element is $e'_0=1$, and the other three can be thought of as different imaginary directions, squaring to $\pm 1$. Split-quaternionic multiplication is non-commutative and associative. The split-quaternions have a real matrix representation relating to the generalized [[Pauli matrices]] and [[skew]],
$$
e'_0 = \si_0 \;\;\;\;\; e'_1 = \si_1 \;\;\;\;\; e'_2 = - i \, \si_2 = \ep \;\;\;\;\; e'_3 = \si_3
$$
with an unconventional ordering chosen to mesh with the [[Pauli matrices]] and [[sl(2)]] basis elements. Their multiplication table is
| $\; e'_0 \;$ | $e'_1$ | $e'_2$ | $\; e'_3 \;$ |
| $e'_1$ | $e'_0$ | $e'_3$ | $e'_2$ |
| $e'_2$ | $-e'_3$ | $-e'_0$ | $e'_1$ |
| $e'_3$ | $-e'_2$ | $-e'_1$ | $e'_0$ |
which can be written using a ''split-quaternion multiplication coefficient matrix'' as
$$
e'_a e'_b = M'_{ab}{}^c e'_c
$$
so, for example, $e'_1 e'_2 = e'_3$ and $M'_{12}{}^3 = 1$. ''Split-quaternionic conjugation'' is given by
$$
\os{e}'_0 = e'_{\os{0}} = e'_0 \;\;\;\;\; \os{e}'_1 = e'_{\os{1}} = -e'_1 \;\;\;\; \os{e}'_2 = e'_{\os{2}} = -e'_2 \;\;\;\; \os{e}'_3 = e'_{\os{3}} = -e'_3
$$
and satisfies $\widetilde{(e'_a e'_b)} = e'_{\os{b}} e'_{\os{a}}$, so we have $M'_{ab}{}^c = M'_{\os{b} \os{a}}{}^{\os{c}}$. The spit-quaternions also have a [[Hermitian]] conjugation,
$$
e'^\da_0 = e'_{0^\da} = e'_0 \;\;\;\;\; e'^\da_1 = e'_{1^\da} = e'_1 \;\;\;\; e'^\da_2 = e'_{2^\da} = -e'_2 \;\;\;\; e'^\da_3 = e'_{3^\da} = e'_3
$$
using the above representation, with this Hermitian conjugation equivalent to the [[transpose]].
The semi-norm on the split-quaternions is
$$
|v'|^2 = \os{v}' v' = v'^0 v'^0 - v'^1 v'^1 + v'^2 v'^2 - v'^3 v'^3 = {\rm Det}(v') \; \in \; \mathbb{R}
$$
and can be obtained from the [[determinant]]. The split-quaternion [[metric]], which can be used to raise or lower split-quaternion indices, is
$$
(u', v') = \ha ( \os{u}' v' + \os{v}' u' ) = n'_{ab} u^a v^b
$$
with $n'_{ab} = diag(+1,-1,+1,-1)$. The combination of metric and quaternionic conjugates, $\os{e}'^a = n'^{ab} \os{e}'_b \in \tilde{\mathbb{H}}'$, are the [[duals|dual space]] to the split-quaternions, $\os{e}'^a e'_b = \de^a_b$, and are their Hermitian conjugates, $\os{e}'^a = e'^\da_a$. The matrix $\de^\os{b}_a = \text{diag}(+1,-1,-1,-1)$ can be used to twiddle or untwiddle indices, such as $M'^\os{b}{}_{ac} = \de^\os{b}_d M'^d{}_{ac} = n^{\os{b}d} M'_{dac}$. If a split-quaternion has nonzero norm, its inverse is $v'^- = \os{v}' / |v'|$. But null split-quaternions don't have inverses, which is why the split-quaternions aren't a [[division algebra]].
The split-quaternions also have a faithful, real matrix representation that, similar to the [[adjoint representation]], comes from their multiplication table,
$$
(e'_c)^b{}_a = M'^\os{b}{}_{ac}
$$
which relates to a [[realification|realify]] of the Pauli matrices.
There are real, $8 \times 8$, [[chiral]] [[representation|Clifford matrix representation]]s of the $Cl(2,2)$ [[Clifford algebra]] related to [[split-quaternion]]s, via [[Clifford division algebra representation]]. Explicitly, from the split-quaternion multiplication coefficient matrix, the cyclic split-quaternionic representation chiral dirac matrix is given by
$$
v^c \Ga'_c =
\lb \begin{array}{cccc}
v^0 & v^1 & -v^2 & v^3 \\
-v^1 & -v^0 & v^3 & -v^2 \\
-v^2 & v^3 & -v^0 & -v^1 \\
-v^3 & v^2 & -v^1 & -v^0 \\
\end{array} \rb
$$
The resulting [[pseudoscalar]] is $\ga' = \ga'_0 \ga'_1 \ga'_2 \ga'_3 = - \si_3 \otimes 1$.
The [[spin group]]s have interesting ''stabilizer subgroup''s, determined by specifying a unit length [[spinor]].
$$
\ba{rcl}
Stab_{Spin(8)} &=& Spin(7) \\
Stab_{Spin(7)} &=& G_2 \\
Stab_{Spin(6)} &=& SU(3) \\
Stab_{Spin(5)} &=& SU(2) \\
Stab_{Spin(4)} &=& SU(2)
\ea
$$
Particle glyphs
$$
\scir{#F2F200}, \ssqu{#F2F200}, \sdia{#F2F200}, \stri{#F2F200}, \sutr{#F2F200}
$$
$$
\mcir{#F2F200}, \msqu{#F2F200}, \mdia{#F2F200}, \mtri{#F2F200}, \mutr{#F2F200}
$$
$$
\bcir{#F2F200}, \bsqu{#F2F200}, \bdia{#F2F200}, \btri{#F2F200}, \butr{#F2F200}
$$
$$
\trip{\mutr{#F2F200}}{\mutr{#F2F200}}{\mutr{#F2F200}},
\trip{\mtri{#F2F200}}{\mtri{#F2F200}}{\mtri{#F2F200}},
\trip{\msqu{#F2F200}}{\msqu{#F2F200}}{\msqu{#F2F200}}
$$
rules
$
\def\hline#1{
\rlap{\hbox{@(hr noshade size="1" style="position:relative; left:-1pt;
width:#1em; border:0px; border-top:1px solid black")}}
}
\def\vline#1#2{\smash{\rule{1px}{#1em}{#2em}}}
$
A table with rules:
$$
\matrix{
\hline{5.238}\\[-.8em]
\vline{1.03}{1.675}&a&\vline{1.03}{1.675}&b&\vline{1.03}{1.675}\\[-.5em]
\hline{5.238}\\[-.8em]
&c&&d\\[-.6em]
\hline{5.238}\\[-.5em]
}$$
Some double lines:
$$
\matrix{
\hline{5.538}\\[-1.2em]
\hline{5.538}\\[-.8em]
\vline{1.13}{1.875}\,\vline{1.13}{1.875}&
a&\vline{1.13}{1.875}&b&
\vline{1.13}{1.875}\,\vline{1.13}{1.875}\\[-.5em]
\hline{5.538}\\[-.8em]
&c&&d\\[-.6em]
\hline{5.538}\\[-1.2em]
\hline{5.538}\\[-.5em]
}$$
Wave equation
$$
\nabla^2 \Ps = \fr{1}{c^2} \fr{\pa^2}{\pa t^2} \Psi
$$
@@display:block;text-align:center;[img[images/png/standard model and gravity small.png]]@@
The ''standard model'' particles are the fermions (leptons and quarks) and [[gauge bosons|connection]] (electromagnetic, weak, and strong bosons), as well as the gravitational [[spin connection]] and [[frame]], and the Higgs boson. The left or right [[chiral]] components of fermions transform under the left-chiral spin connection or its complex conjugate. The left (but not right) chiral doublet of electron neutrino and electron transform under the weak [[su(2)]], as do the other left-chiral lepton and quark doublets. The left and right-chiral quarks are triplets under strong [[su(3)]]. For [[one generation of fermions]], under
$$
(su(2)_W,su(3)_g)_{u(1)_Y}
$$
the [[representation space]] of elementary particles and their [[charge-parity conjugate|conjugates of a massless quantum Dirac spinor]] anti-particles is:
$$
\begin{array}{rclcrcl}
\{ \nu_{eL} , e_L \} & = & (2,1)_{-1} & \s\s & \{ \bar{e}_R, \bar{\nu}_{eR} \} & = & (2,1)_{+1} \\
\{ u_L , d_L \} & = & (2,3)_{\fr{1}{3}} & \s\s & \{ \bar{d}_R, \bar{u}_R \} & = & (2,\bar{3})_{-\fr{1}{3}} \\
\nu_{eR} & = & (1,1)_{0} & \s\s & \bar{\nu}_{eL} & = & (1,1)_{0} \\
e_{R} & = & (1,1)_{-2} & \s\s & \bar{e}_{L} & = & (1,1)_{+2} \\
u_{R} & = & (1,3)_{\fr{4}{3}} & \s\s & \bar{u}_{L} & = & (1,\bar{3})_{-\fr{4}{3}} \\
d_{R} & = & (1,3)_{-\fr{2}{3}} & \s\s & \bar{d}_{L} & = & (1,\bar{3})_{+\fr{2}{3}} \\
\end{array}
$$
Note the existence of the right-chiral neutrino and left-chiral anti-neutrino is considered provisional.
The [[standard model gauge group]] is $U(1)_Y \otimes SU(2)_W \otimes SU(3)_g / {\mathbb Z}_6$
Refs:
*Particle Data Group
**[[PDG - Review of Particle Physics|papers/PDG - Review of Particle Physics.pdf]]
**http://pdg.lbl.gov/
*R. Peccei
**[[Discrete and Global Symmetries in Particle Physics|papers/9807516.pdf]]
*Mark W Hopkins
**[[The Standard Model Lagrangian|papers/Standard Model.pdf]]
*[[standard model (short)|papers/standard model (short).pdf]]
*[[The Standard Model: Physical Basis and Scattering Experiments|papers/0011255.pdf]]
*C. Quigg
**[[The Double Simplex|http://arxiv.org/abs/hep-ph/0509037]]
***speculation on the standard model polytope
The [[Lie group]] under which the [[standard model]] fermions are covariant is $U(1)_Y \otimes SU(2)_W \otimes SU(3)_g / {\mathbb Z}_6$, in which $U(1)_Y$ is the ''hyper-weak'' Lie group, with fermions having ''hypercharge'', $Y$ and ''scaled hypercharge'' (to be integral) of $3Y$, $SU(2)_W$ is the ''weak'' Lie group, and $SU(3)_g$ is the ''strong'' Lie group, mediated by gluons. The [[representation space]] of [[one generation of fermions]] is //miraculously// invariant under a ${\mathbb Z}_6$ subgroup of $U(1)_Y \otimes SU(2)_W \otimes SU(3)_g$, which mods out to give the ''standard model gauge group''. The generating ${\mathbb Z}_6$ element is
$$
g_1 = \al^{3Y} \otimes \al^3 I_2 \otimes \al^2 I_3
$$
with $\al = e^{\fr{2 \pi i}{6}}$ a sixth root of unity. Acting on the fermions this gives
$$
\begin{array}{rclcrcl}
\{ \nu_{eL} , e_L \} & = & (2,1)_{-1} & \s\s & g_1 \{ \nu_{eL} , e_L \} & = & \al^{-3} \al^3 \{ \nu_{eL} , e_L \} = \{ \nu_{eL} , e_L \} \\
\{ u_L , d_L \} & = & (2,3)_{\fr{1}{3}} & \s\s & g_1 \{ u_L , d_L \} & = & \al^{1} \al^3 \al^2 \{ u_L , d_L \} = \{ u_L , d_L \} \\
\nu_{eR} & = & (1,1)_{0} & \s\s & g_1 \nu_{eR} & = & \al^{0} \nu_{eR} = \nu_{eR} \\
e_{R} & = & (1,1)_{-2} & \s\s & g_1 e_R & = & \al^{-6} e_R = e_R \\
u_{R} & = & (1,3)_{\fr{4}{3}} & \s\s & g_1 u_R & = & \al^{4} \al^2 u_R = u_R \\
d_{R} & = & (1,3)_{-\fr{2}{3}} & \s\s & g_1 d_R & = & \al^{-2} \al^2 d_R = d_R \\
\end{array}
$$
and $g_1$ also acts trivially on the anti-fermions.
The choice of a compact [[Cartan subalgebra|Lie algebra structure]] along with designation of positive vs negative weights and weight vectors spanning a real [[representation space]], or positive vs negative roots and root vectors, determines a [[complex structure]] on that space. For a Cartan subalgebra element acting via the adjoint (or appropriately on a representation space) we have a set of [[eigen]]vectors, $V_{\pm i}$, satisfying
$$
\big[ C , V_{\pm i} \big] = \pm \al_i(C) V_{\pm i}
$$
with a consistent choice of positive or negative roots in the [[root system]], and corresponding positive or negative root vectors. Iff $C$ is compact then $\al_i$ is imaginary and the $V_{\pm i}$ are complex conjugates. We can construct a complex structure, $J^C$, compatible with all Cartan operators, $\lb J^C, C \rb = 0$, as
$$
J^C V_{\pm i} = \pm \, i \, V_{\pm i}
$$
This splits the representation space into J-real and J-imaginary halves, $V = V^{\!C}_{\mathbb R} + V^{\!C}_{\mathbb I}$, with $V_\pm = V^{\!C}_{\mathbb R} \mp i V^{\!C}_{\mathbb I}$, $J^C V^{\!C}_{\mathbb R} = V^{\!C}_{\mathbb I}$, and $J^C V^{\!C}_{\mathbb I} = - V^{\!C}_{\mathbb R}$. The explicit matrix representation of the complex structure acting on the representation space can be found from converting via the matrix of eigenvectors,
$$
(J^C)^a{}_b = (V_{\pm j})^a (\pm i) (V^-)^{\pm j}{}_b
$$
Iff the Cartan subalgebra, $\mathfrak{C}$, is split, as in the Chevalley-Serre basis, then $\al_i$ and $V_{\pm i}$ are real, and we can construct a compatible ''split structure'' operator,
$$
J^S V_{\pm i} = \pm V_{\pm i}
$$
This splits the representation space into "positive" and "negative" halves, $V = V^{\!S}_p + V^{\!S}_n$, with $V_\pm = V^{\!S}_p \pm V^{\!S}_n$, $J^S V^{\!S}_p = V^{\!S}_n$, and $J^S V^{\!S}_n = V^{\!S}_p$.
If the Cartan subalgebra is mixed, such as for [[spin(1,3)]], then the eigenspace breaks into four parts,
$$
V = V_{{\mathbb R}p} + V_{{\mathbb I}p} + V_{{\mathbb R}n} + V_{{\mathbb I}n}
$$
with
$$
\begin{array}{rcl}
V_\pm \!\!&\!\!=\!\!&\!\! V_{{\mathbb R}p} \mp i \, V_{{\mathbb I}p} \pm V_{{\mathbb R}n} + i \, V_{{\mathbb I}n} \\
V_\pm^* \!\!&\!\!=\!\!&\!\! V_{{\mathbb R}p} \pm i \, V_{{\mathbb I}p} \pm V_{{\mathbb R}n} - i \, V_{{\mathbb I}n}
\end{array}
$$
satisfying
$$
\big[ C , V_{\pm j} \big] = \pm \al_j(C) V_{\pm j} \s \big[ C , V_{\pm j}^* \big] = \pm \al^*_j(C) V_{\pm j}^*
$$
Note that for an entirely compact Cartan we have $V_-=V_+^*$, but this is not the case for a mixed Cartan. The eigenspace does not evenly partition into $\{V_{{\mathbb R}p}, V_{{\mathbb I}p}, V_{{\mathbb R}n},V_{{\mathbb I}n} \}$ subspaces, since for a real weight $V_{{\mathbb I}p} = V_{{\mathbb I}n} =0$ and for an imaginary weight $V_{{\mathbb R}n} = V_{{\mathbb I}n} =0$. Nevertheless, one may use these to construct a [[real Cartan-Weyl basis]] for a Lie algebra or representation space.
The three [[Lie algebra]] generators, $T_A$, for the ''three dimensional special unitary group Lie algebra'' (//''special unitary group Lie algebra of order two''//), $su(2)$, corresponding to the [[SU(2)]] Lie group, may be [[represented|representation]] by $2 \times 2$ traceless anti-[[Hermitian]] matrices related to the [[Pauli matrices]],
$$
\begin{array}{ccc}
T_1 = i \sigma_{1} = \left[\begin{array}{cc}
0 & i\\
i & 0\end{array}\right] &
T_2 = -i \sigma_{2} = \left[\begin{array}{cc}
0 & -1\\
1 & 0\end{array}\right] &
T_3 = i \sigma_{3} = \left[\begin{array}{cc}
i & 0\\
0 & -i \end{array}\right]\end{array}
$$
From the resulting commutation relations,
$$
\lb T_A, T_B \rb = T_A T_B - T_B T_A = 2 \, \ep_{ABC} T_C
$$
the structure coefficients for this Lie algebra are equal to twice the [[permutation symbol]], $C_{AB}{}^C= 2 \,\ep_{ABC}$. (Note that if we had chosen $T_2 = i \si_2$ then we would have $C_{AB}{}^C = - 2 \,\ep_{ABC}$, which is still $su(2)$ but unconventional.) The generators are orthogonal -- the [[Killing form]] is
$$
\lp T_A, T_B \rp = g_{AB} = C_{AC}{}^D C_{BD}{}^C = 4 \, \ep_{ACD} \ep_{BDC} = - 8 \, \de_{AB} = 4 \, {\rm Tr}\lp T_A T_B \rp
$$
which is proportional to four times the [[trace]] of two multiplied $su(2)$ generators. As a real Lie algebra, $su(2)$ is the compact [[real form]] of $A_1 = su(2,\mathbb{C})$, and related to the split real form, [[sl(2)]]. The [[Cartan-Weyl basis|Lie algebra structure]] is obtained by defining
$$
H = T_3 = i \si_3 = \left[\begin{array}{cc}
i & 0\\
0 & -i \end{array}\right] \s
E_+ = \ha \lp - i T_1 - T_2 \rp
= \left[\begin{array}{cc}
0 & 1 \\
0 & 0 \end{array}\right] \s
E_- = \ha \lp - i T_1 + T_2 \rp =
\left[\begin{array}{cc}
0 & 0 \\
1 & 0\end{array}\right]
$$
resulting in the brackets
$$
\lb H , E_\pm \rb = \pm \, 2 \, i \, E_\pm \s
\lb E_+, E_- \rb = - i \, H
$$
If we further identify
$$
E^\mathbb{R} = (E_+ + E_-) = - \, i \, T_1 = \si_1 \s
E^\mathbb{I} = -i (E_+ - E_-) = i \, T_2 = \si_2
$$
then these satisfy
$$
\lb H , E^\mathbb{R} \rb = -2 \, E^\mathbb{I} \s
\lb H , E^\mathbb{I} \rb = +2 \, E^\mathbb{R} \s
\lb E^\mathbb{R}, E^\mathbb{I} \rb = 2 \, H
$$
consistent with the [[real Chevalley basis|Lie algebra structure]] for a compact Lie algebra. This has an evident [[complex structure|structures on a real representation space]], $J^C E^{\mathbb{R}/\mathbb{I}} = \pm E^{\mathbb{I}/\mathbb{R}}$, compatible with the Cartan subalgebra, $[ H, J^C E^{\mathbb{R}/\mathbb{I}} ] = J^C [ H, E^{\mathbb{R}/\mathbb{I}} ]$.
The $su(2)$ Lie algebra is also the bivector subalgebra, under the cross product, of the three dimensional Clifford algebra, [[Cl(3)]], and isomorphic to the [[spin Lie algebra]] $spin(3)$. Under this interpretation, the generators, $T_A$, are equated with three Clifford bivectors:
$$
\begin{array}{ccc}
T_1 = \si \, \sigma_{1} = \si_{23}
&
T_2 = - \si \, \sigma_{2} = \si_{13}
&
T_3 = \si \, \sigma_{3} = \si_{12}
\end{array}
$$
The 15 [[Lie algebra]] generators for the split [[special unitary group]] of order 4, ''su(2,2)'' may be represented by $4 \times 4$ traceless complex matrices that preserve the $(2,2)$ signature [[Hermitian]] norm, similar to $su(4)$. This Lie algebra is a [[subalgebra]] of [[su(3,3)]] and of [[su(4,2)]], and is isomorphic to the [[conformal algebra]] and to [[spin(2,4)]]. From a [[Clifford matrix representation]] of [[Cl(2,4)]], the negative [[chiral]] part of $spin(2,4)$ bivector generators have the representation
$$
B =
\lb \ba{cc}
B_L - \fr{d}{2} & -2 i p_L \\
2 i k_R & B_R + \fr{d}{2} \\
\ea \rb
=
\lb \ba{cc}
B_{\mathbb{R}}^\va \si_\va - i B_{\mathbb{I}}^\va \si_\va - \fr{d}{2} \si_0 & -2 i p^\mu \bar{\si}_\mu \\
2 i k^\mu \si_\mu & - B_{\mathbb{R}}^\va \si_\va - i B_{\mathbb{I}}^\va \si_\va + \fr{d}{2} \si_0 \\
\ea \rb
$$
while from a subalgebra of $su(3,3)$ the representation of a $su(2,2)$ algebra element, using [[bi-split-quaternion]]s, is
$$
B' =
\lb \ba{cc}
M & v \\
-\os{v}^* & P \\
\ea \rb
=
\lb \ba{cc}
i M_\mathbb{I}^0 \si_0 + M_\mathbb{R}^1 \si_1 -i M_\mathbb{R}^2 \si_2 + M_\mathbb{R}^3 \si_3
&
v_\mathbb{R}^0 \si_0 + i v_\mathbb{I}^0 \si_0
+ v_\mathbb{R}^1 \si_1 + i v_\mathbb{I}^1 \si_1
-i v_\mathbb{R}^2 \si_2 + v_\mathbb{I}^2 \si_2
+ v_\mathbb{R}^3 \si_3 + i v_\mathbb{I}^3 \si_3 \\
-v_\mathbb{R}^0 \si_0 + i v_\mathbb{I}^0 \si_0
+ v_\mathbb{R}^1 \si_1 - i v_\mathbb{I}^1 \si_1
-i v_\mathbb{R}^2 \si_2 - v_\mathbb{I}^2 \si_2
+ v_\mathbb{R}^3 \si_3 - i v_\mathbb{I}^3 \si_3
&
-i M_\mathbb{I}^0 \si_0 + P_\mathbb{R}^1 \si_1 -i P_\mathbb{R}^2 \si_2 + P_\mathbb{R}^3 \si_3
\\
\ea \rb
$$
which has $i$'s and $\si_\ep$'s in different combinations. These $su(2,2)$ representations can be linked by a [[similarity transformation|Dirac matrices]], $B' = U \, B \, U^\da$, with
$$
U = \lb \ba{cc} \si_0 & \si_2 \\ \si_1 & -i \si_3 \ea \rb
$$
giving the messy identification,
$$
\ba{ccc}
\ba{rcl}
M_\mathbb{I}^0 \!\!&\!\!=\!\!&\!\! k_2 + p_2 \\
M_\mathbb{R}^1 \!\!&\!\!=\!\!&\!\! - B_\mathbb{R}^1 - k^3 + p^3 \\
M_\mathbb{R}^2 \!\!&\!\!=\!\!&\!\! B_\mathbb{I}^2 - k^0 + p^0 \\
M_\mathbb{R}^3 \!\!&\!\!=\!\!&\!\! B_\mathbb{R}^3 + k^1 - p^1 \\
& & \\
P_\mathbb{R}^1 \!\!&\!\!=\!\!&\!\! - B_\mathbb{R}^1 + k^3 - p^3 \\
P_\mathbb{R}^2 \!\!&\!\!=\!\!&\!\! - B_\mathbb{I}^2 - k^0 + p^0 \\
P_\mathbb{R}^3 \!\!&\!\!=\!\!&\!\! - B_\mathbb{R}^3 + k^1 - p^1 \\
\ea
&
\s \s
&
\ba{rcl}
v_\mathbb{R}^0 \!\!&\!\!=\!\!&\!\! - k^3 - p^3 \\
v_\mathbb{I}^0 \!\!&\!\!=\!\!&\!\! - B_\mathbb{I}^1 \\
v_\mathbb{R}^1 \!\!&\!\!=\!\!&\!\! - \fr{d}{2} \\
v_\mathbb{I}^1 \!\!&\!\!=\!\!&\!\! k^2 - p^2 \\
v_\mathbb{R}^2 \!\!&\!\!=\!\!&\!\! - k^1 - p^1 \\
v_\mathbb{I}^2 \!\!&\!\!=\!\!&\!\! B_\mathbb{I}^3 \\
v_\mathbb{R}^3 \!\!&\!\!=\!\!&\!\! k^0 + p^0 \\
v_\mathbb{I}^3 \!\!&\!\!=\!\!&\!\! -B_\mathbb{R}^2 \\
\ea
\ea
$$
This similarity transformation also links between the usual representation of [[Cl(2,4)]] and a [[Clifford compound division algebra representation]]. [[Spinor|spinor]]s of $spin(2,4)$ are acted on by $su(2,2)$ as
$$
\Ps' = U \, \Ps =
\lb \ba{cc} \si_0 & \si_2 \\ \si_1 & -i \si_3 \ea \rb
\lb \ba{c} \ps_L \\ \ps_R \ea \rb
=
\lb \ba{c} \ps_L^\wedge - i \ps_R^\vee \\
\ps_L^\vee + i \ps_R^\wedge \\
i \ps_L^\vee - i \ps_R^\wedge \\
i \ps_L^\wedge + i \ps_R^\vee \ea \rb
\; \in \; \mathbb{C}^4
$$
//hey wait, try [[biquaternion]]s with alternative signature in [[Clifford compound division algebra representation]] to get [[Cl(2,4)]]...//
//or try [[Majorana]] [[rep|Dirac matrices]]//
The eight [[Lie algebra]] basis generators for the [[special unitary group]] Lie algebra of order three, ''su(3)'', corresponding to the [[SU(3)]] Lie group, may be [[represented|representation]] by $3 \times 3$ traceless, anti-[[Hermitian]] matrices related to the [[Gell-Mann matrices]], $T_A \sim \pi(T_A) = i \la_A$. The structure constants resulting from the commutation relations, $\lb T_A, T_B \rb = C_{AB}{}^C T_C$, are anti-symmetric in their indices, with nonzero values:
$$
\begin{array}{ccccc}
C_{12}{}^3 = -2 & C_{14}{}^7 = -1 & C_{15}{}^6 = 1 & C_{24}{}^6 = -1 & C_{25}{}^7 = -1 \\
C_{34}{}^5 = -1 & C_{36}{}^7 = 1 & C_{45}{}^8 = -\sqrt{3} & C_{67}{}^8 = -\sqrt{3}
\end{array}
$$
The resulting [[Killing form]] is proportional to the [[trace]] of two multiplied $su(3)$ generators,
$$
\lp T_A, T_B \rp = g_{AB} = C_{AC}{}^D C_{BD}{}^C = - 12 \, \de_{AB} = 6 \, {\rm Tr}(T_A T_B)
$$
Using the Gell-Mann matrices, any Lie algebra element may be represented as a $3 \times 3$ complex matrix,
$$
\ba{rcl}
B^A T_A \!\!&\!\!=\!\!&\!\!
