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[<img[images/png/1-form.png]]A ''1-form'', or ''//cotangent vector//'', $\sf{f}$, 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\n\s[ \sf{f} = f_i \sf{dx^i} \sin T_{p}^{*}M \s]\nIt 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),\n\s[ \sf{f}(\sve{v}) = {\sbf i}_{\sve{v}} \sf{f} = \sve{v} \sf{f} = v^j f_i \sve{\spa_j} \sf{dx^i} = v^j f_i \sde_j^i= v^i f_i \sin \sRe \s]\nThe 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 $\sf{dx^i}$.\n

A ''2-sphere'' embedded in flat 3-space is a two dimensional [[manifold]], $M$, defined by the equation $x^e x^f \sde_{ef} = r^2$ -- it is the surface of constant $r=r$ in [[spherical coordinates]]. The angular spherical coordinates, $(\sth,\sph)$, cover patches of the manifold (other patches are needed for the poles). The [[metric]] induced on the 2-sphere is $g_{ij} = {\srm diag}(r^2, r^2 \ssin^2(\sth))$. The simplest [[frame]] compatible with this metric is\n$$\n\sf{e} = \sf{d \sth} \s, r \ssi_1 + \sf{d \sph} \s, r \ssin(\sth) \ssi_2\n$$\nin which $\ssi_{1/2}$ are the [[Clifford basis vectors]] for [[Cl(2,0)|Clifford matrix representation]]. The coframe is\n$$\n\sve{e} = \ssi^1 \sfr{1}{r} \sve{\spa_\sth} + \ssi^2 \sfr{1}{r \ssin(\sth)} \sve{\spa_\sph}\n$$\nThe [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n$$\n\sf{\som} = - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp = - \sf{d \sph} \scos(\sth) \ssi_{12}\n$$\nThe [[Clifford vector bundle]] curvature is\n$$\n\sff{F} = \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} = \sf{d \sth} \sf{d \sph} \ssin(\sth) \ssi_{12}\n$$\nThe [[Clifford-Ricci curvature]] is\n$$\n\sf{R} = \sve{e} \stimes \sff{F} = \sf{d \sph} \sfr{1}{r} \ssin(\sth) \ssi_2 + \sf{d \sth} \sfr{1}{r} \ssi_1 = \sfr{1}{r^2} \sf{e}\n$$\nshowing that the 2-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\sLa = 0$ (as do all two dimensional spaces). The [[Clifford curvature scalar]] is $R = \sve{e} \scdot \sf{R} = \sfr{2}{r^2}$.