\lb\ba{ccc}
i \, B^3 + \fr{i}{\sqrt{3}} B^8 & i \, B^1 + B^2 & i \, B^4 + B^5 \\
i \, B^1 - B^2 & - i \, B^3 + \fr{i}{\sqrt{3}} B^8 & i \, B^6 + B^7 \\
i \, B^4 - B^5 & i \, B^6 - B^7 & -\fr{2i}{\sqrt{3}} B^8
\ea\rb
=
\lb\ba{ccc}
V-M & v & -m^* \\
-v^* & P-V & p \\
m & -p^* & M-P
\ea \rb \\
\!\!&\!\!=\!\!&\!\! V \, H_v + M \, H_m + P \, H_p + (v \, E^+_v - v^* E^-_v) + (m \, E^+_m - m^* E^-_m) + (p \, E^+_p - p^* E^-_p) \\
\!\!&\!\!=\!\!&\!\! u(1)_3 + u(1)_8 + 2_v + 2_m + 2_p
\ea
$$
with $v$, $m$, $p$ complex numbers, and $M$, $P$, $V$ pure imaginary numbers. Note that the same Lie algebra element is specified by $\{ M+c, P+c, V+c \}$, so there are only two degrees of freedom in those three parameters, $B^3$ and $B^8$. With orthogonal [[Cartan subalgebra|Lie algebra structure]] basis generators $\{ T_3,T_8 \}$, or non-orthogonal $\{ \mathcal{H}_m, \mathcal{H}_p, \mathcal{H}_v \}$, the root vectors, their root coordinates, and their Lie brackets, are
$$
\begin{array}{|rcl|cc|ccc|ccc|}
\hline
& \!\!\!\!\!\!\!\!\!su(3)\!\!\!\!\!\!\!\!\! & & g^3 & g^8 & v & m & p & r & g & b\\
\hline
E^+_v = g^{r\bar{g}} \!\!&\!\!=\!\!&\!\! \ha ( +T_2 - i \, T_1 ) & +2 & 0 & +2 & -1 & -1 & +1 & -1 & 0\\
E^-_v = g^{\bar{r}g} \!\!&\!\!=\!\!&\!\! \ha ( -T_2 - i \, T_1 ) & -2 & 0 & -2 & 1 & +1 & -1 & +1 & 0\\
E^+_m = g^{\bar{r}b} \!\!&\!\!=\!\!&\!\! \ha ( - T_5 - i \, T_4 ) & -1 & -\sqrt{3} & -1 & +2 & -1 & -1 & 0 & +1\\
E^-_m = g^{r\bar{b}} \!\!&\!\!=\!\!&\!\! \ha ( +T_5 - i \, T_4 ) & +1 & +\sqrt{3} & +1 & -2 & +1 & +1 & 0 & -1\\
E^+_p = g^{g\bar{b}} \!\!&\!\!=\!\!&\!\! \ha ( +T_7 - i \, T_6 ) & -1 & +\sqrt{3} & -1 & -1 & +2 & 0 & +1 & -1\\
E^-_p = g^{\bar{g}b} \!\!&\!\!=\!\!&\!\! \ha ( - T_7 - i \, T_6 ) & +1 & -\sqrt{3} & +1 & +1 & -2 & 0 & -1 & +1\\
\hline
\end{array}
\s \s
\ba{lcrcl}
\!\!&\!\! \!\!&\!\! [ T_3, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm i \, g_3 \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ T_8, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm i \, g_8 \, E^\pm_\al \\
\mathcal{H}_v \! \!\!&\!\!=\!\!&\!\! [ E^+_v, E^-_v ] \!\!&\!\!=\!\!&\!\! -i \, T_3 \\
\mathcal{H}_m \! \!\!&\!\!=\!\!&\!\! [ E^+_m, E^-_m ] \!\!&\!\!=\!\!&\!\! -i \, (- \ha T_3 - \fr{\sqrt{3}}{2} T_8) \\
\mathcal{H}_p \! \!\!&\!\!=\!\!&\!\! [ E^+_p, E^-_p ] \!\!&\!\!=\!\!&\!\! -i \, (- \ha T_3 + \fr{\sqrt{3}}{2} T_8) \\
\!\!&\!\! \!\!&\!\! [ \mathcal{H}_v, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm v \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ \mathcal{H}_m, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm m \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ \mathcal{H}_p, E^\pm_\al ] \!\!&\!\!=\!\!&\!\! \pm p \, E^\pm_\al \\
\!\!&\!\! \!\!&\!\! [ E^+_v, E^+_m ] \!\!&\!\!=\!\!&\!\! -E^-_p \\
\ea
$$
This description of [[Lie algebra structure]] is consistent with a Cartan-Weyl basis and Chevalley-Serre basis, with [[simple root|root system]] vectors chosen as any two of $\{ E^+_v, E^+_m, E^+_p \}$. The $su(3)$ [[root system]] is a hexagon, which we can understand as a 2D projection of the roots of [[sp(3)]].
In physics, the $su(3)$ Lie algebra is associated with the strong force, moderated by ''gluon''s, which are $su(3)$ valued 1-form fields that interact with ''quark''s in a fundamental representation space, and ''anti-quark''s in its conjugate space. It is useful to consider yet a different orthogonal set of ''color'' basis elements, $\{ C_r, C_g, C_b \}$, that are in the Cartan subalgebra of $u(3)$ but are not in $su(3)$, such that [>img[images/png/g2rs2.png]]
$$
\s \s \s \mathcal{H}_v = C_r - C_g \s \s \mathcal{H}_m = - C_r + C_b \s \s \mathcal{H}_p = C_g - C_b \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
$$
The corresponding quark weight vectors and weights are
$$
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\begin{array}{|rcl|cc|ccc|ccc|}
\hline
& & & g^3 & g^8 & v & m & p & r & g & b \\
\hline
q^r \!\!&\!\!=\!\!&\!\! [ 1, 0, 0 ]^T & +1 & +\fr{1}{\sqrt{3}} & +1 & -1 & 0 & +1 & 0 & 0\\
\bar{q}^r \!\!&\!\!=\!\!&\!\! [ 1, 0, 0 ] & -1 & -\fr{1}{\sqrt{3}} & -1 & +1 & 0 & -1 & 0 & 0\\
q^g \!\!&\!\!=\!\!&\!\! [ 0, 1, 0 ]^T & -1 & +\fr{1}{\sqrt{3}} & -1 & 0 & +1 & 0 & +1 & 0\\
\bar{q}^g \!\!&\!\!=\!\!&\!\! [ 0, 1, 0 ] & +1 & -\fr{1}{\sqrt{3}} & +1 & 0 & -1 & 0 & -1 & 0\\
q^b \!\!&\!\!=\!\!&\!\! [ 0, 0, 1 ]^T & 0 & -\fr{2}{\sqrt{3}} & 0 & +1 & -1 & 0 & 0 & +1\\
\bar{q}^b \!\!&\!\!=\!\!&\!\! [ 0, 0, 1 ] & 0 & +\fr{2}{\sqrt{3}} & 0 & -1 & +1 & 0 & 0 & -1\\
\hline
\end{array}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
$$
This structure of $su(3)$ is consistent with a [[triality decomposition]], in which each of the three triples, $\{ \mathcal{H}_{v/m/p}, E^+_{v/m/p}, E^-_{v/m/p} \} \sim \{ \mathcal{H} , E^+ , E^- \}$, corresponds to a different [[su(2)]] related to each other by [[complex triality]], with the triality-related $su(2)$ Cartan generators in the $su(3)$ Cartan subalgebra necessarily overlapping. A [[triality automorphism of su(3)]] preserves the structure.
The 35 [[Lie algebra]] basis generators for the split [[special unitary group]] of order 6, ''su(3,3)'', may be represented by $6 \times 6$ traceless complex matrices that preserve the $(3,3)$ signature [[Hermitian]] norm, similar to [[su(6)]]. From the [[exceptional magic square]], the [[compound triality decomposition]] of $su(3,3)$ is
$$
su(3,3) = sl(2)^M + sl(2)^P + sl(2)^V + u(1)_3 + u(1)_8 + 2^m \times 2^p \times 2_v + 2^v \times 2^p \times 2_m + 2^v \times 2^m \times 2_p
$$
combining the structure of [[sp(6,R)]] and [[su(3)]]. This can also be understood as a $3 \times 3$ matrix of [[bi-split-quaternion]]s,
$$
A =
\lb
\begin{array}{ccc}
M & v & -\os{m}^* \\
-\os{v}^* & P & p & \\
m & -\os{p}^* & V \\
\end{array}
\!\!\!\!\! \rb
\;\; \in \; \pi \! \lp su(3,3) \rp
$$
with a restriction on $M$, $P$, and $V$, that they have real coefficients and vanishing scalar parts, except for imaginary scalar parts satisfying $M^0_\mathbb{I} + P^0_\mathbb{I} + V^0_\mathbb{I} = 0$ (removing that last constraint means there's another generator, making $u(3,3)$). The restrictions on $M$, $P$, and $V$ ensure an allowed split-Hermitian signature, $(1,1)$, via $q \, \si_2 + \si_2 q^\da = 0$. A [[su(2,2)]] subalgebra of $su(3,3)$ is the the upper left $2 \times 2$ block of this matrix, which then acts on both columns of $m$ and $p$ as [[spin(2,4)]] [[spinor]]s, with the columns acted on by $V$, corresponding to the decomposition
$$
su(3,3) = su(2,2) + su(1,1) + u(1) + 2 \times 2 \times 2 + 2 \times 2 \times 2
$$
Triality automorphisms of $su(3,3)$ relate directly to [[bi-split-quaternion]] triality. A $su(3,3)$ triality automorphism is innner, $A' = g_T \, A \, g_T^-$, such as
$$
\lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb
\lb
\begin{array}{ccc}
M & v & -\os{m}^* \\
-\os{v}^* & P & p & \\
m & -\os{p}^* & V \\
\end{array}
\!\!\!\!\! \rb
\lb \ba{ccc} & & 1 \\ 1 & & \\ & 1 & \\ \ea \rb
=
\lb
\begin{array}{ccc}
P & p & -\os{v}^* \\
-\os{p}^* & V & m & \\
v & -\os{m}^* & M \\
\end{array}
\!\!\!\!\! \rb
$$
OK... su(3,3) contains [[spin(2,4)]]=su(2,2) the same way e8(24) contains so(12,4)
So maybe the twistors are there in the rest of
su(3,3) = su(2,2) + su(1,1) + u(1) + 2 x 2 x 2 + 2 x 2 x 2
with each 2x2x2 being a [[bi-split-quaternion]] confused with two spinors (one Weyl spinor in each column), and a triality automorphism of su(3,3) rotates them inside [[spin(2,4)]]
into a vector of so(1,3) corresponding to conformal null translations and special conformal transformations. Spacetime is the Hermitian subspace of these vectors. Maybe weak interaction has to do with anti-Hermitian subspace. Put these ideas into [[twistor triality]] and spin(2,4) [[su(2,2)]] isomorphism via a $4 \times 4$ similarity transformation matrix... that comes from relating the Cl(2,4) of su(2,2) to the one from spin(2,4)....
This must also mean that the bi-split-quaternions here are conflated with pairs of Weyl spinors... which agrees I think with the [[table of Clifford matrix representations]]. Work out this conflation explicitly, including [[time conjugate]] (which squares to -1 and anticommutes with i, so should be a quaternion?) and other conjugates.
This actually doesn't work well because left and right chiral Weyl spinor components get all mixed up together in the su(2,2) rep this way. So maybe try su(2,2) with biquaternions?
Maybe use roots and weights to better identify su(3,3) = spin(2,4) + ?
This would all fit inside e8(24) in fun ways, with spinors rotated inside so(4,12) to so(3,11) vectors. su(3,3) is a subalgebra of so(6,6)
The 35 [[Lie algebra]] basis generators for the $(4,2)$ signature [[special unitary group]] of order 6, ''su(4,2)'', may be represented by $6 \times 6$ traceless complex matrices that preserve the $(4,2)$ signature [[Hermitian]] norm, similar to [[su(6)]]. The [[compound triality decomposition]] of $su(4,2)$ is
$$
su(4,2) = su(2)^M + su(2)^P + su(2)^V + u(1)_3 + u(1)_8 + 2^m \times 2^p \times 2_v + 2^v \times 2^p \times 2_m + 2^v \times 2^m \times 2_p
$$
combining the structure of [[sp(3)]] (actually a different non-compact [[symplectic Lie algebra|sp(n)]]) and [[su(3)]]. This can also be understood as a $3 \times 3$ matrix of [[biquaternion]]s,
$$
A =
\lb
\begin{array}{ccc}
M & v & -\os{m}^* \\
\os{v}^* & P & p & \\
m & \os{p}^* & V \\
\end{array}
\!\!\!\!\! \rb
=
\lb
\begin{array}{ccc}
M & v & -m^\da \\
v^\da & P & p & \\
m & p^\da & V \\
\end{array}
\!\!\!\!\! \rb
\;\; \in \; \pi \! \lp su(4,2) \rp
$$
with a restriction on $M$, $P$, and $V$, that they are anti-[[Hermitian]], and so are real vector quaternions, such as
$$
M = M_\mathbb{I}^0 i e_0 + M_\mathbb{R}^\ep e_\ep = M_\mathbb{I}^0 i \si_0 - M_\mathbb{R}^\ep i \si_\ep
$$
except for imaginary scalar parts satisfying $M^0_\mathbb{I} + P^0_\mathbb{I} + V^0_\mathbb{I} = 0$ (removing that last constraint means there's another generator, making $u(4,2)$). The structure of the matrix and restrictions on $M$, $P$, and $V$ give $A \, D + D \, A^\da = 0$ for $D = {\rm diag}(+1, -1, +1)$. A [[su(2,2)]] subalgebra of $su(4,2)$ is the the upper left $2 \times 2$ block of this matrix, which then acts on both columns of $m$ and $p$ as [[spin(2,4)]] [[spinor]]s, with the columns acted on by $V$, corresponding to the decomposition
$$
su(3,3) = su(2,2) + su(1,1) + u(1) + 2 \times 2 \times 2 + 2 \times 2 \times 2
$$
There is sort of a (non-inner) complex [[triality]] [[automorphism|Lie algebra automorphism]] of $su(4,2)$ related directly to [[biquaternion]] triality, $A' = g_T \, A \, g_T^-$, such as
$$
\lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb
\lb
\begin{array}{ccc}
M & v & -m^\da \\
v^\da & P & p & \\
m & p^\da & V \\
\end{array}
\!\!\!\!\! \rb
\lb \ba{ccc} & & 1 \\ 1 & & \\ & 1 & \\ \ea \rb
=
\lb
\begin{array}{ccc}
P & p & v^\da \\
p^\da & V & m & \\
v & -m^\da & M \\
\end{array}
\!\!\!\!\! \rb
$$
that is a complex automorphism, rather than a real automorphism, because if we write this as $T'_C = T_B \, \ph^B{}_C$ then $\ph^B{}_C$ is necessarily complex, and $g_T \notin SU(4,2)$.
su(4,2) contains [[spin(2,4)]] = su(2,2)
So maybe the twistors are there in the rest of
su(4,2) = su(2,2) + su(2) + u(1) + 2 x 2 x 2 + 2 x 2 x 2
with each 2x2x2 being a [[biquaternion]] confused with two spinors (one Weyl spinor in each column), and a complex triality automorphism of su(4,2) rotates the Weyl spinors inside [[spin(2,4)]] into a vector of so(1,3) corresponding to conformal null translations and special conformal transformations. Spacetime is the Hermitian subspace of these vectors. Maybe weak interaction has to do with anti-Hermitian subspace. Put these ideas into [[twistor triality]] and spin(2,4) [[su(2,2)]] isomorphism via a $4 \times 4$ similarity transformation matrix... that comes from relating the Cl(2,4) of biquaternions to the one from spin(2,4)....
This must also mean that the biquaternions here are conflated with pairs of Weyl spinors... which agrees I think with the [[table of Clifford matrix representations]]. Work out this conflation explicitly, including [[time conjugate]] (which squares to -1 and anticommutes with i, so should be a quaternion?) and other conjugates.
Maybe use roots and weights to better identify su(4,2) = spin(2,4) + ?
This would all fit inside e8(24) in fun ways, with spinors rotated inside so(4,12) to so(3,11) vectors. su(4,2) is a subalgebra of so(4,8)
The 35 [[Lie algebra]] basis generators for the [[special unitary group]] Lie algebra of order 6, ''su(6)'', may be represented by $6 \times 6$ traceless, anti-[[Hermitian]] matrices. From the [[exceptional magic square]], the [[compound triality decomposition]] of $su(6)$ is
$$
su(6) = su(2)^M + su(2)^P + su(2)^V + u(1)_3 + u(1)_8 + 2^m \times 2^p \times 2_v + 2^v \times 2^p \times 2_m + 2^v \times 2^m \times 2_p
$$
combining the structure of [[sp(3)]] and [[su(3)]]. This can also be understood as a $3 \times 3$ matrix of [[biquaternion]]s,
$$
\lb
\begin{array}{ccc}
q^M & q_v & -q_m^\da \\
-q_v^\da & q^P & q_p & \\
q_m & -q_p^\da & q^V \\
\end{array}
\!\!\!\!\! \rb
\;\; \in \; \pi \! \lp su(6) \rp
$$
with a restriction that $q^M$, $q^P$, and $q^V$, are anti-Hermitian (so have real coefficients) and $q^{M0}_\mathbb{I} + q^{P0}_\mathbb{I} + q^{V0}_\mathbb{I} = 0$ (removing that last constraint means there's another generator, making $u(6)$). A $su(4)$ subalgebra of $su(6)$ is the the upper left $2 \times 2$ block of this matrix, which then acts on both columns of $q_m$ and $q_p$ as $spin(6)$ [[spinor]]s, with the columns acted on by $su(2)^V$.
Triality [[automorphism|Lie algebra automorphism]]s of $su(6)$ relate directly to [[biquaternion]] triality. A $su(6)$ triality automorphism is innner, $A' = g_T \, A \, g_T^-$, such as
$$
\lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb
\lb
\begin{array}{ccc}
q^M & q_v & -q_m^\da \\
-q_v^\da & q^P & q_p & \\
q_m & -q_p^\da & q^V \\
\end{array}
\!\!\!\!\! \rb
\lb \ba{ccc} & & 1 \\ 1 & & \\ & 1 & \\ \ea \rb
=
\lb
\begin{array}{ccc}
q^P & q_p & -q_v^\da \\
-q_p^\da & q^V & q_m & \\
q_v & -q_m^\da & q^M \\
\end{array}
\!\!\!\!\! \rb
$$
A ''Lie subalgebra'', $\mathfrak{h}$, is a subset of a [[Lie algebra]], $\mathfrak{h} \subset \mathfrak{g}$, that is closed under the Lie bracket, $\lb \mathfrak{h}, \mathfrak{h} \rb = \mathfrak{h}$. The Lie algebra of a Lie [[subgroup]] is necessarily a Lie subalgebra.
There are five ''exceptional'' Lie algebras, [[g2]], [[f4]], [[e6]], [[e7]], and [[e8]].
\begin{eqnarray}
so(2n+2) &=& so(2n+1) + (2n+1) \\
so(2n+1) &=& so(2n) + (2n) \\
sl(2n) &=& sp(2n) + (? 2n - 1 ?) \\
so(8) &=& g2 + 2 \times 7 \\
so(7) &=& g2 + 7 \\
g2 &=& sl(3) + 3 + 3? \\
e6 &=& f4 + 26 \\
f4 &=& so(9) + 16 \\
f4 &=& so(8) + 8 + 8 + 8 \\
\end{eqnarray}
$$
g2 \subset so(7) \subset so(8) \subset so(9) \subset f4
$$
Ref:
*Kostant Brylinski
**[[Nilpotent Orbits, Normality, and Hamiltonian Group Actions|papers/Kostant Brylinski - Nilpotent Orbits, Normality, and Hamiltonian Group Actions.pdf]]
A [[group]], $H$, is a ''subgroup'' of another group, $H \subset G$, iff it is a subset of $G$ and also satisfies the properties of a group: it necessarily contains the identity element and is closed under products and inverses..
A ''submanifold'', $S$, of an $n$ dimensional [[manifold]], $M$, is a subset of points of $M$ that is itself an $s$ dimensional manifold, inheriting a collection of charts from $M$. If points on patches of $M$ are labeled by $z^i$, with coordinate [[indices]] $1 \leq i \leq n$, then patches of $S$ may be labeled by coordinates $x^a$ with $1 \leq a \leq s$, and the points of $S$ in $M$ given as a ''parameterized surface'', $z_S^i = z_S^i(x)$. A submanifold is sometimes also described as a [[diffeomorphism]], $\ph$, from a manifold $N$ to $S$ in $M$,
\begin{eqnarray}
\ph : N &\mapsto& S \subset M \\
z_S^i(x) &=& \ph^i(x)
\end{eqnarray}
in which $x^a$ are coordinates for $N$. Either $S$, $N$, or $\ph$ may be referred to as "the submanifold".
If a [[manifold]], $M$, has a [[tangent bundle]] [[metric]], any [[submanifold]], $N$, inherits a [[metric]] from its immersion in $M$,
$$
\ph : N \mapsto S \subset M
$$
Specifically, if $\ve{u}$ and $\ve{v}=v^a \ve{\pa_a}$ are [[tangent vector]]s to paths on $N$, then they [[pushforward|pullback]] to tangent vectors on $M$,
$$
\ph_* \ve{v} = v^a \fr{\pa \ph^i}{\pa x^a} \ve{\pa_i}
$$
and their scalar product on $N$ is given by the scalar product of their pushforwards,
$$
\lp \ve{u}, \ve{v} \rp = \lp \ph_* \ve{u}, \ph_* \ve{v} \rp = u^a v^b \fr{\pa \ph^i}{\pa x^a} \fr{\pa \ph^j}{\pa x^b} g_{ij}
$$
so the metric on $N$ is given in terms of the metric on $M$ by
$$
g_{ab} = \fr{\pa \ph^i}{\pa x^a} \fr{\pa \ph^j}{\pa x^b} g_{ij}
$$
If there is a [[frame]] of 1-forms, $\f{e^\al}$, on $M$, this frame [[pulls back|pullback]] to give a set of frame 1-forms on $N$,
$$
\f{e'^\al} = \ph^* \f{e^\al} = \f{dx^a} \fr{\pa \ph^i}{\pa x^a} \lp e_i \rp^\al
$$
with metric, $g_{ab} = \lp e'_a \rp^\al \lp e'_b \rp^\be \et_{\al \be}$; but since the dimension, $n$, of $M$ is (typically) larger than the dimension, $s$, of $N$ then there will be $(n-s)$ extra frame 1-forms, $\f{e'^\ph}$, on $N$. This may be remedied by ''adapting'' the frame to the surface by a [[Lorentz rotation]], $L^\al{}_\be(x)$, that ensures the extra frame 1-forms vanish,
$$
\begin{eqnarray}
\f{e''^\mu} &=& L^\mu{}_\be \f{e'^\be} \\
\f{e''^\ph} &=& L^\ph{}_\be \f{e'^\be} = 0
\end{eqnarray}
$$
A ''superconnection'',
$$
\udf{A} = \f{A} + \ud{A}
$$
is the sum of a [[Lie algebra]] valued [[connection]] [[1-form]] field, $\f{A} = \f{dx^i} A_i^{\p{i}B} T_B$, and a Lie algebra valued [[Grassmann number]] field, $\ud{A} = \ud{A^B} T_B$. This construct arises naturally in the [[BRST technique]], in which it is called a BRST extended connection.
//supercurvature//
//geometry?//
Ref:
*Lee and Hwan
**[[BRST Quantization of Gauge Theory in Noncommutative Geometry: Matrix Derivative Approach|papers/9512215.pdf]]
*Shlomo Sternberg
**[[Toronto Lectures on Physics|papers/Sternberg - Toronto Lectures on Physics.pdf]]
Ref:
*[[Krippendorf - Cambridge Lectures on Supersymmetry and Extra Dimensions|http://arxiv.org/abs/1011.1491v1]]
*[[Peskin - Supersymmetry in Elementary Particle Physics|papers/Peskin - Supersymmetry in Elementary Particle Physics.pdf]]
*[[Martin - A Supersymmetry Primer|papers/Martin - A Supersymmetry Primer.pdf]]
*[[Ichinose - Graphical Representation of Supersymmetry|papers/Ichinose - Graphical Representation of Supersymmetry.pdf]]
*Josh Kantor
**[[Supersymmetry|papers/supertiddlers3.pdf]]
**[[Supergeometry|papers/physicstiddlers-supergeo.pdf]]
A ''symmetric space'' is a [[homogeneous space]], $G/H$, with a restriction that its [[Lie algebra]] be invariant under negation of $\mathfrak{g}/\mathfrak{h}$. Specifically, if $H_P$ are the generators of $\mathfrak{h}$ and $K_A \in \mathfrak{g}/\mathfrak{h}$ are the [[coset generators|reductive]], then the homogeneous space, $G/H$, is a symmetric space iff the Lie algebra,
\begin{eqnarray}
\lb H_P, H_Q \rb &=& C_{PQ}{}^R H_R \\
\lb H_P, K_A \rb &=& C_{PA}{}^B K_B + C_{PA}{}^R H_R \\
\lb K_A, K_B \rb &=& C_{AB}{}^C K_C + C_{AB}{}^R H_R
\end{eqnarray}
is invariant under $K_A \mapsto -K_A$. This will be true iff $C_{PA}{}^R=0$ and $C_{AB}{}^C=0$, in which case $H$ is [[symmetric|reductive]] in $G$. So a symmetric space is necessarily [[reductive]],
$$
{\rm Ad}_{\mathfrak{g}/\mathfrak{h}} \mathfrak{h} = \lb \mathfrak{g}/\mathfrak{h} , \mathfrak{h} \rb \subset \mathfrak{g}/\mathfrak{h}
$$
as well as having
$$
{\rm Ad}_{\mathfrak{g}/\mathfrak{h}} \mathfrak{g}/\mathfrak{h} = \lb \mathfrak{g}/\mathfrak{h}, \mathfrak{g}/\mathfrak{h} \rb \subset \mathfrak{h}
$$
Symmetries of a [[Lie algebra]] are described by reflections, $r_\al$, along [[simple roots|Lie algebra structure]]. These symmetries constitute the ''Weyl group''.
As an example, removing the symmetry,
$$
w = r_1 r_8 r_3 r_4
$$
of [[e8]] results in
$$
SO(8) \times SU(3) \times U(1)^2
$$
This is also the Lie group from breaking
$$
r_1 r_2 r_3 r_4 r_5 r_6
$$
Or something like that...
//my understanding of this is very rough right now//
Ref:
*Akin Wingerter
**[[Aspects of Grand Unification in Higher Dimensions|papers/Wingerter - Aspects of Grand Unification in Higher Dimensions.pdf]]
***Nice comprehensive thesis with introductory material.
***see p118
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Every [[Clifford algebra]], $Cl(p,q)$, has many [[Clifford matrix representation]]s, with elements represented as real, complex, or quaternionic matrices. Although a larger representation can always be chosen, the pattern of "minimal" representations is interesting and useful. For a Clifford algebra of $p$ positive and $q$ negative [[signature|Minkowski metric]], this table, with $n=p+q$ running vertically and $\rho = p-q$ horizontally, shows the minimal matrix representations:
| $\p{0}$ || $-8$ | $-7$ | $-6$ | $-5$ | $-4$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| $0$ || | | | | | | | | $\mathbb{R}$ | | | | | | | | |
| $1$ || | | | | | | | $\mathbb{C}$ | | $2 \, \mathbb{R}$ | | | | | | | |
| $2$ || | | | | | | $\mathbb{H}$ | | $\mathbb{R}(2)$ | | $\mathbb{R}(2)$ | | | | | | |
| $3$ || | | | | | $2 \, \mathbb{H}$ | | $\mathbb{C}(2)$ | | $2 \, \mathbb{R}(2)$ | | $\mathbb{C}(2)$ | | | | | |
| $4$ || | | | | $\mathbb{H}(2)$ | | [[$\mathbb{H}(2)$|Dirac matrices]] | | $\mathbb{R}(4)$ | | [[$\mathbb{R}(4)$|Dirac matrices]] | | $\mathbb{H}(2)$ | | | | |
| $5$ || | | | $\mathbb{C}(4)$ | | $2 \, \mathbb{H}(2)$ | | $\mathbb{C}(4)$ | | $2 \, \mathbb{R}(4)$ | | $\mathbb{C}(4)$ | | $2 \, \mathbb{H}(2)$ | | | |
| $6$ || | | $\mathbb{R}(8)$ | | $\mathbb{H}(4)$ | | $\mathbb{H}(4)$ | | $\mathbb{R}(8)$ | | $\mathbb{R}(8)$ | | $\mathbb{H}(4)$ | | $\mathbb{H}(4)$ | | |
| $7$ || | $2 \, \mathbb{R}(8)$ | | $\mathbb{C}(8)$ | | $2 \, \mathbb{H}(4)$ | | $\mathbb{C}(8)$ | | $2 \, \mathbb{R}(8)$ | | $\mathbb{C}(8)$ | | $2 \, \mathbb{H}(4)$ | | $\mathbb{C}(8)$ | |
| $8$ || $\mathbb{R}(16)$ | | $\mathbb{R}(16)$ | | $\mathbb{H}(8)$ | | $\mathbb{H}(8)$ | | [[$\mathbb{R}(16)$|Cl(4,4)]] | | $\mathbb{R}(16)$ | | $\mathbb{H}(8)$ | | $\mathbb{H}(8)$ | | $\mathbb{R}(16)$ |
| $\ga \ga$ || $+$ | $+$ | $-$ | $-$ | $+$ | $+$ | $-$ | $-$ | $+$ | $+$ | $-$ | $-$ | $+$ | $+$ | $-$ | $-$ | $+$ |
For example, the $\mathbb{R}(4)$ at $n=4$ and $\rh=2$ says the $4\times4$ real Majorana representation exists for the [[Cl(3,1)]] [[Dirac matrices]]. The notation $2 \, \mathbb{H}(2)$ means a $2$-block-diagonal matrix, with blocks that are $2 \times 2$ matrices of quaternions. And note that quaternions are often represented as $2\times2$ complex matrices, so $\mathbb{H}(2) \sim \mathbb{C}(4)$. A positive or negative $\ga^2$, the square of the [[pseudoscalar]], determines whether or not these Clifford algebras admit a [[chiral]] decomposition. This table repeats with eight-fold [[Bott periodicity]], so we let $n = n_0 + 8e$ and $\rh = \rh_0 + 8f$ for integer $e$ and $f$, with $0 \le n_0 \le 7$ and $0 \le \rh_0 \le 7$. Because of [[Clifford representation doubling]], $Cl(p+1,q+1) = Cl(p,q) \otimes Cl(1,1)$, this table may be expressed somewhat more succinctly and completely as
| $\rho_0$ || $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
| $n$ || $\mathbb{R}(2^{n/2})$ | $2 \, \mathbb{R}(2^{(n-1)/2})$ | $\mathbb{R}(2^{n/2})$ | $\mathbb{C}(2^{(n-1)/2})$ | $\mathbb{H}(2^{n/2-1})$ | $2 \, \mathbb{H}(2^{(n-1)/2})$ | $\mathbb{H}(2^{n/2-1})$ | $\mathbb{C}(2^{(n-1)/2})$ |
[[Spinor|spinor]] [[representation space]]s are real, [[complex|complex structure]], or [[quaternion]]ic, based on Clifford algebra signature,
| $\rho_0$ || $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
| $\ps$ || $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{C}$ | $\mathbb{H}$ | $\mathbb{H}$ | $\mathbb{H}$ | $\mathbb{C}$ | $\mathbb{R}$ |
Note that spinors inherit a structure from one dimension higher.
Ref:
[[http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras|http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras]]
[[https://arxiv.org/pdf/1101.5690.pdf|https://arxiv.org/pdf/1101.5690.pdf]]
''E8 Theory''
Moscone, 1/15/10
All gravitational and standard model particle fields may be described as parts of a superconnection valued in the Lie algebra of the split real form of E8, with dynamics described by its curvature. The algebra of standard model fields and their embedding in E8 is exhibited explicitly by a matrix representation, and schematically using weight diagrams. The implications for quantum gravity and particle physics are explored, and open questions discussed.
Slides:
#[[E8 Theory summary ]]
#[[E8 Theory ]]
#[[Structure of interactions]]
#[[Fiber bundle]]
#[[Matrix representation]]
#[[Weights]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5)]]
#[[Pati-Salam]]
#[[SO(10) ]]
#[[E6 ]]
#[[Gauge-gravity unification]]
#[[Action]]
#[[Spin]]
#[[Everything together]]
#[[Spin(3 11) ToE]]
#[[Matrix representation of spin(3 11)]]
#[[Spin(3 11) ToE ]]
#[[E8 ToE]]
#[[Superconnection ]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Matrix representation of spin(4 12)]]
#[[Elementary Particle Explorer]]
#[[E8 triality]]
#[[Generations]]
#[[E8 Theory summary ]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''Unification''
7/5/10
Slides:
#[[Unification.]]
#[[Gravity]]
#[[Dirac equation]]
#[[Dirac equation in curved spacetime]]
#[[Dirac equation in curved spacetime with gauge fields and Higgs]]
#[[Grand unification]]
#[[GraviGUT unification]]
#[[Superconnection ]]
#[[E8 unification]]
#[[E8 Theory summary ]]
#[[Structure of Standard Model interactions]]
#[[Fiber bundle]]
#[[Matrix representation of the Standard Model]]
#[[Matrix representation of the Standard Model ]]
#[[Standard Model generator matrices]]
#[[Particle states and charges]]
#[[Weight diagrams]]
#[[Twist]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Torus twist]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5)]]
#[[Pati-Salam]]
#[[SO(10) ]]
#[[E6 ]]
#[[GraviGUT unification from the fermion action]]
#[[Action for bosons]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT]]
#[[Matrix representation of spin(11 3)]]
#[[Standard Model generator matrices]]
#[[Matrix representation of the Standard Model ]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[Superconnection ]]
#[[BRST]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Spin(12 4) GraviGUT]]
#[[E8(-24) ToE]]
#[[Predicted particles]]
#[[Generations]]
#[[Elementary Particle Explorer]]
#[[E8 triality]]
#[[E8(8) ToE]]
#[[E8 Theory summary ]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''A Quest for Unification''
3/6/11
Everything in our universe is composed of elementary particle fields interacting according to the laws of quantum physics. Remarkably, these balanced physical interactions correspond to the geometry of elegant mathematical structures. Even more remarkably, these structures describing our universe appear to be parts of a larger structure, long revered by mathematicians for its complex beauty. In this talk, rich in graphics, Garrett Lisi will describe the geometry of our universe, some new ideas on unification, and a bit of his own story -- including current efforts to create locally optimal living and research environments for conducting theoretical science.
Slides:
#[[A Quest for Unification ]]
#[[HST and LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''A Quest for Unification''
All elementary particles and interactions in the Standard Model of physics may be described as the geometry of fiber bundles -- complicated shapes twisting over spacetime. The charges of each kind of elementary particle correspond to how many times that field excitation twists around the maximal torus associated with each different fundamental force. By plotting the charges of all known particles we see how this pattern fits into the more complete patterns corresponding to Grand Unified Theories of particle physics. By applying this same geometric description to gravity and its interactions we see how it may be unified with the other forces and particles as parts of a single geometric field. Remarkably, the resulting pattern of charges fits into the pattern describing what is perhaps the most beautiful geometric structure known to mathematics, E8.
Collection of slides to use on 4/24/09:
#[[A Quest for Unification]]
#[[What is an electron field, geometrically?]]
#[[Dirac Lagrangian in curved spacetime]]
#[[Unified bosonic connection]]
#[[Periodic table of the standard model]]
#[[Roots, weights, and particle charges]]
#[[Gluons and quarks]]
#[[Strong interactions]]
#[[Lie group geometry.]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Standard model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5) GUT]]
#[[SO(10) GUT]]
#[[SO(10) GUT.]]
#[[Pati-Salam GUT.]]
#[[Pati-Salam GUT]]
#[[Gauge theory geometry]]
#[[Gravity as a gauge field]]
#[[Standard model and gravity]]
#[[Superconnection]]
#[[Standard model and gravity in E8 - sort of]]
#[[Elementary Particle Explorer]]
#[[Maximal tortoise]]
<<slider chkTestSlider 'addenda to talk for CSUF 09' 'Useful addenda >' "Click here to see the addenda">>
''A Geometric Theory of Everything''
10/9/10
Slides:
#[[A Geometric Theory of Everything]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model]]
#[[Georgi-Glashow SU(5)]]
#[[SO(10) ]]
#[[E6 ]]
#[[Lie groups ]]
#[[Spin]]
#[[Geometry of interactions ]]
#[[Spin ]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''A Geometric Theory of Everything''
5/21/12
Slides:
#[[A Geometric Theory of Everything ]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions ]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''A Geometric Theory of Everything''
3/26/11
Slides:
#[[A Geometric Theory of Everything ]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
The Universe as a Beautiful Shape
*The many interacting fields of the standard model and gravity are conventionally represented as elements of several different algebras. Each of these algebras can be expressed using matrices or, equivalently, as parts of a large Lie algebra. When all fields of gravity and the standard model (including three generations of fermions) are efficiently combined, they match just one simple Lie group: the largest exceptional group, E8 -- considered by many to be the most beautiful structure in mathematics. When this shape, E8, twists around our four dimensions of spacetime, the fields of the standard model and gravity are parts of the connection describing this twist -- with all interactions and dynamics described by its curvature. If this new model of reality is successful, everything in our universe will be understood as the symmetry and evolution of a beautiful shape.
Slides for [[FQXi 07 conference]] talk, with some supporting tiddlers:
#[[Standard model and gravity]]
##[[standard model]]
#[[Everything as a principal bundle connection]]
#[[Standard model and gravity in a matrix]]
##[[principal bundle]]
#[[Gravitational part of the connection]]
##[[spacetime frame]]
##[[spacetime spin connection]]
#[[Bosonic part of the connection]]
##[[Clifford algebra]]
##[[Cl(1,7)|Cl(8)]]
##[[connection]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
#[[Curvature of bosonic part]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
#[[Gravitational action]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
#[[Action for everything]]
#[[Why this Lie algebra]]
##[[Lie algebra]]
#[[Real simple compact Lie groups]]
##[[Lie groups]]
#[[Triality decomposition of E8]]
##[[e8]]
##[[e8 triality decomposition]]
#[[E8 T.O.E.]]
#[[Geometry of Yang-Mills theory]]
#[[Cartan subalgebra and charges]]
##[[Lie algebra structure]]
#[[E8 roots]]
##[[e8 root system]]
#[[Reducing E8 to the standard model]]
##[[e6]]
#[[Discussion]]
Extra slides:
#[[BRST gauge fixing]]
#[[BRST extended connection]]
#[[Massive Dirac operator in curved spacetime]]
Here is my [[practice audio|talks/FQXi07/audio/FQXi07.wav]] for the talk. The slides above will change as they are used for other talks. To see the slides as they were presented, they have been put in a [[pdf file of all the slides|talks/FQXi07/FQXi2007.pdf]].
''A Quest for Unification''
4/14/11
Slides:
#[[A Quest for Unification ]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions ]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
A connection with everything
Slides for [[ILQGS|http://relativity.phys.lsu.edu/ilqgs/]] talk on 11/13/07:
#[[Periodic table of the standard model]]
#[[A connection with everything]]
#[[Review of some representation theory]]
#[[Gluon and quark weights]]
#[[Strong G2]]
#[[Exceptional Lie brackets]]
#[[G2 in SO(7) .]]
#[[G2 in SO(7) ..]]
#[[G2 in SO(7) ...]]
#[[G2 in SO(7) ....]]
#[[G2 in SO(7) ....x]]
#[[Pati-Salam model plus gravity]]
#[[Gravitational SO(3,1)]]
#[[Electroweak SU(2) and U(1)]]
#[[Graviweak SO(7,1)]]
#[[Graviweak F4]]
#[[F4 root system]]
#[[F4 and G2 together]]
#[[E8 root system .]]
#[[E8 root system ..]]
#[[E8 root system ...]]
#[[E8 root system ....]]
#[[E8 root system ....x]]
#[[E8 root system ....x.]]
#[[E8 root system ....x.x]]
#[[E8 root system ....x.x.]]
#[[E8 root system ....x.x..]]
#[[E8 root system ....x.x...]]
#[[E8 root system ....x.x....]]
#[[E8 root system ....x.x....x]]
#[[E8 root system ....x.x....x.]]
#[[E8 root system ....x.x....x..]]
#[[E8 root system ....x.x....x..x]]
#[[E8 root system ....x.x....x..x.]]
#[[E8 root system ....x.x....x..x..]]
#[[E8 root system ....x.x....x..x...]]
#[[E8 root system ....x.x....x..x...x]]
#[[E8 periodic table]]
#[[E8 connection]]
#[[E8 curvature]]
#[[Action for everything]]
#[[Gravitational part of the action]]
#[[Fermionic part of the action]]
#[[E8 Theory summary]]
#[[E8 Theory discussion]]
Extra slides:
#[[BRST extended connection]]
#[[Geometry of Yang-Mills theory]]
#[[The Coleman-Mandula theorem]]
The slides above will change as they are used for other talks. To see the slides as they were presented, they are available in a [[pdf file of all the slides|talks/ILQGS07/Lisi.pdf]] along with the [[ILQGSO7 audio|talks/ILQGS07/lisi111307.mp3]].
''A Geometric Theory of Everything''
2/25/11
Using pure geometry, Einstein described gravity as the warping of four-dimensional spacetime, with the measurements of rulers and clocks differing based on their position and motion. Similarly, the other forces and particles of nature are described as the twisting of geometric structures over spacetime, and around each other. Examining the patterns of twists, we'll find that gravity and every elementary particle in our Universe may be facets of the most beautiful structure known to mathematics dancing over spacetime according to the laws of quantum physics.
Slides:
#[[A Geometric Theory of Everything ]]
#[[HST]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Torus twist]]
#[[Electroweak symmetry breaking]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Particle Zoo ]]
#[[Everything together ]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
These are the slide-tiddlers for [[Loops '07|http://www.matmor.unam.mx/eventos/loops07/]] talk, with audio and some supporting tiddlers. //(These may change as I modify them for other talks -- if you want to see them as presented, grab the pdf from the bottom of this tiddler.)//
#[[Standard model and gravity]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide1.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[standard model]]
#[[Standard model and gravity in one big connection]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide2.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[principal bundle]]
#[[Gravitational part of bosonic connection]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide3.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[spacetime frame]]
##[[spacetime spin connection]]
#[[Bosonic connection]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide4.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[Clifford algebra]]
##[[Cl(8)]]
##[[connection]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
#[[Curvature of bosonic connection]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide5.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[curvature]]
##[[Clifford-Riemann curvature]]
#[[Gravitational action]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide6.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
#[[Why this Lie algebra]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide7.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[Lie algebra]]
#[[Real simple compact Lie groups]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide8.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[Lie groups]]
#[[Triality decomposition of E8]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide9.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
#[[Pieces of E8]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide10.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[e8]]
#[[E8 generator conversion]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide11.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
#[[E8 triality structure]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide12.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
##[[e8 triality decomposition]]
#[[E8 TOE]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide13.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
#[[Discussion]]
##<html><embed src="talks/Loops07/Loops2007audio/Loops2007.slide14.wav" loop="false" autoplay="false" width="200" height="20"></embed></html>
Extra slides (include some in talk pdf):
#[[Geometry of Yang-Mills theory]]
#[[BRST gauge fixing]]
#[[BRST extended connection]]
#[[Massive Dirac operator in curved spacetime]]
#[[Plan of attack (old)]]
[[Open all slides as a pdf|talks/Loops07/Loops2007.pdf]]
''The Geometry of Our Universe''
7/28/11
Slides:
#[[The Geometry of Our Universe]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Quantum chromodynamics]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''The Geometry of Particle Physics''
8/17/11
Slides:
#[[The Geometry of Particle Physics]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Twist]]
#[[Electric charges]]
#[[Electroweak charges]]
#[[Torus twist]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Strong charges]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[Geometry of interactions]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[Predicted particles]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''A Geometric Theory of Everything''
5/11/12
Slides:
#[[A Geometric Theory of Everything ]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions ]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
An Exceptionally Simple Theory of Everything
*All fields of the standard model and gravity are unified as an E8 principal bundle connection.
Slides for Perimeter talk on 10/04/07, with some supporting tiddlers:
#[[An Exceptionally Simple Theory of Everything]]
#[[Everything as a principal bundle connection]]
#[[Review of some representation theory]]
#[[Gluon and quark weights]]
#[[Strong G2]]
#[[Exceptional Lie brackets]]
#[[G2 in SO(7)]]
#[[Pati-Salam model plus gravity]]
#[[Gravitational SO(3,1)]]
#[[Electroweak SU(2) and U(1)]]
#[[Graviweak SO(7,1)]]
#[[Graviweak F4]]
#[[F4 root system]]
#[[F4 and G2 together]]
#[[E8 root system]]
#[[E8 periodic table]]
#[[E8 connection]]
#[[E8 curvature]]
#[[Action for everything]]
#[[Gravitational part of the action]]
#[[Fermionic part of the action]]
#[[E.S.T.o.E. summary]]
#[[E.S.T.o.E. discussion]]
Extra slides:
#[[Geometry of Yang-Mills theory]]
#[[BRST gauge fixing]]
The slides above will change as they are used for other talks. To see the slides as they were presented, they have been put in a [[pdf file of all the slides|talks/Perimeter07/Perimeter07.pdf]].
How, why, on what, and where I work.
Collection of slides to use on 7/5/09:
#[[Welcome ]]
#[[Action of matter]]
#[[Geometry of interactions]]
#[[Rotational charges]]
#[[Particle zoo]]
#[[Elementary Particle Explorer]]
''An Exceptionally Simple Theory of Everything''
6/3/12
Slides:
#[[An Exceptionally Simple Theory of Everything]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions ]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
Yale, 10/23/09
Elementary particles, geometry, unification, kitesurfing -- an unusual life in physics.
Slides:
#[[Elementary particles geometry unification]]
#[[Structure of interactions]]
#[[Fiber bundle ]]
#[[Twist]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Torus twist]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5)]]
#[[E6 ]]
#[[Action of matter]]
#[[Geometry of interactions]]
#[[Spin]]
#[[Spin(3 11) ToE]]
#[[Matrix representation of spin(4 12)]]
#[[Superconnection ]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Elementary Particle Explorer]]
#[[Particle zoo]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'talk for Zuckerman 09' 'Zuckerman 09 >' "Click here to see the addenda">>
''The Beauty of Particle Physics''
8/4/10
Slides:
#[[The Beauty of Particle Physics]]
#[[Geometry of interactions ]]
#[[Spin]]
#[[Structure of Standard Model interactions]]
#[[Weight diagrams]]
#[[Twist]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Torus twist]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5)]]
#[[Pati-Salam]]
#[[SO(10) ]]
#[[E6 ]]
#[[Superconnection ]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[Lie groups ]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[Generations]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[Maximilian Tortoise]]
#[[Particle states and charges]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
The Beauty of Particle Physics
Slides for [[TED|http://www.ted.com/]] talk on 2/28/08:
#[[Coral reef]]
#[[Branching coral]]
Alcyonium digitatum
#[[Branching coral.]]
#[[Schrodinger's Cat]]
#[[Quantum Mechanics]]
everything that can happen, does
what can happen
#[[Particle interactions]]
LHC animation
120-180 MW
Monterey = 2539 (10^3/(24 365)) (30161/424842) = 20MW
http://www.energy.ca.gov/electricity/electricity_by_county_2005.html
http://www.co.monterey.ca.us/population.htm
#[[Elementary particle zoo]]
#[[Electric charge]]
#[[Hypercharge and weak charge]]
#[[Hypercharge and weak charge.]]
Higgs
#[[Hypercharge, weak charge, and strong charges]]
#[[Hypercharge, weak charge, strong charges, and spins]]
standard up to here
#[[Two new charges]]
some particles on top of each other -- two more axes to uniquely identify particles
unification in E8
fill 20 holes -- new particles, LHC?
pattern corresponds to a shape
Galileo's assertion that "the book of nature is written in mathematics"
slide of scary math
wipeout picture?
by quantum mechanics, have an E8 coral
branch a spinning E8 with mathematica?
seed with random E8 elements, with small random momenta, move, plot, and branch with random variations. Branch via generated lookup table. Fill screen that way. (keep seeding steadily and randomly, starting from black)
beauty and truth, fun and work, have to balance.
physics, love, surfing -- diagram like three particles, only with pictures
van
How do you spend all your time on physics, love, and surfing and still afford to live by the beach? You live in a van.
[[Maui coral]]
hanging out with coral
The slides above will change as they are used for other talks. To see the slides as they were presented, they are available in a [[pdf file of all the slides|talks/TED08/Lisi.pdf]] along with the practice [[TED08 audio|talks/TED08/Lisi.mp3]].
''A Geometric Theory of Everything''
12/10/11
Slides:
#[[A Geometric Theory of Everything ]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''C, P, T, and Triality''
7/12/24
Slides:
#[[Unification]]
#[[Why three generations?]]
#[[C, P, T, and Triality]]
#[[Groups]]
#[[Quaternion group]]
#[[Pin group]]
#[[Spin eigenvalues]]
#[[Fermion basis states]]
#[[C, P, and T]]
#[[CPT cube]]
#[[Biquaternionic spinors]]
#[[Quaternion triality and the CPTt Group]]
#[[Multi-generational fermion states]]
#[[Fermions in Exceptional Unification]]
Addenda
#[[C, P, and T in QFT]]
''An Exceptionally Simple Theory of Everything''
6/13/13
Slides:
#[[An Exceptionally Simple Theory of Everything ]]
#[[Subatomic particles]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Electroweak torus]]
#[[Electroweak symmetry breaking]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Maximal tori]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Lie group fibers over spacetime]]
#[[Structure of interactions ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Elementary Particle Explorer]]
#[[Science Hostel]]
#[[To do]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
Collection of slides to use on 3/7/08:
#[[Geometry of Yang-Mills theory]]
##[[principal bundle]]
##[[vector-form algebra]]
#[[BRST gauge fixing]]
##[[BRST technique]]
#[[BRST extended connection]]
#[[Massive Dirac operator in curved spacetime]]
#[[A connection with everything]]
#[[E8 connection]]
#[[E8 curvature]]
#[[Action for everything]]
#[[Gravitational part of the action]]
##[[spacetime frame]]
##[[spacetime spin connection]]
##[[modified BF gravity]]
##[[volume form]]
##[[Clifford curvature scalar]]
#[[Bosonic part of the connection]]
##[[Clifford algebra]]
##[[Cl(1,7)|Cl(8)]]
##[[connection]]
##[[spin connection]]
##[[frame]]
##[[su(2)]]
##[[su(3)]]
##[[indices]]
#[[Curvature of bosonic part]]
##[[curvature]]
##[[Clifford-Riemann curvature]]
#[[Fermionic part of the action]]
#[[Periodic table of the standard model]]
#[[Review of some representation theory]]
#[[Gluon and quark weights]]
#[[Strong G2]]
#[[Exceptional Lie brackets]]
#[[G2 in SO(7)]]
#[[Pati-Salam model plus gravity]]
#[[Gravitational SO(3,1)]]
#[[Electroweak SU(2) and U(1)]]
#[[Graviweak SO(7,1)]]
#[[Graviweak F4]]
#[[E8 periodic table]]
#[[E8 Theory summary]]
#[[E8 Theory discussion]]
#[[Real simple compact Lie groups]]
##[[Lie groups]]
#[[The Coleman-Mandula theorem]]
''An Exceptionally Simple Theory of Everything''
6/3/12
defaults write -g NSScrollAnimationEnabled -bool NO
Slides:
#[[An Exceptionally Simple Theory of Everything]]
#[[LHC]]
#[[LHC animation]]
#[[Particle Zoo]]
#[[Structure of interactions ]]
#[[Laser beam]]
#[[Connection field]]
#[[Circular fibers of electromagnetism]]
#[[Fiber bundle ]]
#[[Electric charge]]
#[[Twist]]
#[[Electric charge ]]
#[[Electroweak model]]
#[[Circles twist in a Lie group]]
#[[Electroweak symmetry breaking]]
#[[Torus twist]]
#[[Electroweak model ]]
#[[Weak interaction ]]
#[[Weak interaction]]
#[[Weak interaction ]]
#[[Quantum chromodynamics]]
#[[Quantum chromodynamics ]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Standard Model ]]
#[[Georgi-Glashow SU(5) GUT ]]
#[[Spin(10) GUT]]
#[[E6 SuperGUT]]
#[[Superconnection ]]
#[[Lie groups ]]
#[[E6 SuperGUT ]]
#[[What is spin]]
#[[Geometry of interactions]]
#[[Gravity ]]
#[[Spin]]
#[[Everything together]]
#[[Spin(11 3) GraviGUT ]]
#[[E8 ToE]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Predicted particles]]
#[[E8 triality]]
#[[E8 Theory summary ]]
#[[E8 animation]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
''E8 Theory''
Yale, 10/24/09
All of the gravitational and standard model particle fields of physics may be described as parts of a superconnection valued in the Lie algebra of a noncompact real form of E8, with dynamics described by its curvature. This algebra of standard model fields and its embedding in E8 may be exhibited explicitly by a matrix representation, and schematically using weight diagrams. Several open questions with this model remain, including the mathematical interpretation of the superconnection -- which has an analogue in the BRST formulation of gauge theories.
Slides:
#[[Geometric structure of the standard model and gravity]]
#[[E8 Theory]]
#[[Structure of interactions]]
#[[Fiber bundle]]
#[[Matrix representation]]
#[[Weights]]
#[[Twist]]
#[[Electric charge]]
#[[Electroweak model]]
#[[Torus twist]]
#[[Standard Model]]
#[[Grand Unified Theories]]
#[[Georgi-Glashow SU(5)]]
#[[SO(10) ]]
#[[Pati-Salam]]
#[[E6 ]]
#[[Gauge-gravity unification]]
#[[Action]]
#[[Spin]]
#[[Everything together]]
#[[Spin(3 11) ToE]]
#[[Matrix representation of spin(3 11)]]
#[[Spin(3 11) ToE ]]
#[[E8 ToE]]
#[[Superconnection ]]
#[[BRST]]
#[[E8 ToE (projection to the Coxeter plane)]]
#[[Matrix representation of spin(4 12)]]
#[[Elementary Particle Explorer]]
#[[E8 triality]]
#[[Generations]]
#[[E8 Theory summary ]]
#[[Maximilian Tortoise]]
<<slider chkTestSlider 'addenda to talk for Zuckerman 09' 'Useful addenda >' "Click here to see the addenda">>
The ''tangent bundle'', $TM$, with $n$ dimensional base [[manifold]], $M$, is a [[vector bundle]] with $n$ fiber basis elements identified as the [[coordinate basis vectors]], $\ve{\pa_i}$, for the manifold. The fiber at a base manifold point, $p$, is the $n$ dimensional tangent space, $T_p M$, spanned by the coordinate basis vectors, and the tangent bundle is the union of all tangent spaces, $TM = \bigcup_{p \in M} T_p M$. The transition functions for the basis elements, $\ve{\pa^2_i} = \lp t^{21} \rp_i{}^j \ve{\pa^1_j}$, over overlapping patches, $U_1$ and $U_2$, are given by the ''Jacobian matrix'',
$$\lp t^{21} \rp_i{}^j = \fr{\pa x_1^j}{\pa x_2^i}$$
The structure group is thus the group of general linear transformations, $G = GL(n,\Re)$. A ''tangent vector field'' (sometimes just //''vector field''//), $\ve{v}=\ve{v}(x) = v^i(x) \ve{\pa_i}$, over the manifold is a section of the tangent bundle, and consists of a [[tangent vector]] at each manifold point.
When a [[metric]] exists for the tangent bundle the [[orthonormal basis vectors|frame]], $\ve{e_\al} = \lp e_\al \rp^i \ve{\pa_i}$, may alternatively be used as local fiber basis elements for the tangent bundle. The transition functions are then [[Lorentz transformations|Lorentz rotation]], $\ve{e^2_\al} = \lp L^{21} \rp^\be{}_\al \ve{e^1_\be}$. The value of the Lorentz transformation coefficients may be calculated from the frame components over the different patches and the coordinate basis transition function coefficients,
$$
\ve{e^2_\al} = \lp e^2_\al \rp^i \ve{\pa^2_i} = \lp e^2_\al \rp^i \lp t^{21} \rp_i{}^j \ve{\pa^1_j} = \lp e^2_\al \rp^i \lp t^{21} \rp_i{}^j \lp e^{1-}_j \rp^\be \ve{e^1_\be}
$$
to get $\lp L^{21} \rp^\be{}_\al = \lp e^2_\al \rp^i \lp t^{21} \rp_i{}^j \lp e^{1-}_j \rp^\be$. Carrying this out for all the transition functions gives the [[reduction of the structure group]] for the tangent bundle from the general linear group to the Lorentz group. Note also that, through equating the Lorentz transition functions, $L^\be{}_\al$, and using the [[frame]], $\ve{e_\al} \f{e} = \ga_\al$, the tangent bundle may be [[associated]] to the [[Clifford vector bundle]].
A change of basis due to [[coordinate change]], $\ve{\pa_i} \mapsto \ve{\pa'_i} = \fr{\pa x^j}{\pa x'^i} \ve{\pa_j}$, is the most basic type of [[tangent bundle gauge transformation]], with a different type of gauge transformation given by Lorentz rotation of the orthonormal basis vectors. The [[partial derivative]] is zero when acting on coordinate basis vectors, $\pa_i \ve{\pa_j} = 0$, but this derivative doesn't properly keep track of the local trivialization or gluing between patches. To remedy this, a [[tangent bundle covariant derivative|tangent bundle connection]] is introduced which, via a [[tangent bundle connection]], keeps track of how the basis vectors change over the base when taking the derivative of a vector field -- it co-varies with a gauge transformation. Using the covariant derivative, any vector may be [[parallel transport|tangent bundle parallel transport]]ed along any path on the base to obtain a new vector at any point along the path. For a closed path, the parallel transport of a vector is represented by a [[tangent bundle holonomy]] -- an element of the general linear group which acts on the initial vector. For a small closed path, or loop, the holonomy is given approximately by the [[Riemann curvature]] (//tangent bundle curvature//) -- an important geometric descriptor of the tangent bundle and connection.
The [[vector bundle connection]] for the [[tangent bundle]], $TM$, is defined through the operation of the suitable [[vector bundle covariant derivative|vector bundle connection]] on [[coordinate basis vectors]],
\begin{eqnarray}
\na_i \ve{\pa_j} &=& \Ga^k{}_{ij} \ve{\pa_k} \\
\f{\na} \ve{\pa_j} &=& \f{\Ga}^k{}_j \ve{\pa_k}
\end{eqnarray}
The coefficients, $\Ga^k{}_{ij}$, of the ''tangent bundle connection'', $\f{\Ga}^k{}_j = \f{dx^i} \Ga^k{}_{ij}$, are referred to as the [[Christoffel symbols]], and the index positions are arranged to agree with convention rather than having the usual order for vector bundle connection coefficients.
An alternative expression of the tangent bundle connection may be found by calculating the ''tangent bundle covariant derivative'' of the [[orthonormal basis vectors|frame]],
\begin{eqnarray}
\na_i \ve{e_\al} &=& \na_i \lp \lp e_\al \rp^j \ve{\pa_j} \rp = \lp \pa_i \lp e_\al \rp^j + \lp e_\al \rp^k \Ga^j{}_{ik} \rp \ve{\pa_j} = \lp D_i \lp e_\al \rp^j \rp \ve{\pa_j} = w_{i}{}^\be{}_\al \ve{e_\be} \\
\f{\na} \ve{e_\al} &=& \f{w}^\be{}_\al \ve{e_\be}
\end{eqnarray}
in which the coefficients of the ''tangent bundle spin connection'', $w_{i}{}^\be{}_\al$, are defined. Since the tangent bundle spin connection rotates the orthonormal basis vectors, it is antisymmetric in the last two indices, $w_{i}{}^\be{}_\al=-w_{i\al}{}^\be$. Since the orthonormal basis vectors may be directly related, through the [[frame]], to [[Clifford basis vectors]], $\ve{e_\al} \leftrightarrow \ga_\al$, the tangent bundle spin connection coefficients may be equated with the [[spin connection]] coefficients, $w_i{}^\be{}_\al = \om_i{}^\be{}_\al$. In fact, by working with the [[Clifford vector bundle]] and its connection one may entirely avoid use of a tangent bundle connection.
Nevertheless, we might as well write out the tangent bundle covariant derivative of a tangent vector field using a tangent bundle connection,
\begin{eqnarray}
\na_i \ve{v} &=& \na_i v^j \ve{\pa_j} = \lp \pa_i v^k + \Ga^k{}_{ij} v^j \rp \ve{\pa_k} = \lp D_i v^k \rp \ve{\pa_k} \\
&=& \na_i v^\al \ve{e_\al} = \lp \pa_i v^\be + w_{i}{}^\be{}_\al v^\al \rp \ve{e_\be} = \lp D_i v^\be \rp \ve{e_\be} \\
\f{\na} \ve{v} &=& \f{d} \ve{v} + \f{\Ga}^k{}_j v^j \ve{\pa_k}
= \lp \f{d} v^\be + \f{w}^\be{}_\al v^\al \rp \ve{e_\be}
\end{eqnarray}
Note that, via [[vector-form algebra]], it is reasonable to build things like [[vector valued form]]s, $\f{\na} \ve{v} = \f{dx^i} \na_i \ve{v}$, using the covariant derivative.
Acting on the co[[frame]], $\f{e}{}^\al \in T^* M$, [[dual|dual space]] to the tangent bundle frame, the [[cotangent bundle covariant derivative|cotangent bundle connection]] is
$$
\na_i \f{e}{}^\al = w_{i\be}{}^\al \f{e}{}^\be
\s
\f{\na} \f{e}{}^\al = \f{w}{}_\be{}^\al \f{e}{}^\be
$$
The values of the spin connection coefficients, and hence tangent bundle connection coefficients, may be found by solving [[Cartan's equation]].
One kind of [[vector bundle gauge transformation]] for a [[tangent bundle]] is induced by the action of a position dependent element of the general linear group on the [[coordinate basis vectors]],
$$
\ve{\pa_i} \mapsto \ve{\pa'_i} = t_i{}^j(x) \, \ve{\pa_j}
$$
with the corresponding action on vector fields,
$$
\ve{v} \mapsto \ve{v'} = v^i \ve{\pa'_i} = v^i t_i{}^j \ve{\pa_j}
$$
Such a gauge transformation results from a [[coordinate change]], $t_i{}^j = \fr{\pa x^j}{\pa x'^i}$.
A second kind of ''tangent bundle gauge transformation'' is an arbitrary, position dependent [[Lorentz rotation]] of the [[orthonormal basis vectors|frame]],
$$
\ve{e_\al} \mapsto \ve{e'_\al} = L^\be{}_\al(x) \, \ve{e_\be}
$$
with the corresponding action on vector fields,
$$
\ve{v} \mapsto \ve{v'} = v^\al \ve{e'_\al} = v^\al L^\be{}_\al \ve{e_\be}
$$
Under a coordinate change (the first kind of tangent bundle gauge transformation) the [[tangent bundle connection]] coefficients change as the [[Christoffel symbols]] and the [[Riemann curvature]] transforms as a [[tensor|coordinate change]]. Under a local Lorentz rotation of the ONBV's (the second kind of tangent bundle gauge transformation) the [[tangent bundle covariant derivative|tangent bundle connection]] changes as
\begin{eqnarray}
\f{\na'} \ve{e'_\al} &=& \lp \f{\na} \ve{e_\al} \rp' \\
\f{\na'} \lp L^\be{}_\al \ve{e_\be} \rp &=& \f{w}^\be{}_\al \ve{e'_\be} \\
\lp \f{d} L^\ga{}_\al \rp \ve{e_\ga} + L^\be{}_\al \f{w'}^\ga{}_\be \ve{e_\ga} &=& \f{w}^\be{}_\al L^\ga{}_\be \ve{e_\ga}
\end{eqnarray}
giving the transformation law for the tangent bundle spin connection,
$$
\f{w'}^\ga{}_\de = L_\de{}^\al \f{w}^\be{}_\al L^\ga{}_\be - L_\de{}^\al \f{d} L^\ga{}_\al
$$
For an infinitesimal gauge transformation, $L^\be{}_\al \simeq \de_\al^\be + l^\be{}_\al$, the connection changes to
$$
\f{w'}^\ga{}_\de \simeq \f{w}^\ga{}_\de + \f{w}^\be{}_\de l^\ga{}_\be + l_\de{}^\al \f{w}^\ga{}_\al - \f{d} l^\ga{}_\de
$$
in which $l_\de{}^\al = - l^\al{}_\de$. The Riemann curvature consequently transforms to
$$
\ff{R'}^\ga{}_\de = \f{d} \f{w'}^\ga{}_\de + \f{w'}^\ga{}_\be \f{w'}^\be{}_\de = L_\de{}^\al \ff{R}^\be{}_\al L^\ga{}_\be
$$
The [[tangent bundle parallel transport]] equation may be solved along a path by writing the solution, $\ve{u}(x(t)) = u^\al(t) b_\al = u^\be(0) L^\al{}_\be(t) \ve{e_\al}$, in terms of a path dependent [[Lorentz rotation]],
$$
L^\al{}_\be(t) = Pe^{-\int_0^t \f{\om}^\al{}_\be}
$$
the ''tangent bundle [[path holonomy|vector bundle holonomy]]'', satisfying the ''tangent bundle path holonomy equation'',
$$
0 = \fr{d}{d t} L^\al{}_\be(t) + \ve{v} \f{\om}^\al{}_\ga L^\ga{}_\be(t)
$$
from an initial condition of $L^\al{}_\be(0) = \de^\al_\be$, along a [[path]] with velocity $\ve{v}=\fr{dx^i}{dt} \ve{\pa_i}$. This equation may be readily converted to an integral equation,
$$
L^\al{}_\be(t) - \de_\be^\al = - \int_0^t \f{dt} \fr{dx^i}{dt} \om_i{}^\al{}_\ga L^\ga{}_\be(t)
$$
For small displacements along the path, $x^i = x^i_0 + \va^i(t)$, the solution may be found to any order. To first order,
$$
L^\al{}_\be(t) \simeq \de_\be^\al - \int_0^t \f{dt} \fr{d \va^i}{dt} \om_i{}^\al{}_\ga L^\ga{}_\be(t) \simeq \de_\be^\al - \va^i \om_i{}^\al{}_\be
$$
and to second order,
\begin{eqnarray}
L^\al{}_\be(t) &\simeq& \de_\be^\al - \int_0^t \f{dt} \fr{d \va^i}{dt} \lb \om_i{}^\al{}_\ga + \va^j \pa_j \om_i{}^\al{}_\ga \rb \lb \de_\be^\ga - \va^k \om_k{}^\ga{}_\be \rb \\
&\simeq& \de_\be^\al - \va^i \om_i{}^\al{}_\be + \va^{ij} \lb - \pa_j \om_i{}^\al{}_\be + \om_i{}^\al{}_\ga \om_j{}^\ga{}_\be \rb
\end{eqnarray}
with the [[second order path dependence|path holonomy]] above equal to
$$
\va^{ij} = \lb \int_0^t \f{dt} \fr{d \va^i}{dt} \va^j \rb
$$
The ''tangent bundle [[holonomy]]'' is the tangent bundle path holonomy, $L^\al{}_\be= Pe^{-\oint \f{\om}^\al{}_\be}$, for an arbitrary closed path on the base manifold. A small, square-ish path may be specified by choosing two orthonormal tangent vectors, $\ve{u}$ and $\ve{v}$, at a point $x_{0}$ and making a closed path by going $\va$ in the $\ve{u}$ direction, then $\va$ along $\ve{v}$, $\varepsilon$ along $-\ve{u}$, then $\va$ along $-\ve{v}$ back to $x_{0}$. This path gives an anti-symmetric second order path dependence,
$$
\va^{ij} =\va^{2} \lp v^{i}u^{j}-v^{j}u^{i} \rp
$$
implying a [[loop|vector-form algebra]] described by a tangent 2-vector, $\vv{L} = \va^{2}\ve{v}\,\ve{u}$. The holonomy around this small loop is approximately the path holonomy to second order,
$$
L^\al{}_\be \simeq \de_\be^\al + \va^{ij} \lb \pa_i \om_j{}^\al{}_\be + \om_i{}^\al{}_\ga \om_j{}^\ga{}_\be \rb
=1 + \ha \va^{ij} R_{ij}{}^\al{}_\be
=1 - \vv{L} \ff{R}^\al{}_\be
$$
with the (defining) appearance of the [[Riemann curvature]],
$$
\ff{R}^\al{}_\be = \f{d} \f{\om}^\al{}_\be + \f{\om}^\al{}_\ga \f{\om}^\ga{}_\be
= \f{dx^i} \f{dx^j} \lp \pa_i \om_j{}^\al{}_\be + \om_i{}^\al{}_\ga \om_j{}^\ga{}_\be \rp
= \ha \f{dx^i} \f{dx^j} R_{ij}{}^\al{}_\be
$$
(The contraction of the loop with the curvature, $\vv{L} \ff{R}^\al{}_\be$, is a nice example of [[vector-form algebra]].) Any vector, $\ve{v}=v^\al \ve{e_\al}$, parallel transported around a small loop, $\vv{L}$, is transformed to
$$
\ve{v} \mapsto \ve{v'} = v^\be L^\al{}_\be{} \ve{e_\al} \simeq \ve{v} - \vv{L} \ff{R}^\al{}_\be v^\be \ve{e_\al}
$$
to first order in loop area, $\va^2$. This provides a nice alternative definition of curvature in terms of parallel transport around small closed paths.
A vector (or vector field) of a [[tangent bundle]] may be [[vector bundle parallel transport]]ed along a path. When an observer travels along a [[path]], $x(t)$, on the base she perceives the coordinate basis vector elements to vary according to the [[tangent bundle connection]],
$$
\ve{v} \f{\na} \ve{\pa_j} = \ve{v} \f{\Ga}^k{}_j \ve{\pa_k} = v^i \Ga^k{}_{ij} \ve{\pa_k}
$$
with $\ve{v}$ the [[tangent vector]] (path velocity) to the parameterized curve. Another tangent vector, $\ve{u}(x(t)) = u^j \ve{\pa_j}$, is parallel transported along the path if it is perceived to be unmoving by an observer traveling along with it,
$$
0 = \ve{v} \f{\na} \ve{u} = v^i \lp \pa_i u^k + u^j \Ga^k{}_{ij} \rp \ve{\pa_k} = \lp \fr{d}{d t} u^k + \Ga^k{}_{ij} v^i u^j \rp \ve{\pa_k}
$$
This ''tangent bundle parallel transport equation'' may also be expressed using the tangent bundle spin connection,
$$
0 = \ve{v} \f{\na} \ve{u} = v^i \lp \pa_i u^\be + u^\al w_i{}^\be{}_\al \rp \ve{e_\be} = \lp \fr{d}{d t} u^\be + w_i{}^\be{}_\al v^i u^\al \rp \ve{e_\be}
$$
Equivalently, a tangent vector field, $\ve{u}(x)$, having values along a path, is parallel transported iff it is [[horizontal|connection]] along the path.
[<img[images/png/tangent vector.png]]The //''velocity''//, or ''tangent vector'', $\ve{v}$, with respect to some parameter, $t$, of a [[path]], $c(t)$, at a point is defined via the ''directional derivative'' of a [[function]], $f(x)$, along the path,
$$
\ld \frac{df(c(t))}{d t} \rl_{t=0}
= \ld \frac{dc^{i}(t)}{d t} \rl_{t=0} \ld \frac{\partial f}{\partial x^{i}} \rl_{c(0)}
= \ld v^{i} \ve{\frac{\partial}{\partial x^i}}[f] \rl_{c(0)}
= \ve{v}[f]
$$
A tangent vector, or simply "//''vector''//", can also be visualized in the pseudo-Euclidean embedding space containing the manifold. For a manifold parameterized by coordinates, the [[coordinate basis vectors]] are
$$
\ve{\pa_i} = \ve{\frac{\partial}{\partial x^i}} = \frac{\partial \ve{p}}{\partial x^i} = \pa_i \ve{p}
$$
with the parameterized manifold points, $\ve{p}(x)$, vectors from some arbitrary origin in the flat embedding space.
[<img[images/png/tangent vector p.png]]A vector at a point, $p$, may be written in terms of coordinate basis vectors,
$$
\ve{v} = \fr{d c^i(t)}{d t} \ve{\pa_i} = v^i \ve{\pa_i}
$$
The real valued quantities, $v^i$, are the velocity components. (Summation over repeated [[indices]] is implied)
[<img[images/png/tangent space.png]]The space of all tangent vectors to a [[manifold]], $M$, at a point, $p$, is a [[vector space]], $T_p M$, the ''tangent space'' to $M$ at $p$.
The directional derivative of a function may also be be written using the [[exterior derivative]] and [[vector-form algebra]] as
$$
\ve{v}[f] = \ve{v} \f{d} f = v^i \pa_i f
$$
A good [[theory of everything]] provides a nice top-down description of how the kinematics and dynamics of the standard model and gravity arise from a beautiful and concise geometric picture. This tiddler describes my best attempt at building such a theory.
Here is a periodic table of the standard model and gravitational fields (//Everything//):$\p{{}_{\big(}}$
@@display:block;text-align:center;[img[images/png/standard model and gravity.png]]@@
My guess is that everything is described by a broken [[e8]] valued [[superconnection]] over our four dimensional base [[manifold]]:
\begin{eqnarray}
\udf{A} &=& \f{H_1} + \f{H_2} + \ud{\Ps}{}_{I} + \ud{\Ps}{}_{II} + \ud{\Ps}{}_{III} \\
&=& \big( {\scriptsize \frac{1}{2}} \f{\om} + {\scriptsize \frac{1}{4}} \f{e} \ph + \f{W} + \f{B_1} \big) + \big( \f{B_2} + \f{G} \big) \\
&& + \big( \ud{\nu^e} + \ud{e} + \ud{u} + \ud{d} \big)
+ \big( \ud{\nu^\mu} + \ud{\mu} + \ud{c} + \ud{s} \big)
+ \big( \ud{\nu^\ta} + \ud{\ta} + \ud{t} + \ud{b} \big)
\end{eqnarray}
I think these fields, plus a few others, can be matched up with this symmetry breaking of [[e8]]:
\begin{eqnarray}
e8 &=& f4 + g2 + 26 \! \times \! 7 \\
&=& so(1,7) + su(3) + (8+8+8)\times(1+1+3+3) + 3\times(3+3) + 2\times1 \\
&=& so(1,3) + su(2) + su(2) + 4 \! \times \! 4 + su(3) + (8+8+8)\times(1+1+3+3) + 3\times(3+3) + 2\times1 \\
&\to& {\scriptsize \frac{1}{2}} \om + W + B_1 + {\scriptsize \frac{1}{4}} e \ph + G + 3 \! \times \! \Ps + w \Ph + B_2
\end{eqnarray}
(I'm working on this now.) For the dynamics, all standard model interactions (and gravity) come from the [[curvature]] of the superconnection:
\begin{eqnarray}
\udff{F} &=& \f{d} \udf{A} + {\scriptsize \frac{1}{2}} \big[ \udf{A}, \udf{A} \big] \\
&=& \big( \f{d} \f{H_1} + \f{H_1} \f{H_1} \big) + \big( \f{d} \f{H_2} + \f{H_2} \f{H_2} \big) \\
&& + \big( \f{d} \ud{\Ps}{}_{I} + \f{H_1} \ud{\Ps}{}_{I} + \ud{\Ps}{}_{I} \f{H_2} \big)
+ \big( \f{d} \ud{\Ps}{}_{II} + \f{H_1} \ud{\Ps}{}_{II} + \ud{\Ps}{}_{II} \f{H_2} \big)
+ \big( \f{d} \ud{\Ps}{}_{III} + \f{H_1} \ud{\Ps}{}_{III} + \ud{\Ps}{}_{III} \f{H_2} \big) \\
\end{eqnarray}
The $H_1$ part of this curvature is:
\begin{eqnarray}
\ff{F_1} &=& \f{d} \f{H_1} + \f{H_1} \f{H_1} \\
&=& \Big( \ha ( \f{d} \f{\om} + \ha \f{\om} \f{\om} ) - \fr{1}{16} \ph^2 \f{e} \f{e} \Big)
+ \Big( \fr{1}{4} \big( \f{d} \f{e} + \ha [ \f{\om}, \f{e} ] \big) \ph - \fr{1}{4} \f{e} \big( \f{d} \ph + [ \f{B} + \f{W}, \ph ] \big) \Big)
+ \Big( \f{d} \f{B} + \f{d} \f{W} + \f{W} \f{W} \Big) \\
&=& \ha \big( \ff{R} - \fr{1}{8} \ph^2 \f{e} \f{e} \big)
+ \fr{1}{4} \big( \ff{T} \ph - \f{e} \f{D} \ph \big)
+ \big( \ff{F_B} + \ff{F_W} \big) \\
\end{eqnarray}
The action for everything is written as a modified BF theory, with the ''super-B'' field defined as $\ff{\od{B}} = \ff{B} + \fff{\od{B}} \,$:
\begin{eqnarray}
S &=& \int \big< \ff{\od{B}} \udff{F} + \nf{\Phi} ( \f{H_1}, \f{H_2}, \ff{B} ) \big> \\
&=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\Ps} + \f{H_1} \ud{\Ps} + \ud{\Ps} \f{H_2} \big)
+ \ff{B} \ff{F} - {\scriptsize \frac{1}{4}} \ff{B^s} \ff{B^s} \ga + \ff{B^{m,h,G}} \ff{*B^{m,h,G}} \big>
\end{eqnarray}
Up to a boundary term, after varying $\ff{B}$, this is
$$
S = \int \big< \fff{\od{B}} \big( \f{d} \ud{\Ps} + \f{H_1} \ud{\Ps} + \ud{\Ps} \f{H_2} \big)
- \nf{e} \fr{1}{16} \ph^2 \big( R - \fr{3}{2} \ph^2 \big) + \fr{1}{64} \big( \ff{T} \ph - \f{e} \f{D} \ph \big) * \big( \ff{T} \ph - \f{e} \f{D} \ph \big) + \fr{1}{4} \ff{F^{W,B,G}} \ff{*F^{W,B,G}} \big>
$$
The cosmological constant is defined in terms of the Higgs vev, $\La = - \fr{3}{4} \ph^2$. The fermionic part of this action, using $\fff{\od{B}} = \nf{e} \od{\Ps} \ve{e}$, is
\begin{eqnarray}
S_f &=& \int \big< \fff{\od{B}} \big( \f{d} \ud{\Ps} + \f{H_1} \ud{\Ps} + \ud{\Ps} \f{H_2} \big) \big> \\
&=& \int \big< \nf{e} \od{\Ps} \ve{e} \big( \f{d} \ud{\Ps} + {\scriptsize \frac{1}{2}} \f{\om} \ud{\Ps} + {\scriptsize \frac{1}{4}} \f{e} \ph \ud{\Ps} + \f{B} \ud{\Ps} + \f{W} \ud{\Ps} + \ud{\Ps} \f{G} \big) \big> \\
&=& \int \nf{d^4 x} |e| \, \big< \od{\Ps} \ga^\mu (e_\mu)^i \big( \pa_i \ud{\Ps} + {\scriptsize \frac{1}{4}} \om_i^{\p{i} \mu \nu} \ga_{\mu \nu} \ud{\Ps} + B_i \ud{\Ps} + W_i \ud{\Ps} - \ud{\Ps} G_i \big) + \od{\Ps} \, \ph \, \ud{\Ps} \big>
\end{eqnarray}
the Dirac action in curved spacetime.
Some open questions:
*What is the action, including the symmetry breaking mechanism and triality?
A ''theory of everything'' (//''TOE''//) is a mathematical structure concisely describing both [[general relativity|spacetime]] and the [[standard model]] of particle physics.
The ''[[unitary]] time conjugate'' of a [[Dirac spinor]] field, $\Ps$, using the Weyl representation of the [[Dirac matrices]], is a [[Lorentz transformation]] involving a [[Cl(1,3)]] [[Clifford reflection]] along the time direction,
$$
T_U : \Ps \, \to \, \Ps(t,x)^{T_U} = i \ga_0 \ga \, \Ps(-t,x)
$$
in which the ''unitary time conjugation operator'', $T_U$, is an element of the [[Lorentz group]] that flips [[chiral]]ity and squares to $T_U^2 = -1$. The [[Dirac equation]] with a time conjugate [[gauge field|principal bundle]],
$$
\f{A} = \f{dt} A^0(t,x) + \f{dx^\ep} A_\ep(t,x) \, \to \, \f{A}^{T_U} = - \f{dt} A^0(-t,x) + \f{dx^\ep} A_\ep(-t,x)
$$
is satisfied by the time conjugate spinor,
$$
\begin{array}{rcl}
\lp i \ga^\mu (\pa_\mu + A_\mu^{T_U} ) - m \rp \Ps^{T_U}
\!\!&\!\!=\!\!&\!\! \lp i \ga^\mu ( \pa_\mu + A_\mu^{T_U} ) - m \rp i \ga_0 \ga \, \Ps(-t,x) \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \ga \lp - i \ga^0 ( \pa_0 - A_0(-t,x) ) + i \ga^\va ( \pa_\va + A_\va(-t,x) ) - m \rp \Ps(-t,x) \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \ga \lp - i \ga^0 ( - \pa'_0 - A_0(t',x) ) + i \ga^\va ( \pa_\va + A_\va(t',x) ) - m \rp \Ps(t',x) \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \ga \lp i \ga^\mu ( \pa'_\mu + A_\mu(t',x) ) - m \rp \Ps(t',x) = 0
\end{array}
$$
Note that if there is a [[chiral]] coupling between gauge field and spinor, such as in the weak interaction, $W_\mu P_L \Ps$, in which $P_{L/R}$ is the [[left/right chirality projector]], this will not be invariant under unitary time conjugation since $\ga_0 P_L \ga_0 = P_R$. Unitary time conjugation of a [[quantum Dirac spinor]] results from the action of a corresponding [[unitary]] operator in the [[infinite-dimensional unitary representation]],
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^{T_U} = \hat{\cal{T}}_{\!\! U} \ud{\hat{\Ps}} \hat{\cal{T}}_{\!\! U}^-
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp ( \ud{\hat{a}}_p^{\wedge/\vee})^{T_U} {u}_p^{\wedge/\vee} e^{-i p_\mu x^\mu} + (\ud{\hat{b}}_p^{\wedge/\vee \, \da})^{T_U} {v}_p^{\wedge/\vee} e^{+i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \ga \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{+i (Et+p \cdot x_s)} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{-i (Et + p \cdot x_s)} \rp \\
\!\!&\!\!=\!\!&\!\! i \ga_0 \ga \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{-p}^{\wedge/\vee} {u}_{-p}^{\wedge/\vee} e^{+i (Et-p \cdot x_s)} + \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} {v}_{-p}^{\wedge/\vee} e^{-i (Et - p \cdot x_s)} \rp \\
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \pm \ud{\hat{a}}_{-p}^{\wedge/\vee} v_p^{\vee/\wedge} e^{+i (Et- p \cdot x_s)} \pm \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} u_p^{\vee/\wedge} e^{-i (Et- p \cdot x_s)} \rp
\end{array}
$$
using some [[Dirac solution identities]], $i \ga_0 \ga \, u_{-p}^{\wedge/\vee} = \pm v_p^{\vee/\wedge}$ and $i \ga_0 \ga \, v_{-p}^{\wedge/\vee} = \pm u_p^{\vee/\wedge}$. The corresponding unitary time conjugation transformation of the creation and annihilation operators is thus
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^{T_U} = \mp \ud{\hat{b}}_{-p}^{\vee/\wedge \, \da}
\s\; \s
\lp \ud{\hat{b}}_p^{\wedge/\vee \, \da} \rp^{T_U} = \mp \ud{\hat{a}}_{-p}^{\vee/\wedge}
$$
The unitary time conjugate of creating a particle with [[momentum]] $p$ is annihilating an antiparticle with momentum $-p$.
Because $T_U$ transforms positive energy states to nonphysical negative energy states, we don't consider unitary time conjugation alone as a symmetry. We can, however, combine it with [[charge conjugation|charge conjugate]], $C$, and [[creation conjugation|creation conjugate]], $K$, to construct the ''time conjugate'' of a [[Dirac spinor]] field,
$$
T : \Ps \, \to \, \Ps(t,x)^T = - \lp \lp \Ps^K \rp^C \rp^{T_U} = - i \ga_0 \ga \, i \ga_2 \Ps(-t,x) = \ga_{13} \Ps(-t,x)
$$
in which the ''time conjugation operator'', $T = -K C T_U$, squares to $T^2 = -1$. Time conjugation of a [[quantum Dirac spinor]] results from the action of a corresponding [[antiunitary]] operator in the [[infinite-dimensional unitary representation]],
$$
\begin{array}{rcl}
\ud{\hat{\Ps}}^T = \hat{\cal{T}} \ud{\hat{\Ps}} \hat{\cal{T}}^-
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp ( \ud{\hat{a}}_p^{\wedge/\vee})^T {u}_p^{\wedge/\vee \, *} e^{+i p_\mu x^\mu} + (\ud{\hat{b}}_p^{\wedge/\vee \, \da})^T {v}_p^{\wedge/\vee \, *} e^{-i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! \ga_1 \ga_3 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_p^{\wedge/\vee} {u}_p^{\wedge/\vee} e^{-i (-Et-p \cdot x_s)} + \ud{\hat{b}}_p^{\wedge/\vee \, \da} {v}_p^{\wedge/\vee} e^{+i (-Et - p \cdot x_s)} \rp \\
\!\!&\!\!=\!\!&\!\! \ga_1 \ga_3 \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \ud{\hat{a}}_{-p}^{\wedge/\vee} {u}_{-p}^{\wedge/\vee} e^{+i p_\mu x^\mu} + \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} {v}_{-p}^{\wedge/\vee} e^{-i p_\mu x^\mu} \rp \\
\!\!&\!\!=\!\!&\!\! \int{\fr{d^3p}{(2\pi)^3 (2E)}} \lp \mp \ud{\hat{a}}_{-p}^{\wedge/\vee} u_{p}^{\vee/\wedge \, *} e^{+i p_\mu x^\mu} \mp \ud{\hat{b}}_{-p}^{\wedge/\vee \, \da} v_{p}^{\vee/\wedge \, *} e^{-i p_\mu x^\mu} \rp \\
\end{array}
$$
using some [[Dirac solution identities]], $\ga_{13} \, u_{-p}^{\wedge/\vee} = \mp \, u_p^{\vee/\wedge \, *}$ and $ \ga_{13} \, v_{-p}^{\wedge/\vee} = \mp \, v_p^{\vee/\wedge \, *}$. The corresponding time conjugation transformation of the creation and annihilation operators is thus
$$
\lp \ud{\hat{a}}_p^{\wedge/\vee} \rp^T = \pm \ud{\hat{a}}_{-p}^{\vee/\wedge}
\s \s
\lp \ud{\hat{b}}_p^{\wedge/\vee \da} \rp^T = \pm \ud{\hat{b}}_{-p}^{\vee/\wedge \, \da}
$$
The time conjugate of a particle with [[momentum]] $p$ and spin $s$ is the same particle with momentum $-p$ and spin $-s$. For massless particles, represented by a [[massless quantum Dirac spinor]], the time conjugate of a left-handed fermion is the left-handed fermion with opposite momentum. Since time conjugation does not conjugate between creation and annihilation, but only reverses the momentum and spin, it is more accurately described as ''motion conjugation''.
The ''antiunitary time conjugation operator'', $T' = - C T_U = \ga_{13} K$, with $\Ps^{T'} = \ga_{13} \Ps(-t,x)^*$, is useful as a generator for the [[CPT group]].
//Speculation://
Since $i^T = - i$, time conjugation of a Dirac spinor in the Weyl representation, which is a [[quaternionic representation space|table of Clifford matrix representations]], may be equated algebraically with a [[quaternion]]ic multiplication, $\ps \to \ps \, j$ or $\ps \to \ps \, k$ or some phased combination, $\ps \to \ps (\al j + \be k)$, where $\ps$ must be a Weyl spinor as a quaternion.
The ''torsion'' is defined to be a Clifford vector valued 2-form (a [[Clifform]]) equal to the [[Clifford vector bundle]] [[covariant derivative|spin connection]] of the [[frame]],
$$
\ff{T} = \f{\na} \f{e} = \f{d} \f{e} + \f{\om} \times \f{e}
$$
In terms of components, this equation is
\begin{eqnarray}
\ha T_{ij}{}^\al &=& \pa_{\lb i \rd} \lp e_{\ld j \rb} \rp^\al + \om_{\lb i \rd}{}^{\al \be} \lp e_{\ld j \rb} \rp_\be \\
\ff{T^\al} &=& \f{d} \f{e^\al} + \f{\om}^\al{}_\be \f{e^\be}
\end{eqnarray}
A [[spin connection]], $\f{\om}$, naturally splits into two parts,
$$
\f{\om} = \f{\nu} + \f{\ka}
$$
a torsionless part, $\f{\nu}$, satisfying [[Cartan's equation]],
$$
0 = \f{d} \f{e} + \f{\nu} \times \f{e}
$$
and the ''contorsion'', $\f{\ka}$, satisfying
$$
\ff{T} = \f{\ka} \times \f{e}
$$
These equations may be solved in closed form for $\f{\nu}$ and $\f{\ka}$. Similar to the solution to Cartan's equation, the solution for the contorsion components is
$$
\ka_i{}^{\be \ga} = -\ha \lp e_i \rp^\de \et_{\de \al} \lp T^{\al \be \ga} - T^{\be \ga \al} + T^{\ga \al \be} \rp
$$
A connection, $\f{\om}$, with $\ff{T}=0$ is said to be ''torsionless''.
The torsion components can also be written economically in terms of the [[Christoffel symbols]],
$$
\Ga^k{}_{ij} = \lp e_\al \rp^k \lp \pa_i \lp e_j \rp^\al + w_{i}{}^\al{}_\be \lp e_j \rp^\be \rp
$$
as
$$
T_{ij}{}^\al = \lp e_k \rp^\al \lp \Ga^k{}_{ij} - \Ga^k{}_{ji} \rp = 2 \lp e_k \rp^\al \Ga^k{}_{\lb ij \rb}
$$
The ''trace'' of an $n \times n$ [[matrix|linear operator]], $A^i{}_j$, is the sum of its diagonal matrix elements,
$$
{\rm Tr}\lp A \rp = A^i{}_j \de_j^i = A^i{}_i
$$
It is equivalent to the [[scalar part]] operator, $\left< A \right> = \fr{1}{n} {\rm Tr}(A)$ of a [[Clifford element]] when using a matrix representation.
For any two matrices, ${\rm Tr}(AB) = {\rm Tr}(BA)$, even when they are not square matrices.
A [[group]], $G$, acts ''transitively'' on a set, $S$, via the left action iff for every $x$ and $y$ in $S$ there is some $g \in G$ such that $g x = y$.
The ''transpose'', $A^T$, of a general [[linear operator]], $A$, acting on a [[vector space]] is defined by conjugation within [[metric]]s,
$$
( v, A^T u ) = ( u , A \, v )
$$
Using basis vectors,
$$
A^T{}_{ab} = g_{ad} A^T{}^d{}_b = ( e_a, A^T e'_b) = ( e'_b , A \, e_a ) = g'_{bc} A^c{}_a = A_{ba}
$$
giving the usual matrix transpose for the components. This works even for non-square matrices. For a composition of linear operators, $(A B)^T = B^T A^T$, and as a conjugation, $(A^T)^T=A$. For a tensor, raising and lowering indices with a metric, we have
$$
A^T{}^i{}_j = A{}_j{}^i
$$
Note that one needs to be careful in calculations with Mathematica, because its Transpose just swaps $i \leftrightarrow j$, which works just fine when it's a tensor with indices on the same level, $\text{Transpose}(A_{ij}) = A_{ji} = A^T_{ij}$, but is dangerous for indices on different levels,
$$
\text{T}(A^i{}_j) = \text{Transpose}(A^i{}_j) = A^j{}_i = g^{j m} g_{i n} A_m{}^n = g^{j m} g_{i n} A^T{}^n{}_m \ne A^T{}^i{}_j = A{}_j{}^i
$$
Although it is consistent if we consider it as a map to tensor components in the [[dual space]], such as $\text{T}(v^i) = v_i$, or, as matrices, $\text{T}(v) = v^T \in V^*$. Using this dual space notion for the transpose, we have
$$
v^T A^T u = ( v, A^T u ) = ( u , A \, v ) = u^T A \, v
$$
Each $n$ dimensional [[division algebra]] (and its split-version) has a ''cubic form'' -- a real, cyclic, trilinear ''triality function'', $T(v, \ps, \ch)$, of three elements, or, equivalently, of vectors and chiral spinors,
$$
\ch^b v^c \ps^a \Ga_{cba} = \ch^T v \, \ps = T(v, \ps, \ch) = \lp \os{\ch}, v \, \ps \rp = \ha \lp \ch (v \ps) + \widetilde{\ch (v \ps)} \rp = \ch^b v^c \ps^a M_{ca\os{b}}
$$
This triality function is invariant under the [[triality group]], ${\rm Tri}(\mathbb{D})$ -- a subgroup of $Pin(n) \times Pin(n) \times Pin(n)$, with elements $r = \{r_v, r_-, r_+\}$ satisfying
$$
T(r_v v, r_- \ps, r_+ \ch) = T(v, \ps, \ch)
$$
[[Reflections|Clifford reflection]], $R^u_v$, through a unit length vector, $u$, acting on vectors, $R^u_{vv}$, and on spinors, $R^u_{v-}$ and $R^u_{v+}$, can be expressed using Clifford algebra or division algebra,
$$
\ba{rclcrclcrclcrcl}
v' \!\!&\!\!=\!\!&\!\! R^u_v v = - u v u & \s & v' \!\!&\!\!=\!\!&\!\! R^u_{vv} v = - u \os{v} u & \s & \ch' \!\!&\!\!=\!\!&\!\! R^u_{- v} v = \os{v} \os{u} & \s & \ps' \!\!&\!\!=\!\!&\!\! R^u_{+ v} v = \os{u} \os{v} \\
\Ps' \!\!&\!\!=\!\!&\!\! R^u_v \Ps = u \Ps & & \ch' \!\!&\!\!=\!\!&\!\! R^u_{v-} \ps = \os{\ps} \os{u} & \s & \ps' \!\!&\!\!=\!\!&\!\! R^u_{- -} \ps = - u \os{\ps} u & \s & v' \!\!&\!\!=\!\!&\!\! R^u_{+ -} \ps = \os{\ps} \os{u} \\
\!&\! \!&\! & & \ps' \!\!&\!\!=\!\!&\!\! R^u_{v+} \ch = \os{u} \os{\ch} & \s & v' \!\!&\!\!=\!\!&\!\! R^u_{- +} \ch = \os{u} \os{\ch} & \s & \ch' \!\!&\!\!=\!\!&\!\! R^u_{+ +} \ch = - u \os{\ch} u
\ea
$$
and, since triality is cyclic, we can also have ''extended reflections'', $R^u_{-}$ or $R^u_{+}$, with $u$ a unit chiral spinor, acting on vectors and spinors. These extended reflections leave triality anti-invariant, such as
$$
T' = T(R^u_{vv} v, R^u_{v-} \ps, R^u_{v+} \ch) = -T(v, \ps, \ch)
$$
and, allowing $u$ to be non-unit length, give ''duality functions'' (also called ''incidence relations'', relating to the [[twistor]] program), such as $v' = R^\ch_{+-} \ps = \widetilde{\ch \ps}$. Combining two reflections of the same type gives a rotation, an element of the triality group. Combining two extended rotations of different types gives an element of the triality group that isn't a rotation. Combining four extended reflections through elements equal to $1$ can give a ''triality automorphism'', $t = R^1_+ R^1_- R^1_v R^1_+$, which is an element of the triality group that takes vectors to negative spinors, negative spinors to positive spinors, and positive spinors to vectors,
$$
\ps' = t \, v \s \s \ch' = t \, \ps \s \s v' = t \, \ch
$$
consistent with the invariance of triality under cyclic permutation of its arguments.
From [[division algebra confusion]], a triplet of division algebra elements can be related to a vector, $v \in n_v$, a negative chiral spinor, $\ps \in n_-$, and a positive chiral spinor, $\ch \in n_+$. A triality automorphism then maps between these. When these elements are part of the [[triality decomposition]] of a [[Lie algebra]] in the [[exceptional magic square]], they are similarly mapped by a triality automorphism of that [[Lie algebra|Lie algebra automorphism]], such as the [[triality automorphism of f4]], in which the [[triality subalgebra|triality group]], $\mathfrak{h} = {\rm tri}(\mathbb{D})$, undergoes a triality automorphism (inner or outer) that can be calculated using the invariance of the Lie brackets, such as
$$
\mathfrak{h}' = [v',v'] = [\ps,\ps] = T(\mathfrak{h})
$$
which, for example, gives an outer automorphism of [[spin(8)]].
A triality automorphism, $T$, can also be described by a [[triality matrix]], mapping between the [[roots|Lie algebra structure]] or weights corresponding to vectors and spinors, as well as between the other Lie algebra roots and between Lie algebra generators, including between the generators of its [[Cartan subalgebra|Lie algebra structure]].
Compositions of a triality automorphism produce the cyclic group of order three, $\mathbb{Z}_3 = \{ 1, T, T^2 \}$, the finite symmetry group of a triangle.
The [[triality]] automorphism, and its constituent duality automorphisms, produce inner [[Lie algebra automorphism]]s of [[f4]] via [[division algebra confusion]]. As a matrix operation on the generators, $T'_A = T_B \ph^B{}_A $, the first duality, $D_1$, is
$$
\lb
\begin{array}{cccc}
\ga'_{\al \be} &
\ga'_{0 \al} &
Q'^-_a &
Q'^+_b
\end{array}
\rb
=
\lb
\begin{array}{cccc}
\ga_{\ga \de} &
\ga_{0 \be} &
Q^-_a &
Q^+_b
\end{array}
\rb
\lb
\begin{array}{cccc}
{\underset{bb}{D}}^{\ga \de}{}_{\al \be} & & \\
& - {\underset{vv}{D}}{\,}^{\be}{}_{\al} & & \\
& & & {\underset{-+}{D}}^a{}_b \\
& & {\underset{+-}{D}}^b{}_a & \\
\end{array}
\rb
$$
in which ${\underset{vv}{D}}={\underset{-+}{D}}={\underset{+-}{D}}=diag(+1,-1,-1,...)=K$ is the ''octonion conjugation operator'', and the duality automorphism of $spin(8)$ generators can be explicitly calculated from the [[f4]] Lie algebra brackets of the transformed vector generators,
$$
\ga'_{\al \be} = \ga_{\ga \de} \, {\underset{bb}{D}}^{\ga \de}{}_{\al \be} = - \ha \lb \ga'_{0 \al}, \ga'_{0 \be} \rb
= - \ha \lb \ga_{0 \ga}, \ga_{0 \de} \rb {\underset{vv}{D}}{\,}^{\ga}{}_{\al} {\underset{vv}{D}}{\,}^{\de}{}_{\be}
= \ga_{\ga \de} K\,{}^{\ga}_{\al} K\,{}^{\de}_{\be}
$$
Combining a bunch of these, the triality automorphism of f4 corresponding to $T=D_3 D_1 D_2 D_3$ is
$$
\lb
\begin{array}{cccc}
\ga'_{\al \be} &
\ga'_{0 \al} &
Q'^-_a &
Q'^+_b
\end{array}
\rb
=
\lb
\begin{array}{cccc}
\ga_{\ga \de} &
\ga_{0 \be} &
Q^-_a &
Q^+_b
\end{array}
\rb
\lb
\begin{array}{cccc}
{\underset{bb}{T}}^{\ga \de}{}_{\al \be} & & \\
& & & \de^{\be}{}_{b} \\
& \de^a{}_\al & & \\
& & \de^b{}_a & \\
\end{array}
\rb
$$
in which, from f4, we calculate the [[spin(8)]] triality, $ {\underset{bb}{T}}^{\ga \de}{}_{\al \be} = \ha ( \Ga{}^\ga \overline{\Ga}{}^\de )_{\al \be}$, via
$$
\ga'_{\al \be} = \ga_{\ga \de} \, {\underset{bb}{T}}^{\ga \de}{}_{\al \be} = - \ha \lb \ga'_{0 \al}, \ga'_{0 \be} \rb
= - \ha \lb Q^+_\al, Q^+_\be \rb
= \ga_{\ga \de} \ha ( \Ga^\ga \overline{\Ga}{}^\de )_{\al \be}
$$
These automorphism matrices can be applied to the [[matrix of f4 structure constants]] to confirm they give a Lie algebra automorphism. This automorphism corresponds to an automorphic action of a [[triality matrix]] on the [[f4 structure]].
The [[triality]] automorphism, and its constituent duality automorphisms, produce inner [[Lie algebra automorphism]]s of [[f4(4)]] via [[division algebra confusion]]. As a matrix operation on the generators, $T''_A = T_B \ph^B{}_A $, the first duality, $D_1$, is
$$
\lb
\begin{array}{cccc}
\ga''_{\al \be} &
\ga''_{ \al} &
Q''^-_a &
Q''^+_b
\end{array}
\rb
=
\lb
\begin{array}{cccc}
\ga'_{\ga \de} &
\ga'_{ \be} &
Q'^-_a &
Q'^+_b
\end{array}
\rb
\lb
\begin{array}{cccc}
{\underset{bb}{D}}^{\ga \de}{}_{\al \be} & & \\
& - {\underset{vv}{D}}{\,}^{\be}{}_{\al} & & \\
& & & {\underset{-+}{D}}^a{}_b \\
& & {\underset{+-}{D}}^b{}_a & \\
\end{array}
\rb
$$
in which ${\underset{vv}{D}}={\underset{-+}{D}}={\underset{+-}{D}}=diag(+1,-1,-1,...)=K$ is the [[split-octonion]] conjugation operator, and the duality automorphism of $spin(4,4)$ generators can be explicitly calculated from the [[f4(4)]] Lie algebra brackets of the transformed vector generators,
$$
\ga''_{\al \be} = \ga'_{\ga \de} \, {\underset{bb}{D}}^{\ga \de}{}_{\al \be} = - \ha \lb \ga''_{ \al}, \ga''_{ \be} \rb
= - \ha \lb \ga'_{ \ga}, \ga'_{ \de} \rb {\underset{vv}{D}}{\,}^{\ga}{}_{\al} {\underset{vv}{D}}{\,}^{\de}{}_{\be}
= \ga'_{\ga \de} K\,{}^{\ga}_{\al} K\,{}^{\de}_{\be}
$$
Combining a bunch of these, the triality automorphism of f4 corresponding to $T=D_3 D_2 D_1 D_3$ is
$$
\lb
\begin{array}{cccc}
\ga'''_{\al \be} &
\ga'''_{ \al} &
Q'''^-_a &
Q'''^+_b
\end{array}
\rb
=
\lb
\begin{array}{cccc}
\ga'_{\ga \de} &
\ga'_{ \be} &
Q'^-_a &
Q'^+_b
\end{array}
\rb
\lb
\begin{array}{cccc}
{\underset{bb}{T}}^{\ga \de}{}_{\al \be} & & \\
& & \de^{\be}{}_{a} & \\
& & & \de^{a}{}_{b} \\
& \de^b{}_\al & & \\
\end{array}
\rb
$$
in which, from $f_{4(4)}$, we calculate $ {\underset{bb}{T}}^{\ga \de}{}_{\al \be} = \ha ( \overline{\Ga}{}'^\ga \Ga'^\de )_{\al \be}$ via
$$
\ga'''_{\al \be} = \ga'_{\ga \de} \, {\underset{bb}{T}}^{\ga \de}{}_{\al \be} = - \ha \lb \ga'''_{ \al}, \ga'''_{ \be} \rb
= - \ha \lb Q'^+_\al, Q'^+_\be \rb
= \ga'_{\ga \de} \ha ( \Ga'^\ga \overline{\Ga}{}'^\de )_{\al \be}
$$
These automorphism matrices can be applied to the $f_{4(4)}$ generators to confirm they give a [[Lie algebra automorphism]]. The dualities and triality above work only for the [[split-octonionic representation of Cl(4,4)]] and [[f4(4)]], but directly relate to dualities and triality in other representations via a [[similarity transformation for Cl(4,4)]]. If the $Cl(4,4)$ similarity transformation, $U$, is extended to act on $f_{4(4)}$ generators, $T_a \to T_b \, U_f{}^b{}_a$, with block diagonal $ U_f = \lb 1, 1, U_-, U_+ \rb$, then the triality automorphism in the non-split-octonion representation will be by $U_f^T \,T \, U_f$.
From the decomposition of $f_{4(4)}$ with respect to [[sp(3)]], this triality automorphism relates to the [[triality automorphism of sp(3)]].
The [[sl(3)]] Lie algebra has an inner [[triality]] [[automorphism|Lie algebra automorphism]], $T : A \to g_T \, A \, g_T^-$, with
$$
g_T = \lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb \; \in \; SL(3)
$$
a reduced [[triality matrix]] in the $3 \times 3$ representation of the Lie group, that transforms the root vectors and Cartan subalgebra elements as
$$
T \;\;\; : \;\;\; E^\pm_v \to E^\pm_m \to E^\pm_p \to E^\pm_v \;\;\; \;\;\; H_v \to H_m \to H_p \to H_v
$$
This corresponds to a rotation on root space coordinates, and on the Cartan subalgebra basis elements, by
$$
\lb \ba{cc} -\ha & -\fr{\sqrt{3}}{2} \\ \fr{\sqrt{3}}{2} & -\ha \ea \rb
$$
and the corresponding triality automorphism matrix, $T : T'_A = \ph_A{}^B T_B = g_T \, T_A \, g_T^-$, with
$$
\ph_A{}^B =
\lb \ba{cccccccc}
& & & 1 & & & & \\
& & & & -1 & & & \\
& & -\ha & & & & & -\fr{\sqrt{3}}{2} \\
& & & & & 1 & & \\
& & & & & & -1 & \\
1 & & & & & & & \\
& 1 & & & & & & \\
& & \fr{\sqrt{3}}{2} & & & & & -\ha \\
\ea \rb
$$
This matches the [[triality automorphism of su(3)]].
From [[quaternion]]ic [[triality]] we construct duality [[Lie algebra automorphism]]s of [[sp(3)]] generators,
$$
\begin{array}{rclrclrcl}
&\!\! D_1 \!\!& & &\!\! D_2 \!\!& & &\!\! D_3 \!\!& \\
T^v_a{}' \!\!\!&\!\! = \!\!&\!\! -T^v_\os{a} \;\;&\;\; T^v_a{}' \!\!\!&\!\! = \!\!&\!\! T^m_\os{a} \;\;&\;\; T^v_a{}' \!\!\!&\!\! = \!\!&\!\! T^p_\os{a} \\
T^m_a{}' \!\!\!&\!\! = \!\!&\!\! T^p_\os{a} \;\;&\;\; T^m_a{}' \!\!\!&\!\! = \!\!&\!\! T^v_\os{a} \;\;&\;\; T^m_a{}' \!\!\!&\! = \!\!&\!\! -T^m_\os{a} \\
T^p_a{}' \!\!\!&\!\! = \!\!&\!\! T^m_\os{a} \;\;&\;\; T^p_a{}' \!\!\!&\!\! = \!\!&\!\! -T^p_\os{a} \;\;&\;\; T^p_a{}' \!\!\!&\!\! = \!\!&\!\! T^v_\os{a} \\
T^V_A{}' \!\!\!&\!\! = \!\!&\!\! T^V_A \;\;&\;\; T^V_A{}' \!\!\!&\!\! = \!\!&\!\! T^M_A \;\;&\;\; T^V_A{}' \!\!\!&\!\! = \!\!&\!\! T^P_A \\
T^M_A{}' \!\!\!&\!\! = \!\!&\!\! T^P_A \;\;&\;\; T^M_A{}' \!\!\!&\!\! = \!\!&\!\! T^V_A \;\;&\;\; T^M_A{}' \!\!\!&\!\! = \!\!&\!\! T^M_A \\
T^P_A{}'\!\!\!&\!\! = \!\!&\!\! T^M_A. \;\;&\;\; T^P_A{}' \!\!\!&\!\! = \!\!&\!\! T^P_A \;\;&\;\; T^P_A{}' \!\!\!&\!\! = \!\!&\!\! T^V_A \\
\end{array}
$$
Combining dualities in pairs can give a ''triality automorphism of sp(3)'', such as $T=D_3 D_1 D_2 D_3$:
$$
\begin{array}{rclrcl}
T^v_a{}' \!\!\!&\!\! = \!\!&\!\! T^p_a{} \;\;\;\; & \;\;\;\; T^V_A{}' \!\!\!&\!\! = \!\!&\!\! T^P_A{} \\
T^m_a{}' \!\!\!&\!\! = \!\!&\!\! T^v_a{} \;\;\;\; & \;\;\;\; T^M_A{}' \!\!\!&\!\! = \!\!&\!\! T^V_A{} \\
T^p_a{}' \!\!\!&\!\! = \!\!&\!\! T^m_a{} \;\;\;\; & \;\;\;\; T^P_A{}' \!\!\!&\!\! = \!\!&\!\! T^M_A{}\\
\end{array}
$$
These duality and triality automorphisms hold for both the compact, $sp(3)$, or split real form, [[sp(6,R)]].
This is also describable as an inner triality automorphism, $T : A \to g_T \, A \, g_T^-$, with
$$
g_T = \lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb \; \in \; SP(3)
$$
a reduced [[triality matrix]] in the $3 \times 3$ representation of the Lie group. This corresponds to a triality automorphism matrix, $T : T'_A = \ph_A{}^B T_B = g_T \, T_A \, g_T^-$, matching the description above.
Consider the rank 3 [[symplectic Lie algebra|sp(n)]], [[sp(6,R)]]. The Cartan subalgebra can be constructed from a basis generator from each of $sl(2)^V$, $sl(2)^-$, and $sl(2)^+$, with the $e'_1$ split-quaternion generators being compact Cartan generators and the $e'_2$ and $e'_3$ split-quaternion generators being split Cartan generators. As an important example, we choose a compact Cartan subalgebra, $C_c = C_c^1 T^V_1 + C_c^2 T^M_1 + C_c^3 T^P_1$, and a mixed Cartan subalgebra, $C_m = C_m^1 T^V_1 + C_m^2 T^M_3 + C_m^3 T^P_3$. The root coordinates and (non-normalized) root vectors are then:
$$
\begin{array}{|c|ccc|c|}
\hline
\al & \al_c^V & \al_c^M & \al_c^P & V_\al \\
\hline
T^V_1, T^M_1, T^P_1 & 0 & 0 & 0 & T^V_1, T^M_1,T^P_1 \\
\om^V_+ & +2i & 0 & 0 & +i T^V_2 + T^V_3 \\
\om^V_- & -2i & 0 & 0 & -i T^V_2 + T^V_3 \\
\om^M_+ & 0 & +2i & 0 & +i T^M_2 + T^M_3 \\
\om^M_- & 0 & -2i & 0 & -i T^M_2 + T^M_3 \\
\om^P_+ & 0 & 0 & +2i & +i T^P_2 + T^P_3 \\
\om^P_- & 0 & 0 & -2i & -i T^P_2 + T^P_3 \\
v^v_{+-} & 0 & +i & -i & +i T^v_0 + T^v_1 \\
v^v_{-+} & 0 & -i & +i & -i T^v_0 + T^v_1 \\
v^v_{++} & 0 & +i & +i & +i T^v_2 + T^v_3 \\
v^v_{--} & 0 & -i & -i & -i T^v_2 + T^v_3 \\
v^m_{+-} & +i & 0 & -i & -i T^m_0 + T^m_1 \\
v^m_{-+} & -i & 0 & +i & +i T^m_0 + T^m_1 \\
v^m_{++} & +i & 0 & +i & +i T^m_2 + T^m_3 \\
v^m_{--} & -i & 0 & -i & -i T^m_2 + T^m_3 \\
v^p_{+-} & +i & -i & 0 & +i T^p_0 + T^p_1 \\
v^p_{-+} & -i & +i & 0 & -i T^p_0 + T^p_1 \\
v^p_{++} & +i & +i & 0 & +i T^p_2 + T^p_3 \\
v^p_{--} & -i & -i & 0 & -i T^p_2 + T^p_3 \\
\hline
\end{array}
\s\;\;\;
\begin{array}{|c|ccc|c|}
\hline
\al' & \al_m^V & \al_m^M & \al_m^P & V'_\al \\
\hline
T^V_1, T^M_3, T^P_3 & 0 & 0 & 0 & T^V_1, T^M_3,T^P_3 \\
\om'{}^V_+ & +2i & 0 & 0 & +i T^V_2 + T^V_3 \\
\om'{}^V_- & -2i & 0 & 0 & -i T^V_2 + T^V_3 \\
\om'{}^M_+ & 0 & +2 & 0 & + T^M_1 + T^M_2 \\
\om'{}^M_- & 0 & -2 & 0 & - T^M_1 + T^M_2 \\
\om'{}^P_+ & 0 & 0 & +2 & + T^P_1 + T^P_2 \\
\om'{}^P_- & 0 & 0 & -2 & - T^P_1 + T^P_2 \\
v'{}^v_{+-} & 0 & +1 & -1 & + T^v_0 + T^v_3 \\
v'{}^v_{-+} & 0 & -1 & +1 & - T^v_0 + T^v_3 \\
v'{}^v_{++} & 0 & +1 & +1 & + T^v_1 + T^v_2 \\
v'{}^v_{--} & 0 & -1 & -1 & - T^v_1 + T^v_2 \\
v'{}^m_{+-} & +i & 0 & -1 & - T^m_0 \! - \! i T^m_1 \! + \! i T^m_2 \! + \! T^m_3 \\
v'{}^m_{-+} & -i & 0 & +1 & - T^m_0 \! + \! i T^m_1 \! - \! i T^m_2 \! + \! T^m_3 \\
v'{}^m_{++} & +i & 0 & +1 & + T^m_0 \! + \! i T^m_1 \! + \! i T^m_2 \! + \! T^m_3 \\
v'{}^m_{--} & -i & 0 & -1 & + T^m_0 \! - \! i T^m_1 \! - \! i T^m_2 \! + \! T^m_3 \\
v'{}^p_{+-} & +i & -1 & 0 & + T^p_0 \! - \! i T^p_1 \! + \! i T^p_2 \! + \! T^p_3 \\
v'{}^p_{-+} & -i & +1 & 0 & + T^p_0 \! + \! i T^p_1 \! - \! i T^p_2 \! + \! T^p_3 \\
v'{}^p_{++} & +i & +1 & 0 & - T^p_0 \! + \! i T^p_1 \! + \! i T^p_2 \! + \! T^p_3 \\
v'{}^p_{--} & -i & -1 & 0 & - T^p_0 \! - \! i T^p_1 \! - \! i T^p_2 \! + \! T^p_3 \\
\hline
\end{array}
$$
Either of these [[different Cartans]] can be used to form a different [[Cartan-Weyl basis|Lie algebra structure]] for sp(6,R). We can also transform from the mixed-Cartan basis to the compact-Cartan basis, (//need to fix some of these +- labels to PM//)
$$
\begin{array}{ccc}
\lb \begin{array}{c}
T^V_1 \\ V^V_+ \\ V^V_- \
\end{array} \rb
=
\lb \begin{array}{ccc}
1 & & \\
& 1 & \\
& & 1
\end{array} \rb
\lb \begin{array}{c}
T^V_1 \\ V'{}^V_+ \\ V'{}^V_- \\
\end{array} \rb
& &
\lb \begin{array}{c}
T^\pm_1 \\ V^\pm_+ \\ V^\pm_- \\
\end{array} \rb
= \ha
\lb \begin{array}{ccc}
\sqrt{2} & -\sqrt{2} & 0 \\
i & i & \sqrt{2} \\
-i & -i & \sqrt{2}
\end{array} \rb
\lb \begin{array}{c}
V'{}^\pm_+ \\ V'{}^\pm_- \\ T{}^\pm_3
\end{array} \rb
\\
& & \\
\lb \begin{array}{c}
V^v_{+-} \\ V^v_{-+} \\ V^v_{++} \\ V^v_{--}
\end{array} \rb
= \ha
\lb \begin{array}{cccc}
i & -i & 1 & -1 \\
-i & i & 1 & -1 \\
1 & 1 & i & i \\
1 & 1 & -i & -i
\end{array} \rb
\lb \begin{array}{c}
V'{}^v_{+-} \\ V'{}^v_{-+} \\ V'{}^v_{++} \\ V'{}^v_{--}
\end{array} \rb
& \s &
\lb \begin{array}{c}
V^{m/p}_{+-} \\ V^{m/p}_{-+} \\ V^{m/p}_{++} \\ V^{m/p}_{--}
\end{array} \rb
= \fr{1}{\sqrt{2}}
\lb \begin{array}{cccc}
i & 0 & -i & 0 \\
0 & -i & 0 & i \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1
\end{array} \rb
\lb \begin{array}{c}
V'{}^{m/p}_{+-} \\ V'{}^{m/p}_{+-} \\ V'{}^{m/p}_{++} \\ V'{}^{m/p}_{--}
\end{array} \rb
\end{array}
$$
Complex structure at all relevant?
The ''triality automorphism of sp(6,R)'', $T : T_i \mapsto T''_i$, matches the [[triality automorphism of sp(3)]],
$$
\begin{array}{rclrcl}
T^v_a{}'' \!\!\!&\!\! = \!\!&\!\! T^p_a{} \;\;\;\; & \;\;\;\; T^V_A{}'' \!\!\!&\!\! = \!\!&\!\! T^P_A{} \\
T^m_a{}'' \!\!\!&\!\! = \!\!&\!\! T^v_a{} \;\;\;\; & \;\;\;\; T^M_A{}'' \!\!\!&\!\! = \!\!&\!\! T^V_A{} \\
T^p_a{}'' \!\!\!&\!\! = \!\!&\!\! T^m_a{} \;\;\;\; & \;\;\;\; T^P_A{}'' \!\!\!&\!\! = \!\!&\!\! T^M_A{}\\
\end{array}
$$
which maps compact-Cartan roots and root vectors, such as
$$
\begin{array}{c}
\om^-_+ \mapsto \om^P_+ \mapsto \om^V_+ \mapsto \om^M_+ \\
v^m_{+-} \mapsto v^p_{-+} \mapsto v^v_{-+} \mapsto v^m_{+-} \\
v^m_{++} \mapsto v^p_{++} \mapsto v^v_{++} \mapsto v^m_{++} \\
\end{array}
\s
\s
\s
\begin{array}{c}
V^-_+ \mapsto V^P_+ \mapsto V^V_+ \mapsto V^M_+ \\
V^m_{+-} \mapsto V^p_{-+} \mapsto V^v_{-+} \mapsto V^m_{+-} \\
V^m_{++} \mapsto V^p_{++} \mapsto V^v_{++} \mapsto V^m_{++} \\
\end{array}
$$
corresponding to a compact-Cartan root coordinate transformation similar to the transformation of the Cartan subalgebra generators,
$$
\lb
\begin{array}{c}
\al''{}_c^V \\ \al''{}_c^M \\ \al''{}_c^P
\end{array}
\rb
=
\lb
\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}
\rb
\lb
\begin{array}{c}
\al_c^V \\ \al_c^M \\ \al_c^P
\end{array}
\rb
\s
\s
\s
\lb
\begin{array}{c}
T''{}_1^V \\ T''{}_1^M \\ T''{}_1^P
\end{array}
\rb
=
\lb
\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}
\rb
\lb
\begin{array}{c}
T_1^V \\ T_1^M \\ T_1^P
\end{array}
\rb
$$
The triality transformation of mixed-Cartan root vectors is kinda ugly.
The [[su(3)]] Lie algebra has an inner [[triality]] [[automorphism|Lie algebra automorphism]], $T : A \to g_T \, A \, g_T^-$, with
$$
g_T = \lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb \; \in \; SU(3)
$$
a reduced [[triality matrix]] in the $3 \times 3$ representation of the Lie group, that transforms the root vectors and Cartan subalgebra elements as
$$
T \;\;\; : \;\;\; E^\pm_v \to E^\pm_m \to E^\pm_p \to E^\pm_v \;\;\; \;\;\; \mathcal{H}_v \to \mathcal{H}_m \to \mathcal{H}_p \to \mathcal{H}_v
$$
This corresponds to a rotation on root space coordinates, and on the Cartan subalgebra basis elements, by
$$
\lb \ba{cc} -\ha & -\fr{\sqrt{3}}{2} \\ \fr{\sqrt{3}}{2} & -\ha \ea \rb
$$
and the corresponding triality automorphism matrix, $T : T'_A = \ph_A{}^B T_B = g_T \, T_A \, g_T^-$, with
$$
\ph_A{}^B =
\lb \ba{cccccccc}
& & & 1 & & & & \\
& & & & -1 & & & \\
& & -\ha & & & & & -\fr{\sqrt{3}}{2} \\
& & & & & 1 & & \\
& & & & & & -1 & \\
1 & & & & & & & \\
& 1 & & & & & & \\
& & \fr{\sqrt{3}}{2} & & & & & -\ha \\
\ea \rb
$$
Consider a [[reductive]] decomposition of a [[Lie algebra]], $\mathfrak{g} = \mathfrak{h} + \mathfrak{g}/\mathfrak{h}$, which has a [[triality]] [[automorphism]], $T$, that breaks the [[symmetric space]], $\mathfrak{g}/\mathfrak{h}$, into a triplet, which can have different labels,
$$
\mathfrak{g}/\mathfrak{h} = v + m + p = v + \ps + \ch = n_v + n_m + n_p = n_v + n_- + n_+
$$
in which the triplet elements are a [[vector|vector space]], negative [[chiral]] [[spinor]], and positive chiral spinor [[representation space]]s of $\mathfrak{h}$ of equal dimension, corresponding to three elements of a [[division algebra]], related by $T$. The brackets between elements are
$$
\ba{c}
[ \mathfrak{h}, \mathfrak{h} ] = \mathfrak{h} \\
[ \mathfrak{h}, v ] = v \s [ \mathfrak{h}, \ps ] = \ps \s [ \mathfrak{h}, \ch ] = \ch \\
[ v, v ] = \mathfrak{h} \s [ \ps, \ps ] = \mathfrak{h} \s [ \ch, \ch ] = \mathfrak{h} \\
[ v, \ps ] = \ch \s [ \ps, \ch ] = v \s [ \ch, v ] = \ps \\
\ea
$$
corresponding to non-vanishing structure constants,
$$
C_{\mathfrak{h}\mathfrak{h}}{}^\mathfrak{h} \;\; C_{\mathfrak{h}v}{}^v \;\; C_{\mathfrak{h}m}{}^m \;\; C_{\mathfrak{h}p}{}^p \;\; C_{vv}{}^\mathfrak{h} \;\; C_{mm}{}^\mathfrak{h} \;\; C_{pp}{}^\mathfrak{h} \;\; C_{vm}{}^p \;\; C_{mp}{}^v \;\; C_{pv}{}^m
$$
A triality decomposition, which is reductive but not [[symmetric|reductive]], implies a symmetric decomposition, with $\mathfrak{h} + v$ the symmetric subalgebra.
The ''triality subalgebra'', $\mathfrak{h} = {\rm Tri}(\mathbb{D})$, varies for each division algebra. The simple triality decompositions of the Lie algebras related to the non-trivial division algebras, [[su(3)]], [[sl(3)]], [[sp(3)]], [[sp(6,R)]], [[f4]], and [[f4(4)]], are
$$ \ba{rcl}
su(3) \!\!&\!\!=\!\!&\!\! u(1) + u(1) + 2_v + 2_m + 2_p \\
sp(3) \!\!&\!\!=\!\!&\!\! su(2)_v + su(2)_m + su(2)_p + 2^m \otimes 2^p + 2^v \otimes 2^p + 2^v \otimes 2^m \\
sp(6,\mathbb{R}) \!\!&\!\!=\!\!&\!\! sl(2)_v + sl(2)_m + sl(2)_p + 2^m \otimes 2^p + 2^v \otimes 2^p + 2^v \otimes 2^m \\
f_{4(-52)} \!\!&\!\!=\!\!&\!\! spin(8) + 8_v + 8_m + 8_p \\
f_{4(4)} \!\!&\!\!=\!\!&\!\! spin(4,4) + 8_v + 8_m + 8_p \\
\ea
$$
A triality decomposition can also be done using the outer product of two division algebras, as in the [[exceptional magic square]], forming a [[compound triality decomposition]].
The [[triality]] function is invariant under the ''triality group'', ${\rm Tri}(\mathbb{D})$ -- a subgroup of $Pin(n) \times Pin(n) \times Pin(n)$, with elements $r = \{r_v, r_-, r_+\}$ satisfying
$$
T(r_v v, r_- \ps, r_+ \ch) = T(v, \ps, \ch)
$$
The ''triality algebra'', ${\rm tri}(\mathbb{D})$, is the Lie algebra of the triality group, and its elements, $R = \{R_v, R_-, R_+ \}$, satisfy
$$
T(R_v v, \ps, \ch) + T(v, R_- \ps, \ch) + T(v,\ps, R_+ \ch) = 0
$$
For the [[complex number]]s, [[quaternion]]s, and [[octonion]]s the triality algebras are
$$
\ba{rcl}
{\rm tri}(\mathbb{C}) &=& u(1) + u(1) \\
{\rm tri}(\mathbb{H}) &=& su(2) + su(2) + su(2) \\
{\rm tri}(\mathbb{O}) &=& spin(8)
\ea
$$
The best way to understand how these act on the corresponding triplets, $\{v,s-,s+\}$, is via their embedding in the corresponding Lie algebras, [[su(3)]], [[sp(3)]], and [[f4]],
$$
\ba{rcccl}
su(3) &=& {\rm tri}(\mathbb{C}) + \mathbb{C} + \mathbb{C} + \mathbb{C} &=& u(1) + u(1) + (1+\bar{1})_v + (1+\bar{1})_{s-} + (1+\bar{1})_{s+} \\
sp(3) &=& {\rm tri}(\mathbb{H}) + \mathbb{H} + \mathbb{H} + \mathbb{H} &=& su(2) + su(2) + su(2) + (2,2,1)_v + (2,1,2)_{s-} + (1,2,2)_{s+} \\
f_4 &=& {\rm tri}(\mathbb{O}) + \mathbb{O} + \mathbb{O} + \mathbb{O} &=& spin(8) + 8_v + 8_{s-} + 8_{s+}
\ea
$$
A ''triality matrix'', such as
$$
T =
\lb \begin{array}{cccc}
- \ha & \ha & - \ha & \ha \\
- \ha & \ha & \ha & - \ha \\
\ha & \ha & \ha & \ha \\
- \ha & - \ha & \ha & \ha \\
\end{array} \rb
\s \mathrm{or} \s
T =
\lb \begin{array}{cccc}
- \ha & - \ha & - \ha & - \ha \\
\ha & \ha & - \ha & - \ha \\
\ha & - \ha & \ha & - \ha \\
\ha & - \ha & - \ha & \ha \\
\end{array} \rb
$$
satisfying $T^3=1$, is a linear operator mapping between the [[weights]] of 8-dimensional vectors and spinors, $8^v \mapsto 8^- \mapsto 8^+ \mapsto 8^v$, and mixing between the roots of [[spin(8)]], as in [[f4 structure]], or equivalently as a [[triality]] mixing of [[octonion]]s. If we spectrally decompose such a matrix, we see that a triality matrix is a rotation by an angle of $\fr{2\pi}{3}$ in a plane in 4D. This corresponds to a ''reduced triality matrix'', acting on lower dimensional spaces, such as
$$
T = \lb \ba{ccc} & 1 & \\ & & 1 \\ 1 & & \\ \ea \rb
\s \mathrm{or} \s
T = \lb \ba{cc} -\ha & -\fr{\sqrt{3}}{2} \\ \fr{\sqrt{3}}{2} & -\ha \ea \rb
$$
[[triality]] is an outer [[automorphism]] of Spin(8) and an inner automorphism of [[F4]] and [[E8]].
Ref:
*Jorg Schray and Corinne Manogue
**[[Octonionic representations of Clifford algebras and triality|https://arxiv.org/abs/hep-th/9407179]]
***Gives Clifford rep from octonionic multiplication table, and octonion (but not split-octonion) triality.
*Andrad Rojas Toppan
**[[The Signature Triality of Majorana-Weyl Spacetimes|http://arxiv.org/abs/hep-th/0005035]]
***Gives explicit $\Si_3$ rep, and transformation matrices between $V$, $S^+$, and $S^-$.
*Louis Crane
**[[An Octonionic Geometric (Balanced) state Sum Model|http://arxiv.org/abs/gr-qc/9806060]]
***$so(8)$ is $7$ copies of its Cartan subalgebra -- triality rotates all of them the same.
***Talks about building spin networks using these groups.
*Stephen Adler
**[[Frustrated SU(4) as the Preonic Precursor of the Standard Model|http://arxiv.org/abs/hep-th/9610190]]
***How coupling constants run for this kind of group.
*Z.K. Silagadze
**[[SO(8) Colour as possible origin of generations|http://arxiv.org/abs/hep-ph/9411381]]
***Talks loosely about a lot of these possibilities.
*Lounesto
**[[Octonions and Triality|papers/loun112.pdf]]
*Frederick Witt
**[[Special metrics and Triality|papers/0602414.pdf]]
**[[Special metric structures and closed forms|papers/0502443.pdf]]
Just as a [[Lie algebra involution]] splits a [[Lie algebra]] into two separate spaces, a [[triality]] automorphism, $T : \mathfrak{g} \mapsto \mathfrak{g}$, splits a complex Lie algebra into three separate [[eigen]]spaces satisfying
$$
T \, \mathfrak{h}_+ = e^{+ \fr{2 \pi i}{3}} \mathfrak{h}_+ \s T \, \mathfrak{h}_0 = \mathfrak{h}_0 \s T \, \mathfrak{h}_- = e^{- \fr{2 \pi i}{3}} \mathfrak{h}_-
$$
in which $\mathfrak{h}_0$ is the invariant subalgebra and $\mathfrak{h}_\pm$ are [[complex conjugate|complex structure]] [[representation space]]s. Working this through the Lie bracket, we see that elements of these subspace triplets must satisfy
\begin{eqnarray}
\left[ \mathfrak{h}_0 , \mathfrak{h}_+ \right] &=& \mathfrak{h}_+ \\
\left[ \mathfrak{h}_0 , \mathfrak{h}_- \right] &=& \mathfrak{h}_- \\
\left[ \mathfrak{h}_+ , \mathfrak{h}_- \right] &=& \mathfrak{h}_0 \\
\left[ \mathfrak{h}_+ , \mathfrak{h}_+ \right] &=& \mathfrak{h}_- \\
\left[ \mathfrak{h}_- , \mathfrak{h}_- \right] &=& \mathfrak{h}_+ \\
\end{eqnarray}
A real Lie algebra can have a triality automorphism, $T : \mathfrak{g}_{\mathbb R} \mapsto \mathfrak{g}_{\mathbb R}$, iff the [[complex structure]], $J : \mathfrak{g}_{\mathbb C} \mapsto \mathfrak{g}_{\mathbb C}$, of its corresponding complex Lie algebra is compatible with its triality automorphism, $[J,T]=0$. Any [[eigen]]vector, labeled by $a$, in the positive triality eigenspace, $T_{a+} = T_a + i \, T'_a \in \mathfrak{h}_+$, can be written in terms of a vector, $T_a$ in the real Lie algebra plus $i$ (or $J$) times its ''T-complement'', $T'_a$. By writing $T_a$ and $T'_a$ in terms of $T_{a+}$ and $T_{a-} = T_a - i \, T'_a \in \mathfrak{h}_-$ as
\begin{eqnarray}
T_a &=& \ha (T_{a+} + T_{a-}) \\
T'_a &=& \fr{1}{2i} (T_{a+} - T_{a-}) \\
\end{eqnarray}
we can compute the triality transformation of any $T_a$ and its complement $T'_a$ within real $\mathfrak{g}_{\mathbb R}$ as a $\fr{2 \pi}{3}$ rotation in the J-complex plane,
\begin{eqnarray}
T \, T_a &=& - \ha T_a - \fr{\sqrt{3}}{2} T'_a \\
T \, T'_a &=& \fr{\sqrt{3}}{2} T_a - \ha T'_a \\
\end{eqnarray}
Note that ''T-invariant'' real generators are in a real [[subalgebra]], $\mathfrak{h}_{{\mathbb R}0}$, of $\mathfrak{g}_{{\mathbb R}}$, and the ''T-active'' generators (the $T_a$ and $T'_a$) are in its complement. Within this real subspace, $\mathfrak{h}_{{\mathbb R}+} \cup \mathfrak{h}_{{\mathbb R}-}$, we can re-identify the action of a ''T-compatible'' complex structure, $J_T : T_a \to T'_a, J_T : T'_a \to - T_a$, compatible with our triality, $[T,J_T]=0$, that splits the real space. If $T$ is thought of as a rotation of $\fr{2\pi}{3}$ in the $T_a \times T'_a$ plane, then $J_T$ is a rotation by $\fr{\pi}{2}$. If $T$ transforms only between positive xor between negative root vectors under a choice of Cartan subalgebra, then it is compatible with the corresponding complex or split [[structure|structures on a real representation space]].
The ''tribimaximal mixing matrix'',
$$
\lb
\begin{array}{ccc}
\fr{-1}{\sqrt{3}} & \fr{-1}{\sqrt{3}} & \fr{-1}{\sqrt{3}}\\
\fr{-1}{\sqrt{2}} & \fr{1}{\sqrt{2}} & 0 \\
\fr{-1}{\sqrt{6}} & \fr{-1}{\sqrt{6}} & \fr{\sqrt{2}}{\sqrt{3}}
\end{array}
\rb
$$
may relate to the PMNS matrix describing the neutrino masses.
This same matrix rotates the $SO(6)$ root system to obtain an embedded [[G2]] orthogonal to an axis. After transformation by this matrix, the three generations of fermions are obtained by a triality rotation in the plane of this G2, suggesting an intriguing relationship that warrants investigation.
The most direct way to proceed is to introduce an $SU(3)$ gauge group relating the three families. This has been described in a recent paper by Pierre Ramond.
Ref:
*S. King and C. Luhn
**[[A new family symmetry for SO(10) GUTs|http://arxiv.org/abs/0905.1686]]
***New, and very good description of obtaining PMNS from finite subgroup of su(3)
*C. Luhn and P. Ramond
**[[Anomaly Conditions for Non-Abelian Finite Family Symmetries|http://arxiv.org/abs/0805.1736]]
*http://en.wikipedia.org/wiki/Tribimaximal_mixing
*P. Kovtun, A. Zee
**[[A schematic model of neutrinos|http://www.arxiv.org/abs/hep-ph/0604169]]
*E. Ma
**[[Lepton Family Symmetry and the Neutrino Mixing Matrix|http://arxiv.org/abs/hep-ph/0606024]]
*C. Brannen and H. de Vries
**[[PF thread|http://www.physicsforums.com/showthread.php?p=1501369#post1501369]]
*Y. Koide
**[[Tribimaximal Neutrino Mixing and a Relation Between Neutrino- and Charged Lepton-Mass Spectra|http://arxiv.org/abs/hep-ph/0605074]]
A ''twistor'' is a collection of two ordered [[Cl(1,3)]] [[Weyl spinor]]s,
$$
\Ps_T = \{ \ps_L, \ps_R \} = \{ \mu, \la \} = \{ a \g , b ] \} = \{ \{ Z^1, Z^2 \} , \{ Z^3, Z^4 \} \} \in \mathbb{T} = \mathbb{C}^4
$$
that transform as a [[Dirac spinor]] in the [[Weyl representation|Dirac matrices]], $\Ps_T = \Ps$, under [[spin(1,3)]] but are actually [[chiral]] [[spin(2,4)]] spinors, $\Ps_T=\Ps_+$, and have a different relationship to particles and spacetime. Twistors have a [[Hermitian form]], from the $spin(2,4)$ [[conjugate spinor]], that (by choice of [[Cl(2,4)]] representation) can be the same [[Dirac adjoint]] as for Dirac spinors,
$$
\bar{\Ps}_T \Ps_T = \Ps_T^\da \ga^0 \Ps_T = \ps^\da_R \ps_L + \ps_L^\da \ps_R
$$
that is real (or imaginary if these are anticommuting [[Grassmann number]] spinors, which they usually aren't) and invariant under [[conformal transformations|conformal group]] by $Spin(2,4) = SU(2,2) \subset Sl(4,\mathbb{C})$.
The left and right Weyl spinors of a twistor may be considered in partnership with a [[right chiral vector|Dirac matrices]] signifying a location in [[Minkowksi spacetime|rest frame]],
$$
x_R = x_R^0 \si_0 + x_R^\va \si_\va
$$
written using [[Pauli matrices]], that has a norm of $|x_R|^2 = \text{Det } x_R = x_R^0 x_R^0 - x_R^\va x_R^\va$. These three (location, left spinor, and right spinor) satisfy the ''incidence relation'',
$$
\ps_R = i \, x_R \, \ps_L
$$
If we know any of the two variables in the incidence relation, it uniquely determines the third, unless $x_R$ is null. Through the incidence relation, a point in Minkowski spacetime corresponds to a 2D complex surface in $T$. Alternatively, specifying the left and right spinors in the twistor, the incidence relation for the position is solved by a null line in Minkowski spacetime (If $x_R$ and $y_R$ both solve the incidence relation, they are separated by a null vector). The incidence relation is invariant under [[chiral Clifford rotation]],
$$
\ps_R \to U_R \ps_R \s
x_R \to U_R x_R U_R^\da \s
\ps_L \to U_L \ps_L
$$
and it implies a twistor (specified by the left Weyl spinor and location) transforms under a [[translation|conformal group]] as
$$
\Ps_T = \lb \ba{c} \ps_L \\ i \, x_R \ps_L \ea \rb
\;\;\;\; \to \;\;\;\;
U^P_+ \Ps_T
= \lb \ba{cc} 1 & 0 \\ i \, y_R & 1 \ea \rb \lb \ba{c} \ps_L \\ i \, x_R \ps_L \ea \rb
= \lb \begin{array}{c} \ps_L \\ i \, (x_R + y_R) \ps_L \end{array} \rb
$$
If the location in the incidence relation, $x_R$, is null, we can use [[helicity notation]] and write $x_R = x ] \l x$ for some ''location Weyl spinor'', $x ]$, so that the incidence relation,
$$
b ] = i \, x ] \l x \, a \g
$$
tells us the right-handed Weyl spinor in our twistor is our location Weyl spinor times a complex number. Note that we have not required our position to be real -- and sometimes it will be useful to consider complex position coordinates $x^\mu \in \mathbb{C}$.
//work in progress below here...//
If we use a twistor to define a complex vector,
$$
v_R = i \ps_R \ps_L^\da
$$
then it is necessarily null, $\det{v_R} = 0$, and is a real null vector if $\ps_L = i \ps_R$.
Relationship of $\ps_L$ to a left [[helicity state]] for a massless fermion traveling with momentum $v_R$?
A right chiral vector is related to a [[biquaternion]], or a [[quaternion]] with an imaginary time component,
$$
- i \, v_R = v = v^0 \si_0 - v^\va i \si_\ep = v^a e_a
$$
with $v^0 = -i v_R^0$ and $v^\va = v_R^\va$.
One may speculate that there is a ''twistor triality'', similar to Euclidean twistor triality from [[Euclidean twistor confusion]],
\begin{eqnarray}
T(v_R,\ps_L,\ps_R) &=& \langle \ps_R | -i \, v_R \, \ps_L \rangle = \ps_R^\da (- i \, v_R ) \ps_L + \ps_L^\da ( i \, v_R^\da ) \ps_R \\
&=& \ch^T M_R^\da (- i \, v_R ) M_L \ps + \ps^T M_L^\da ( i \, v_R^\da ) M_R \, \ch \\
&=& \ps^a \ch^b \, 2\, \text{Re} \lp M_R^\da (-i v_R^0 \si_0 - v_R^\va i \, \si_\va) M_L \rp_{ba} \\
&=& v_R^c \ps^a \ch^b \Ga'_{cba} \; \in \; \mathbb{R}
\end{eqnarray}
\begin{eqnarray}
T(v_R,\ps_L,\ps_R) &=& \langle \ps_R | -i \, v_R \, \ps_L \rangle = - i \ps_R^\da v_R \ps_L + i \ps_L^\da v_R \ps_R \\
&=& 2\, \text{Re} \lp \ps_R^\da (-i v_R^0 \si_0 - v_R^\va i \, \si_\va) \ps_L \rp \\
&=& v_R^c \ps^a \ch^b \Ga'_{cba} \; \in \; \mathbb{R}
\end{eqnarray}
Is $\Ga'_{cba}$ cyclic? Only $\Ga'_0$ is different, but it's pretty wacky. How does it relate to quaternion multiplication?
A [[linear operator]], $\hat{U}$, on a [[unitary representation]] space is ''unitary'', iff its [[inverse]] is its [[Hermitian]] conjugate, $\hat{U}^- = \hat{U}^\da = \hat{U}^{*T}$, equal to the complex conjugate of its [[transpose]]. The complex scalar product, using the [[Hermitian form]], of two complex vectors is invariant under unitary transformation, $\langle \hat{U} u | \hat{U} v \rangle = \langle u | v \rangle$. The [[exponentiation]], $\hat{U}=e^\hat{A}$, of an anti-[[Hermitian]] operator is unitary, $\hat{U}^\da = e^{\hat{A}}{}^\da = e^{\hat{A}^\da} = e^{-\hat{A}} = \hat{U}^-$.
An ''anti-unitary'' operator satisfies $\langle \hat{U}' u | \hat{U}' v \rangle = \langle u | v \rangle^*$.
A [[Lie group]] ''unitary representation'', $\pi$, is a [[representation]] of Lie group elements as ''[[unitary]] operator''s,
$$
\Pi : G \mapsto GL(V)
$$
on a [[representation space]], $V$, with $\hat{\pi}(g) = \hat{U}$ and $ \hat{\pi}(g^-) =\hat{U}^-=\hat{U}^\da$. If a vector $v \in V$ is transformed by a Lie group element in a unitary representation to $v' = \hat{U} \, v$, then linear operators, $\hat{A}$, on $V$ must transform as $\hat{A}' = \hat{U} \hat{A} \, \hat{U}{}^-$, such that
$$
\hat{A}' v' = \hat{U} \hat{A} \, \hat{U}^- \, \hat{U} \, v = \hat{U} \hat{A} \, v
$$
The unitary [[Lie algebra]] generators, $\hat{A} \in \Pi(\mathfrak{g})$, satisfy $\hat{A}^\da = \hat{\pi}(\om(A))$, relating the [[Hermitian]] conjugate to an [[antilinear involution]]. A contravariant [[Hermitian form]] satisfies
$$
\langle \hat{A}^\da u | v \rangle = \langle u | \hat{A} v \rangle \;\;\; \forall \; \hat{A}
$$
This inspires bra-[[ket]] notation, with ''ket''s, $\hat{A} \, | v \rangle = | \hat{A} v \rangle$, and ''bra''s, $\langle u | \, \hat{A}^\da = \langle \hat{A} u | = \lp \hat{A} | u \rangle \rp^\da$, as vectors in a [[Hilbert space]]. The Hermitian form between two vectors is invariant under the action of unitary operators,
$$
\langle u' | v' \rangle = \langle \hat{U} u | \hat{U} v \rangle = \langle u | \hat{U}^\da \hat{U} | v \rangle = \langle u | v \rangle
$$
If a Lie group, $G$, is compact, then every finite-dimensional representation is equivalent to a unitary one, with $\hat{A} \in \Pi(\mathfrak{g})$ anti-Hermitian, $\hat{A}^\da = -\hat{A}$. Non-compact Lie groups can have an [[infinite-dimensional unitary representation]].
The five ''dimensional units'' used in physics, and their unit in the ''international system'', are
| $L$ |length || $L_s=m$ | meter |
| $M$ |mass || $M_s=kg$ | kilogram |
| $T$ |time || $T_s=s$ | second |
| $C$ |charge || $C_s=C$ | coulomb |
| $K$ |temperature || $K_s=K$ | degrees Kelvin |
All ''derived units'' can be expressed in terms of these, such as energy, $E=\fr{ML^2}{T^2}$.
Some [[Symbols]] that carry no units are
| $\de_i^j \;\; \ga_{\al \dots \be} \;\; \et_{\al \be} \;\; \ep_{\al \dots \be} \;\; \lp e^s_i\rp^\al$ |
while some, including coordinates and parameters, may or may not have units of powers of length and time,
| $x^i \;\; \ve{\pa_i} \;\; \f{dx^i} \;\; t \;\; \lp e_i\rp^\al \;\; g_{ij}$ |
Units in coordinates and parameters are written for suggestive convenience only -- for mathematical consistency, coordinates and parameters take values in the real numbers, with no units. This is true because physics is invariant under arbitrary diffeomorphisms of these parameters. Nevertheless, in some situations it is calculationaly convenient to put length or time units in coordinates, even though these units are really only carried by physical variables, such as the [[frame]]. Some physical quantities and the units they carry are
| $x^\mu \;\; \f{e} \;\; \f{e^\al} \;\; s \;\; \tau$ | $T$ |
| $\ll e\rl$ | $T^n$ |
Keeping track of units helps as a bookkeeping check in calculations, and improves physical intuition. Nevertheless, it is often convenient to ''nondimensionalize'' physical quantities through multiplication with a set of appropriate physical constants. One complete set of constants and their dimensions is
| $[c]=\fr{L}{T}$ | $[\hbar]=\fr{ML^2}{T}$ | $[G]=\fr{L^3}{MT^2}$ | $[\fr{1}{4\pi \ep_0}]=\fr{M L^3}{C^2 T^2}$ | $[k]=\fr{M L^2}{K T^2}$ |
By expressing the dimensional units in terms of these particular quantities, one obtains ''Planck units'', such as the ''Planck time'', $T_P = \sqrt{\fr{\hbar G}{c^5}} = 5.39\times10^{-44} s$. A temporal quantity, such as a frame component, could then be nondimensionlized with respect to this scale, $[\lp e_i \rp^\al / T_P]=1$.
It is also a common practice to nondimensionalize with one or two constants, and then "set those constants equal to one". In this way, some units are "converted" into others. For ''natural units'', employed by particle physicists, all quantities are nondimensionalized using $c$ and $\hbar$ to convert into energies, $E$, which are then expressed in mega-electron-volts, $1 E_n = 1 MeV$. Particle physicists say "set $c=1$ and $\hbar=1$", then the units for length, mass, and time are all expressed in terms of energy,
| $L_n=\fr{\hbar c}{E}=E^{-1}$ | $M_n=\fr{E}{c^2}=E$ | $T_n=\fr{\hbar}{E}=E^{-1}$ |
For example, a particle physicist would say $1 cm$ of length is $\fr{1 cm}{\hbar c} = 5.1\times10^{14} MeV^{-1}$, or write the units of a momentum in terms of energy as $[l]=\fr{M_n L_n}{T_n}=\fr{EE^{-1}}{E^{-1}}=E$. For ''geometric units'', employed by relativists, all quantities are nondimensionalized using $c$ and $G$ to convert into lengths, $L$, which are expressed in meters or possibly ''Planck length''s, $L_p = \sqrt{\fr{\hbar G}{c^3}} = 1.6\times10^{-35} m$. Settting $c=1$ and $G=1$, the units of length, mass, and time in terms of length are
| $L_g=L$ | $M_g=\fr{c^2L}{G}=L$ | $T_g=\fr{L}{c}=L$ |
For example, a relativist would say $1 J$ of energy is $1 \fr{kg m^2}{s^2 } \fr{G}{c^4} = 8.2\times10^{-45} m$, or write the units of a momentum in terms of length as $[l]=\fr{M_g L_g}{T_g}=\fr{LL}{L}=L$.
This practice of automatically nondimensionalizing physical quantities allows expressions to be simplified, since you don't have to keep writing constants like $c$, but some information is hidden. By using natural or geometric units, one bit of that information is saved, since quantities are nondimensionalized only up to one unit, $E$ or $L$. This is useful for "power counting" in perturbative expansions. But it's probably better for establishing intuition and context to use and write physical constants explicitly in expressions, and do nondimensionalization explicitly when quantities are defined, such as using $c$ in the [[rest frame]] to write all "distances" in time units, $T$.
//Do variations or do via configuration space manifold?//
A ''vector bundle'' is a [[fiber bundle]] with a [[vector space]] as the typical fiber, $F=V$. A local trivialization is provided by specifying the local sections, $b^a_\al(x) = \ph^-_a(x,b_\al)$, corresponding to each basis element. These allow local sections of the vector bundle (''local vector field''s) to be written locally over $M$ with components that are functions of base manifold points, $v(x) = v_a^\al(x \in U_a) b^a_\al$. On patch overlaps the transition functions map the basis elements from one patch, $U_b$, to the other, $U_a$, as
$$
b^a_\al = \lp t_{ab} \rp_\al{}^\be(x) \, b^b_\be
$$
The transition functions, $t_{ab}$, are elements of some subgroup of the general linear group, $GL$, the structure group of the vector bundle. A section, $v(x)$, a ''vector field'' over the whole base manifold, is created using the transition functions to glue local sections together over patch overlaps,
$$
v(x) = v^\al_a(x) b^a_\al = v^\be_b(x) b^b_\be = v^\al_a(x) \lp t_{ab} \rp_\al{}^\be(x) \, b^b_\be
$$
so vector coefficients over different patches are related by $v^\be_b(x) = v^\al_a(x) \lp t_{ab} \rp_\al{}^\be(x)$ (keep in mind that patch labels, $a,b$, are not being summed). By abuse of notation, the sections corresponding to the basis elements (the ''vector basis elements'' or //''fiber basis elements''//) over the various patches can be used to heuristically express a vector field as $v(x) = v^\al(x) b_\al$ over the whole base.
A change of basis, $b_\al \mapsto b'_\al = g_\al{}^\be(x) b_\be$, is the most basic type of [[vector bundle gauge transformation]]. The [[partial derivative]] is zero when acting on basis vectors, $\f{\pa} b_\al = 0$, but this derivative doesn't properly keep track of the local trivialization or gluing between patches. To remedy this, a [[vector bundle covariant derivative|vector bundle connection]] is introduced which, via a [[vector bundle connection]], keeps track of how the basis vectors change over the base when taking the derivative of a vector field -- it co-varies with a gauge transformation. Using the covariant derivative, any vector may be [[parallel transport|vector bundle parallel transport]]ed along any path on the base to obtain a new vector at any point along the path. For a closed path, the parallel transport of a vector is represented by a [[vector bundle holonomy]] -- an element of the general linear group which acts on the initial vector. For a small closed path, or loop, the holonomy is given approximately by the [[vector bundle curvature]] -- an important geometric descriptor of the fiber bundle and connection.
The [[connection]] and [[covariant derivative]] for a [[vector bundle]] describe how the fiber basis elements, $b_\al$, transform as one moves around on the base manifold. From any base point, the infinitesimal transformation of the basis elements in any direction is described by the ''vector bundle covariant derivative'' of the basis elements,
\begin{eqnarray}
\na_i b_\al &=& A_{i\al}{}^\be b_\be \\
\f{\na} b_\al &=& \f{A}{}_\al{}^\be b_\be \\
\end{eqnarray}
in which the ''vector bundle connection'', $\f{A}{}_\al{}^\be = \f{dx^i } A_{i\al}{}^\be$, completely encodes the local geometry of the vector bundle. The covariant derivative of a vector field, $v = v^\al b_\al$, using this connection, is
\begin{eqnarray}
\na_i v &=& \lp \pa_i v^\al \rp b_\al + v^\al \na_i b_\al = \lp \pa_i v^\al \rp b_\al + A_{i\al}{}^\be v^\al b_\be = \lp \pa_i v^\be + A_{i\al}{}^\be v^\al \rp b_\be \\
\f{\na} v &=& \f{\pa} v + \f{A}{}_\al{}^\be v^\al b_\be
\end{eqnarray}
Occasionally the covariant derivative for a vector bundle is written as an operator on tensors with indices representing components multiplying the basis elements:
$$
D_i v^\be = \pa_i v^\be + A_{i\al}{}^\be v^\al
$$
but we will mostly avoid this notation.
The [[curvature]] for a [[vector bundle]] describes the local geometry. Applying the [[vector bundle covariant derivative|vector bundle connection]] twice gives the ''vector bundle curvature'',
$$
\f{\na}\f{\na}v = \f{d} \lp \f{A}{}_\al{}^\be v^\al b_\be \rp + \f{A}{}_\al{}^\be \lp \f{d} v^\al \rp b_\be + \f{A}{}_\al{}^\be \lp \f{A}{}_\ga{}^\al v^\ga \rp b_\be
= \lp \f{d} \f{A}{}_\ga{}^\be + \f{A}{}_\al{}^\be \f{A}{}_\ga{}^\al \rp v^\ga b_\be
= \lp \ff{F}{}_\ga{}^\be \rp v^\ga b_\be
$$
The ''vector bundle curvature'', $\ff{F}{}_\ga{}^\be = \f{d} \f{A}{}_\ga{}^\be - \f{A}{}_\ga{}^\al \f{A}{}_\al{}^\be$, may alternatively be obtained from the [[vector bundle holonomy]].
A [[gauge transformation]] for a [[vector bundle]] is induced by the action of an arbitrary, position dependent element of the structure group, $g \in GL$, on the basis elements,
$$
b_\al \mapsto b'_\al = g_\al{}^\be(x) b_\be
$$
with the corresponding action on vector fields,
$$
v \mapsto v' = v^\al b'_\al = v^\al g_\al{}^\be b_\be
$$
Under a gauge transformation the [[vector bundle covariant derivative|vector bundle connection]] changes to
\begin{eqnarray}
\f{\na'} b'_\al &=& \lp \f{\na} b_\al \rp' \\
\f{\na'} \lp g_\al{}^\be b_\be \rp &=& \f{A}{}_\al{}^\be b'_\be \\
\lp \f{d} g_\al{}^\be \rp b_\be + g_\al{}^\be \f{A'}{}_\be{}^\ga b_\ga &=& \f{A}{}_\al{}^\be g_\be{}^\ga b_\ga
\end{eqnarray}
giving the transformation law for the connection,
$$
\f{A'}{}_\al{}^\be = g^-_\al{}^\de \f{A}{}_\de{}^\ga g_\ga{}^\be - g^-_\al{}^\de \lp \f{d} g_\de{}^\be \rp
$$
For an infinitesimal gauge transformation, $g_\al{}^\be \simeq \de_\al^\be + \va_\al{}^\be$, the connection changes to
$$
\f{A'}{}_\al{}^\be \simeq \f{A}{}_\al{}^\be - \f{d} \va_\al{}^\be - \f{A}{}_\al{}^\ga \va_\ga{}^\be + \va_\al{}^\de \f{A}{}_\de{}^\be
$$
The [[vector bundle curvature]] consequently transforms to
$$
\ff{F'}{}_\al{}^\be = \f{d} \f{A'}{}_\al{}^\be + \f{A'}{}_\ga{}^\be \f{A'}{}_\al{}^\ga = g^-_\al{}^\de \ff{F}{}_\de{}^\ga g_\ga{}^\be
$$
The solution, $u(x(t)) = u^\al(t) b_\al = u^\be(0) U_\be{}^\al(t) b_\al$, to the [[vector bundle parallel transport]] equation may be written via the ''vector [[path holonomy]]'',
$$
U_\be{}^\al(t) = Pe^{-\int_0^t \f{A}{}_\be{}^\al}
$$
satisfying the ''vector bundle path holonomy equation'',
$$
0 = \fr{d}{d t} U_\be{}^\al(t) + \ve{v} \f{A}{}_\ga{}^\al U_\be{}^\ga(t)
$$
from an initial condition of $U_\be{}^\al(0) = \de_\be^\al$, along a [[path]] with velocity $\ve{v}=\fr{dx^i}{dt} \ve{\pa_i}$. This equation may be readily converted to an integral equation,
$$
U_\be{}^\al(t) - \de_\be^\al = - \int_0^t \f{dt} \fr{dx^i}{dt} A_{i\ga}{}^\al U_\be{}^\ga(t)
$$
For small displacements along the path, $x^i = x^i_0 + \va^i(t)$, the solution may be found to any order. To first order,
$$
U_\be{}^\al(t) \simeq \de_\be^\al - \int_0^t \f{dt} \fr{d \va^i}{dt} A_{i\ga}{}^\al U_\be{}^\ga(t) \simeq \de_\be^\al - \va^i A_{i\be}{}^\al
$$
and to second order,
\begin{eqnarray}
U_\be{}^\al(t) &\simeq& \de_\be^\al - \int_0^t \f{dt} \fr{d \va^i}{dt} \lb A_{i\ga}{}^\al + \va^j \pa_j A_{i\ga}{}^\al \rb \lb \de_\be^\ga - \va^k A_{k\be}{}^\ga \rb \\
&\simeq& \de_\be^\al - \va^i A_{i\be}{}^\al + \va^{ij} \lb - \pa_j A_{i\be}{}^\al + A_{i\ga}{}^\al A_{j\be}{}^\ga \rb
\end{eqnarray}
with the [[second order path dependence|path holonomy]] above equal to
$$
\va^{ij} = \lb \int_0^t \f{dt} \fr{d \va^i}{dt} \va^j \rb
$$
The ''vector bundle [[holonomy]]'' is the vector bundle path holonomy, $U_\be{}^\al= Pe^{-\oint \f{A}{}_\be{}^\al}$, for an arbitrary closed path on the base manifold. A small, square-ish path may be specified by choosing two orthonormal tangent vectors, $\ve{u}$ and $\ve{v}$, at a point $x_{0}$ and making a closed path by going $\va$ in the $\ve{u}$ direction, then $\va$ along $\ve{v}$, $\varepsilon$ along $-\ve{u}$, then $\va$ along $-\ve{v}$ back to $x_{0}$. This path gives an anti-symmetric second order path dependence,
$$
\va^{ij} =\va^{2} \lp v^{i}u^{j}-v^{j}u^{i} \rp
$$
implying a [[loop|vector-form algebra]] described by a tangent 2-vector, $\vv{L} = \va^{2}\ve{v}\,\ve{u}$. The holonomy around this small loop is approximately the path holonomy to second order,
$$
U_\be{}^\al \simeq \de_\be^\al + \va^{ij} \lb \pa_{i} A_{j\be}{}^\al + A_{i\ga}{}^\al A_{j\be}{}^\ga \rb
=1 + \ha \va^{ij} F_{ij\be}{}^\al
=1 - \vv{L} \ff{F}{}_\be{}^\al
$$
with the (defining) appearance of the [[vector bundle curvature]],
$$
\ff{F}{}_\be{}^\al = \f{d} \f{A}{}_\be{}^\al + \f{A}{}_\ga{}^\al \f{A}{}_\be{}^\ga = \f{dx^i} \f{dx^j} \lp \pa_i A_{j\be}{}^\al + A_{i\ga}{}^\al A_{j\be}{}^\ga \rp = \ha \f{dx^i} \f{dx^j} F_{ij\be}{}^\al
$$
(The contraction of the loop with the curvature, $\vv{L} \ff{F}{}_\be{}^\al$, is a nice example of [[vector-form algebra]].) Any vector, $v=v^\al b_\al$, parallel transported around a small loop, $\vv{L}$, is transformed to
$$
v \mapsto v' = v^\be U_\be{}^\al b_\al \simeq v - \vv{L} \ff{F}{}_\be{}^\al v^\be b_\al
$$
to first order in loop area, $\va^2$. This provides a nice alternative definition of curvature in terms of parallel transport around small closed paths.
A vector (or vector field) of a [[vector bundle]] may be [[parallel transport]]ed along a path. When an observer travels along a [[path]], $x(t)$, on the base she perceives the basis elements to vary according to the [[vector bundle connection]],
$$
\ve{v} \f{\na} b_\al = \ve{v} \f{A}{}_\al{}^\be b_\be = v^i A_{i\al}^{\p{{i\al}}\be} b_\be
$$
with $\ve{v}=\fr{dx^i}{dt} \ve{\pa_i}$ the [[tangent vector]] (path velocity) to the [[path]]. A vector, $u(x(t)) = u^\al b_\al$, is parallel transported along the path if it is perceived to be unmoving by an observer traveling along with it, satisfying the ''vector bundle parallel transport equation'',
$$
0 = \ve{v} \f{\na} u = v^i \lp \pa_i u^\be + u^\al A_{i\al}^{\p{{i\al}}\be} \rp b_\be = \lp \fr{d}{d t} u^\be + v^i A_{i\al}^{\p{{i\al}}\be} u^\al \rp b_\be
$$
Equivalently, a vector field, $u(x)$, having values along a path, is parallel transported iff it is [[horizontal|connection]] along the path.
A ''vector projection'' is a [[vector valued 1-form|vector valued form]] satisfying
$$
\f{\ve{P}} = \f{\ve{P}} \f{\ve{P}} = \f{dx^i} P_i{}^j P_j{}^k \ve{\pa_k}
$$
and hence $P_i{}^j P_j{}^k = P_i{}^k$. One example of a projection is the ''identity projection'', $\f{\ve{I}}=\f{dx^i} \ve{\pa_i}$, and another is the ''null projection'', $P_i{}^j=0$. (For a vector, $\ve{v}$, and $p$-form, $\nf{F}$, the identity projection gives $\f{\ve{I}} \nf{F} = p \nf{F}$ and $\ve{v} \f{\ve{I}} = \ve{v}$ via [[vector-form algebra]].) A projection splits the vector space, $V$, of tangent vectors into two subspaces: the range, $V_r$, and the kernel, $V_0$ -- with $V=V_r+V_0$. Projection leaves vectors in the range space unchanged, $\ve{v} \f{\ve{P}} = \ve{v}$ for $\ve{v} \in V_r$, while taking vectors in the kernel to zero, $\ve{v} \f{\ve{P}} = 0$ for $\ve{v} \in V_0$. The range and kernel of a projection determine it uniquely.
A vector projection, $\f{\ve{P}}$, naturally gives rise to a ''dual projection'', $\f{\ve{N}}=\f{\ve{I}}-\f{\ve{P}}$, satisfying $\f{\ve{N}}\f{\ve{P}}=\f{\ve{P}}\f{\ve{N}}=0$. Using these, any vector or form may be split into range and kernel (parallel and orthogonal) parts,
\begin{eqnarray}
\ve{v} &=& \ve{v}\f{\ve{P}} + \ve{v}\f{\ve{N}} = \ve{v_r} + \ve{v_0} \\
\f{f} &=& \f{\ve{P}}\f{f} + \f{\ve{N}}\f{f} = \f{f^v} + \f{f^0}
\end{eqnarray}
A vector projection, $\f{\ve{P}}$, defined over a manifold naturally gives two [[distribution]]s over the manifold, the ''range distribution'', $\ve{\De_r}$, and the ''kernel distribution'', $\ve{\De_0}$, corresponding to the range and kernel subspaces defined at each point. If the range distribution is involutive it integrates to a foliation of the manifold and $\f{\ve{P}}$ projects vectors from $T_x M$ onto the tangent bundle of the foliating [[submanifold]] at $x$.
There is a natural way to build a [[vector projection]] that maps [[vector valued form]]s at a [[fiber bundle]] section to vector valued forms on the base. If local coordinates $x^a$ are chosen for a patch on the base, $M$, and $y^p$ for a patch on the typical fiber, $F$, a section, $\si:M \rightarrow E$, can be written locally as $\si(x) = (x,y_\si^p(x))$. The [[coordinate basis vectors]] on the base [[push forward|pullback]] to tangent vectors of the section in $E$,
$$
\ve{t_a} = \si_* \ve{\pa_a} = \ve{\pa_a} + \fr{\pa y_\si^p}{\pa x^a} \ve{\pa_p}
$$
Conversely, those vectors at the section on $E$ [[pullback]] to the base vectors, $\si^* \ve{t_a} = \ve{\pa_a}$. Some of the [[coordinate basis 1-forms]] on $E$ [[pullback]] to the corresponding ones on the base, $\si^* \f{dx^a} = \f{dx^a}$, while the others pull back to $\si^* \f{dx^p} = \f{dx^a} \fr{\pa y_\si^p}{\pa x^a}$ on the base. The ''vector projection onto a section'',
$$
\f{\ve{P_\si}} = \f{dx^a} \ve{t_a}
$$
pulls back to the [[identity projection|vector projection]] on the base, $\si ^* \f{\ve{P_\si}} =\si^* \f{dx^a} \si^* \ve{t_a} = \f{dx^a} \ve{\pa_a} = \f{\ve{I_M}}$. The basis vectors along the fiber are in the kernel of the vector projection onto a section, $\ve{\pa_p} \f{\ve{P_\si}} = 0$, so a vector projection onto a section projects "along" the fibers onto the section's tangent space. Contracting with a [[1-form]] on the total space at the section,
$$
\f{f} = \f{dx^a} f_a + \f{dy^p} f_p
$$
using [[vector-form algebra]], gives
$$
\f{\ve{P_\si}} \f{f} = \f{dx^a} f_a + \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} f_p
$$
which pulls back to
$$
\f{f_M} = \si^* \f{f} = \f{dx^a} f_a + \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} f_p = \si^* \lp \f{\ve{P_\si}} \f{f} \rp = \si^* \f{\ve{P_\si}} \si* \f{f} = \f{\ve{I_M}} \si^* \f{f}
$$
The vector projection onto a section can also be used to do something the pullback can't do alone -- map a [[tangent vector]], $\ve{v}$, at the section on $E$ to one on $M$,
$$
\ve{v_M} = \si^* \lp \ve{v} \f{\ve{P_\si}} \rp = v^a \si^* \ve{t_a} = v^a \ve{\pa_a}
$$
This gives a nice way to map a vector valued form, $\nf{\ve{K}}$, at the section on $E$ to one on $M$,
$$
\nf{\ve{K_M}} = \si^* \lp \nf{\ve{K}} \f{\ve{P_\si}} \rp = \lp \si^* \nf{K^a} \rp \ve{\pa_a}
$$
The vector projection onto a section also provides a nice way of mapping vector valued 1-forms onto the section,
\begin{eqnarray}
\f{\ve{P_\si}} \f{\ve{K}} &=& \lp \f{dx^a} \ve{\pa_a} + \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} \ve{\pa_p} \rp \lp \f{dx^b} K_b{}^c \ve{\pa_c} + \f{dx^b} K_b{}^q \ve{\pa_q} + \f{dy^q} K_q{}^b \ve{\pa_b} + \f{dy^q} K_q{}^r \ve{\pa_r} \rp \\
&=& \f{dx^a} \lp \lp K_a{}^b + \fr{\pa y_\si^p}{\pa x^a} K_p{}^b \rp \ve{\pa_b} + \lp K_a{}^q + \fr{\pa y_\si^p}{\pa x^a} K_p{}^q \rp \ve{\pa_q} \rp
\end{eqnarray}
with the resulting vector valued 1-form having a form part on the section but vector parts that are in $TE$ and not necessarily tangent to the section.
The ''section dual projection'' is
$$
\f{\ve{N_\si}} = \f{\ve{I}} - \f{\ve{P_\si}} = \f{dy^p} \ve{\pa_p} - \f{dx^a} \fr{\pa y_\si^p}{\pa x^a} \ve{\pa_p}
$$
It projects onto vectors tangent to the fibers, $\ve{\pa_p} \f{\ve{N_\si}} = \ve{\pa_p}$, and has the tangent vectors in its kernel, $\ve{t_a} \f{\ve{N_\si}} = 0$. A section is horizontal iff the [[Ehresmann connection]] at the section is the same as the section dual projection, $\f{\ve{\cal A}}=\f{\ve{N_\si}}$.
A ''vector space'', $V$, (which may also be an algebra and/or a [[manifold]]) is spanned by basis elements, $e_a$, allowing elements of the space to be written as $v = v^a e_a \in V$, with a sum implied over repeated [[indices]], $a$. A different basis, $e'_a$, may be chosen and used to write the original basis elements as
$$
e_a = t_a{}^b e'_b
$$
using a square, invertible [[matrix|linear operator]] of transition coefficients, $t_a{}^b$. The same vector may be written using the new basis as
$$
v = v^a e_a = v^a t_a{}^v e'_v = v'^b e'_b
$$
in which the components of the vector, in the new basis, are $v'^b = v^a t_a{}^b$.
A ''vector valued form'' (//''VVF''//), $\nf{\ve{A}}$, is an antisymmetric linear operator that maps one or more [[tangent vector]]s at a [[manifold]] point to a tangent vector. Or, considered as a field over the manifold, a vector valued form maps one or more [[vector fields|tangent bundle]] to a vector field. It is a [[differential form]] in the sense that tangent vectors contract with it, but it is vector valued. Using the [[vector-form algebra]], in which vectors don't act on forms to their left, a ''vector valued p-form'' may be written as
$$
\nf{\ve{A}} = \f{dx^i} \dots \f{dx^j} \fr{1}{p!} A_{i \dots j}{}^k \ve{\pa_k}
$$
It has overall form grade $(p-1)$, and its purpose in life is to eat vectors applied from its left and return a vector, which may contract with a 1-form on its right -- i.e. the basis 1-forms and vectors in the vector valued form itself do not contract. For example, a vector contracted with a ''vector valued 1-form'' is a vector,
$$
\ve{v} \f{\ve{A}} = v^i \ve{\pa_i} \f{dx^j} A_j{}^k \ve{\pa_k} = v^i A_i{}^k \ve{\pa_k} = \ve{u}
$$
and one vector valued 1-form contracted with another is another,
$$
\f{\ve{A}} \f{\ve{B}} = \f{dx^i} A_i{}^k \ve{\pa_k} \f{dx^j} B_j{}^m \ve{\pa_m} = \f{dx^i} A_i{}^k B_k{}^m \ve{\pa_m} = \f{\ve{C}}
$$
Using vector valued forms within vector-form algebra gives a number of useful [[vector valued form identities]]. Also, a vector valued 1-form often has an interesting [[spectral decomposition|eigen]].
A [[vector valued form]] may be handled nicely using [[vector-form algebra]]. When used within a graded [[commutator]] bracket, a vector valued $k$-form, $\nf{\ve{K}}$, has form grade $k-1$. Contracting a VVF with two [[differential form]]s acts as a [[derivation]],
$$
\nf{\ve{K}} \lp \nf{F} \nf{G} \rp = \lp \nf{\ve{K}} \nf{F} \rp \nf{G} + \lp -1 \rp^{f \lp k-1 \rp} \nf{F} \lp \nf{\ve{K}} \nf{G} \rp
$$
but contracting a VVF with a VVF times a form (or another VVF) does not,
\begin{eqnarray}
\nf{\ve{K}} \lp \nf{\ve{L}} \nf{F} \rp &=& \lp \nf{\ve{K}} \nf{\ve{L}} \rp \nf{F} + \lp -1 \rp^{\lp k-1 \rp \lp l-1 \rp} \nf{\ve{L}} \lp \nf{\ve{K}} \nf{F} \rp - \lp -1 \rp^{\lp k-1 \rp \lp l-1 \rp} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \nf{F} \\
\nf{\ve{K}} \lp \nf{\ve{L}} \nf{\ve{M}} \rp &=& \lp \nf{\ve{K}} \nf{\ve{L}} \rp \nf{\ve{M}} + \lp -1 \rp^{\lp k-1 \rp \lp l-1 \rp} \nf{\ve{L}} \lp \nf{\ve{K}} \nf{\ve{M}} \rp - \lp -1 \rp^{\lp k-1 \rp \lp l-1 \rp} \lp \nf{\ve{L}} \nf{\ve{K}} \rp \nf{\ve{M}}
\end{eqnarray}
When $k=0$ these give
\begin{eqnarray}
\ve{v} \lp \nf{\ve{L}} \nf{F} \rp &=& \lp \ve{v} \nf{\ve{L}} \rp \nf{F} - \lp -1 \rp^l \nf{\ve{L}} \lp \ve{v} \nf{F} \rp \\
\ve{v} \lp \nf{\ve{L}} \nf{\ve{M}} \rp &=& \lp \ve{v} \nf{\ve{L}} \rp \nf{\ve{M}} - \lp -1 \rp^l \nf{\ve{L}} \lp \ve{v} \nf{\ve{M}} \rp
\end{eqnarray}
which agree with the product rules for vectors on forms when $k=l=0$. (It also raises the interesting possibility of the existence of loop valued forms -- but I don't know what to do with such things yet.) Keep in mind that, to be precise with vector operators in parenthesis, $\lp \nf{\ve{K}} \nf{\ve{L}} \rp \nf{F} = \lp \nf{\ve{K}} \nf{L^i} \rp \ve{\pa_i} \nf{F}$ -- only the vector on the right end of the parenthesis comes out and contracts with $\nf{F}$ or $\nf{\f{M}}$, just as one would use the same notation for derivatives. (With such use of parenthesis, one may get the correct impression that vector-form algebra is the [[Lisp|http://en.wikipedia.org/wiki/Lisp_programming_language]] of differential geometry...)
The [[exterior derivative]] and [[partial derivative]] are not derivations when acting on products with VVF's,
\begin{eqnarray}
\f{d} \lp \nf{\ve{K}} \nf{F} \rp &=& \lp \f{\pa} \nf{\ve{K}} \rp \nf{F} - \lp -1 \rp^k \nf{\ve{K}} \lp \f{d} \nf{F} \rp + \lp -1 \rp^k \lp \nf{\ve{K}} \f{d} \rp \nf{F} \\
\f{\pa} \lp \nf{\ve{K}} \nf{\ve{L}} \rp &=& \lp \f{\pa} \nf{\ve{K}} \rp \nf{\ve{L}} - \lp -1 \rp^k \nf{\ve{K}} \lp \f{\pa} \nf{\ve{L}} \rp + \lp -1 \rp^k \lp \nf{\ve{K}} \f{\pa} \rp \nf{\ve{L}}
\end{eqnarray}
The shorthand notation used here for [[tangent vector]] and [[differential form]] products is slightly unconventional, but leads to a consistent vector-form algebra. The first notational simplification is to discard the [[wedge|wedge product]] between forms and instead simply require that any two [[coordinate basis 1-forms]] anti-commute,
$$
\f{dx^i} \f{dx^j} = - \f{dx^j} \f{dx^i}
$$
which implies that any two [[1-form]]s anti-commute, $\f{a} \f{b} = - \f{b} \f{a}$. The second simplification is to write the ''vector-form interior product'' contraction (aka //''vector-form interior derivation''//) as a [[natural]] vector-form product, producing a form one grade lower,
$$
\ve{v} \nf{b} = {\bf i}_{\ve{v}} \nf{b}
$$
Computations may be carried out using the fact that the "product" of a [[coordinate basis vector|coordinate basis vectors]] and coordinate basis 1-form is either $0$ or $1$ (a real number), $\ve{\pa_i} \f{dx^j} = \de_i^j$. The "product" of a vector and a 2-form is a 1-form
\begin{eqnarray}
\ve{v} \lp \f{a} \f{b} \rp &=& \lp \ve{v} \f{a} \rp \f{b} - \f{a} \lp \ve{v} \f{b} \rp = \lp \ve{v} \f{a} \rp \f{b} - \f{a} \ve{v} \f{b} \\
\ve{\pa_k} \lp \f{dx^i} \f{dx^j} \rp
&=& \lp \ve{\pa_k} \f{dx^i} \rp \f{dx^j} - \f{dx^i} \lp \ve{\pa_k} \f{dx^j} \rp
= \de_k^i \f{dx^j} - \f{dx^i} \de_k^j
= 2 \delta_{k}^{\lb i \rd} \f{dx^{\ld j \rb}}
\end{eqnarray}
As a grade $\lp -1 \rp$ [[derivation]], a vector contracts with all forms to its right via the above ''vector-form distributive rule'' unless it's restricted to act inside parenthesis -- just like for any derivative. (The vector-form algebra is non-associative.) The product of a vector and a scalar field (a 0-form) vanishes, $\ve{v} f = 0$. We use the following notation:
* A vector only acts on forms to its right, with distributive actions guided by parenthesis.
When in doubt, vectors act on all forms to their right until there's another vector or nothing, but parenthesis should be used when there is ambiguity in the operation of vectors on collections of vectors and forms. A vector acting on the product of a $p$-form, $\nf{a}$, and a $q$-form gives a $(p+q-1)$-form via the graded Liebniz distribution rule,
$$
\ve{v} \lp \nf{a} \nf{b} \rp = \lp \ve{v} \nf{a} \rp \nf{b} + \lp -1 \rp^p \nf{a} \lp \ve{v} \nf{b} \rp
$$
These algebraic rules extend to handle multiple vectors provided [[tangent vector]]s also anti-commute,
$$
\ve{u} \ve{v} = - \ve{v} \ve{u}
$$
This object, $\vv{l}=\ve{u} \ve{v}$, a (-2)-form, may be interpreted as a ''loop'' — the infinitesimal, closed, directed path lying in the plane spanned by the two vectors. Contracting it with a 2-form gives a scalar,
$$
\vv{l} \ff{F}
= \ve{u} \ve{v} \ff{F}
= \ve{u} \lp \ve{v} \ff{F} \rp = - \ve{v} \lp \ve{u} \ff{F} \rp
$$
which may be computed using the contraction of a loop of basis vectors with a basis 2-form,
$$
\ve{\pa_i} \ve{\pa_j} \lp \f{dx^k} \f{dx^m} \rp = \ve{\pa_i} \lp \ve{\pa_j} \lp \f{dx^k} \f{dx^m} \rp \rp = - 2 \de_i^{\lb k \rd} \de_j^{\ld m \rb}
$$
Note in the above expressions that both vectors in a loop, $\ve{u}\ve{v}$, act on things to their right, since there is no confining parenthesis -- i.e.
$$
\lp \ve{u} \ve{v} \rp \ff{F} = 0 \not= \ve{u} \ve{v} \ff{F} = \ve{u} \lp \ve{v} \ff{F} \rp
$$
Since vectors don't contract with forms on their left, it's possible to write [[vector valued form]]s, $\f{\ve{P}}$, that don't self-destruct. The rules combine to give many useful [[vector valued form identities]].
It's worth noting that this vector-form notation is analogous to Dirac's bra-ket notation:
$$
\begin{array}{rcl}
\ve{a} \f{b} & \leftrightarrow & < \! a \, | \, b \! > \\
\f{b} \ve{a} & \leftrightarrow & | \, b \! > < \! a \, |
\end{array}
$$
Also, it is related to the [[Hodge dual]] and the [[codifferential]], through
$$
\ve{v} \nf{a} = * \, \f{v} * \nf{a}
$$
The scalar valued volume n-form, or ''volume form'', is the product of the $n$ [[frame]] 1-forms representing a differential volume element on the manifold,
\begin{eqnarray}
\nf{e} &=& \f{e^0}\f{e^1}\dots\f{e^{n-1}} = \fr{1}{n!} \ep_{\al \be \dots \ga} \f{e^\al}\f{e^\be}\dots\f{e^\ga}\\
&=&\f{dx^i}\f{dx^j}\dots\f{dx^k} \lp e_i \rp^0 \lp e_j \rp^1 \dots \lp e_k \rp^{n-1}\\
&=&\f{dx^0}\f{dx^1}\dots\f{dx^{n-1}} \va^{ij\dots k} \lp e_i \rp^0 \lp e_j \rp^1 \dots \lp e_k \rp^{n-1}\\
&=&\nf{d^n x} \ll e \rl
\end{eqnarray}
using [[permutation symbol]]s and the [[determinant]] of the frame, $|e|$. Integrating this over the manifold gives the volume of the manifold, in $T^n$. Note that this only works for an orientable manifold, because of a sign ambiguity that arises for $\nf{e}$ over unorientable manifolds. (//There are fancy ways to fix this that I haven't dealt with yet.//)
The volume form can be written, using the geometric expression for the permutation symbol, as
\[ \nf{e} = \fr{1}{n!} \li \ga_\al \ga_\be \dots \ga_\ga \ga^- \ri \f{e^\al}\f{e^\be}\dots\f{e^\ga} = \fr{1}{n!} \li \lp \f{e} \rp^n \ga^- \ri \]
with
$$
\f{e^\al}\f{e^\be}\dots\f{e^\ga} = \nf{e} \, \ep_{\al \be \dots \ga} = \nf{e} \ll \et \rl \ep^{\al \be \dots \ga}
$$
compatible with a [[permutation identity|permutation identities]].
The volume form can also be written in terms of the conformal scalar, $s$, by using the [[special frame]],
\[ \nf{e} = \nf{d^n x} \ll e \rl = \nf{d^n x} \, s^n \]
The partial derivative of the frame determinant may be related to the [[trace]] of the [[spin connection]],
$$
\pa_i \ll e \rl = \ll e \rl \om_{\al\p{\al}i}^{\p{\al}\al} = \ll e \rl \om_{j\p{\al}\be}^{\p{j}\al} \lp e_\al \rp^j \lp e_i \rp^\be
$$
A quantum ''wavefunction'' is a complex valued function in some basis, $\ps(x) = \langle x | \ps \rangle \in {\mathbb C}$, resulting from the [[Hermitian form]] contraction of some [[Hilbert space]] basis [[ket]]s, $| x \rangle$, with some [[quantum state|quantum mechanics]], $| \ps \rangle$.
If $x \in {\mathbb R}$ is the one dimensional position of a particle, then each $| x \rangle$ is an [[eigen]]vector of the ''quantum position operator'', $\hat{x}$, with eigenvalue $x$, normalized such that $\langle x' | x \rangle = \de(x-x')$. Using this basis, any quantum state has a corresponding normalized wavefunction,
$$
1 = \langle \ps | \ps \rangle = \int_{-\infty}^\infty dx \, \langle \ps | x \rangle \langle x | \ps \rangle = \int_{-\infty}^\infty dx \, \ps^*(x) \, \ps(x)
$$
We can switch to a different basis, such as the normalized eigenvectors,
$$
| p \rangle = \int_{-\infty}^\infty dx \, \fr{1}{\sqrt{2 \pi \hbar}} e^{\fr{i}{\hbar} p x} | x \rangle
$$
allowing us to see that the ''quantum momentum operator'', $\hat{p}$, satisfying $\hat{p} | p \rangle = p | p \rangle$, in the position basis in which the momentum eigenfunctions are
$$
\ps_p(x) = \langle x | p \rangle = \fr{1}{\sqrt{2 \pi \hbar}} e^{\fr{i}{\hbar} p x}
$$
has a ''quantum momentum operator in the position basis'', $\hat{p}_x = - i \hbar \fr{d}{dx}$, satisfying
$$
\hat{p}_x \ps_p(x) = p \, \ps_p(x)
$$
[<img[images/png/wedge product.png]]The ''wedge product'' of two [[1-form]]s is a 2-form, $\f{a} \wedge \f{b} = \f{a} \f{b}$. (A grade 2 [[differential form]].) It is never necessary to write the wedge, "$\wedge$", since any [[differential form]]s multiplied together are always assumed to be wedged. This behavior is best treated by properly defining the [[vector-form algebra]].
The "wedge product" of two or more [[Clifford element]]s, often seen in the literature, is replaced by the functionality of the [[antisymmetric bracket]]. The use of the Clifford wedge is deprecated because, unlike the [[Clifford algebra]] dot product, cross product, and bracket, it is not [[grade|Clifford grade]] independent. The same may hold for grade dependent definitions of the dot product seen elsewhere.
The "$\wedge$" will therefore not be seen here outside of this note.
[[Lie algebra structure]] describes a Lie algebra via an [[eigen]]vector decomposition,
$$
\big[ C , V_{\pm i} \big] = \pm \al_i(C) V_{\pm i}
$$
with regard to a Cartan subalgebra, $C = C^J T^\mathfrak{C}_J \in \mathfrak{C}$, with root vectors, $V_{\pm i}$, with their corresponding roots, $\pm \al_i(C)$, spanning a [[root system]] with a pattern corresponding to the Lie algebra. More generally, a [[representation space]], $V$, has a ''representation space structure'' described by eigenvector decomposition,
$$
\pi(C) V_i = \al_i(C) V_i
$$
with ''weight vectors'', $V_i$, with their corresponding ''weights'', $\al_i(C)$, forming a ''weight system'' acted on by the [[root system]] of the Lie algebra via root and weight addition.
The ''fundamental weights'', $\om^a$, satisfy $<\! \om^a, \al_b \!>\, = \de^a_b$, with $\al_b$ the simple roots -- so, in terms of the inverse of the symmeterized Cartan matrix, the fundamental weights are $\om^a = (\om^a)^b \al_b = S^{-ab} \al_b$. Any weight in a representation can be written as a combination of fundamental weights, $\la = \la_a \om^a = \La - \la'^a \al_a$, or as the representation's ''highest weight'', $\La$, minus a positive integral combination of simple roots. A weight is ''dominant'' iff all $\la_a$ are positive integral. The highest weight is always dominant, and uniquely identifies the representation space. Furthermore, each different dominant weight corresponds to the highest weight of a different irreducible representation space.
The ''Weyl vector'' is a vector in root space equal to the sum of fundamental weights,
$$
\rh = \sum_a \om^a = \rh^b \al_b = \sum_a S^{-ab} \al_b
$$
and is useful for describing the ''character'' of a representation, or for determining a ''principal subalgebra''.
*[[Just Alerting You|http://xkcd.com/c15.html]]
*[[Science|http://xkcd.com/c54.html]]
*[[Gravitational Mass|http://xkcd.com/c89.html]]
*[[Centrifugal Force|http://xkcd.com/c123.html]]
*[[Beliefs|http://xkcd.com/c154.html]]
*[[Donald Knuth|http://xkcd.com/c163.html]]
*[[String Theory|http://xkcd.com/c171.html]]
*[[e to the pi times i|http://xkcd.com/c179.html]]
*[[Nash|http://xkcd.com/c182.html]]
*[[Matrix Transform|http://xkcd.com/c184.html]]
*[[Unscientific|http://xkcd.com/397/]]
*[[Large Hadron Collider|http://xkcd.com/401/]]
*[[Purity|http://xkcd.com/435/]]
*[[Height|http://xkcd.com/482/]]
*[[Depth|http://xkcd.com/485/]]
*[[Outreach|http://xkcd.com/585/]]
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