A ''3-sphere'' embedded in flat 4-space of positive signature is a three dimensional [[manifold]], $M$, defined by the equation $x^w x^x \sde_{wx} = r^2$ -- it is the surface of constant $r=r$ in $4d$ [[hyperspherical coordinates]],\n\sbegin{eqnarray}\nx^1 &=& r \scos(a^1) \s\s\nx^2 &=& r \ssin(a^1) \scos(a^2) \s\s\nx^3 &=& r \ssin(a^1) \ssin(a^2) \scos(a^3) \s\s\nx^4 &=& r \ssin(a^1) \ssin(a^2) \ssin(a^3)\n\send{eqnarray}\nThe 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 $g_{ij} = {\srm diag}(r^2, r^2 \ssin^2(a^1),r^2 \ssin^2(a^1) \ssin^2(a^2))$. The simplest [[frame]] compatible with this metric is\n$$\n\sf{e} = \sf{d a^1} \s, r \ssi_1 + \sf{d a^2} \s, r \ssin(a^1) \ssi_2 + \sf{d a^3} \s, r \ssin(a^1) \ssin(a^2) \ssi_3\n$$\nin which $\ssi_{1/2/3}$ are the [[Clifford basis vectors]] for [[Cl(3,0)|Cl(3)]]. The coframe is\n$$\n\sve{e} = \ssi^1 \sfr{1}{r} \sve{\spa_1} + \ssi^2 \sfr{1}{r \ssin(a^1)} \sve{\spa_2} + \ssi^3 \sfr{1}{r \ssin(a^1) \ssin(a^2)} \sve{\spa_3}\n$$\nThe [[torsion]]less [[spin connection]], found by solving [[Cartan's equation]], $0=\sf{d} \sf{e} + \sf{\som} \stimes \sf{e}$, is\n\sbegin{eqnarray}\n\sf{\som} &=& - \sve{e} \stimes \sf{d} \sf{e} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sf{d} \sf{e} \srp \s\s\n&=& - \sf{d a^2} \scos(a^1) \ssi_{12} - \sf{d a^3} \scos(a^1) \ssin(a^2) \ssi_{13} - \sf{d a^3} \scos(a^2) \ssi_{23}\n\send{eqnarray}\nThe [[Clifford vector bundle]] curvature is\n\sbegin{eqnarray}\n\sff{F} &=& \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} \s\s\n&=& \sf{d a^1} \sf{d a^2} \ssin(a^1) \ssi_{12} + \sf{d a^1} \sf{d a^3} \ssin(a^1) \ssin(a^2) \ssi_{13} + \sf{d a^2} \sf{d a^3} \ssin^2(a^1) \ssin(a^2) \ssi_{23}\n\send{eqnarray}\nThe [[Clifford-Ricci curvature]] is\n\sbegin{eqnarray}\n\sf{R} &=& \sve{e} \stimes \sff{F} \s\s\n&=& \sf{d a^1} \sfr{2}{r} \ssi_1 + \sf{d a^2} \sfr{2}{r} \ssin(a^1) \ssi_2 + \sf{d a^3} \sfr{2}{r} \ssin(a^1) \ssin(a^2) \ssi_3 \s\s\n&=& \sfr{2}{r^2} \sf{e}\n\send{eqnarray}\nshowing that the 3-sphere is an [[Einstein space|Einstein's equation]] with cosmological constant $\sLa = \sfr{1}{2 r^2}$. The [[Clifford curvature scalar]] is $R = \sve{e} \scdot \sf{R} = \sfr{8}{r^2}$.

<<note HideTags>>$$\n\sbegin{array}{rclclc}\n\sf{\som} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} \sha \som_k^{\sp{k}\smu\snu} \sga_{\smu\snu} \s!&\s!\s! \sin \s!\s!&\s! \sf{Cl}^2(3,1)\n&\n\squad\n\sf{e} = \sf{dx^k} (e_k)^\smu \sga_\smu \s, \sin \s, \sf{Cl}^1(3,1) \svp{|_{(}} \s\s\n\n\sf{W} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} W_k^{\sp{i}\spi} \sfr{i}{2} \ssi_\spi \s!&\s!\s! \sin \s!\s!&\s! \sf{su}(2)\n&\n\squad\n\slb \smatrix{\n\sph_+ \s\s \sph_0\n} \srb\n\sqquad\n\slb \smatrix{\n\snu_{eL} \s\s e_L\n} \srb\n\s\s\n\n\sf{B} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} B_k i \s!&\s!\s! \sin \s!\s!&\s! \sf{u}(1)\n&\n\squad\nY \s\s\n\n\sf{g} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^k} g_k^{\sp{k}A} \sfr{i}{2} \sla_A \s!&\s!\s! \sin \s!\s!&\s! \sf{su}(3)\n&\n\squad\n\slb u^r, u^g, u^b \srb \svp{|^{(^(}_{(}}\n\send{array}\n\sbegin{array}{c}\n\squad\n\slb \smatrix{\ne_L^\swedge \s\s e_L^\svee \s\s e_R^\swedge \s\s e_R^\svee\n} \srb\n\s; \s\s\n\s; \s\s\n\send{array}\n$$\n$$\n\supdownarrow \svp{{\shuge(}_{\sbig(}}\n$$\n$$\n\sbegin{array}{rcl}\n\sudf{A} \s!\s!&\s!\s!=\s!\s!&\s!\s! {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{W} + \sf{B} + \sf{g} + ( \sud{\snu}{}_e + \sud{e} + \sud{u} + \sud{d} ) \s\s\n&& + \s, (\sud{\snu}{}_\smu + \sud{\smu} + \sud{c} + \sud{s}) + (\sud{\snu}{}_\sta + \sud{\sta} + \sud{t} + \sud{b}) \svp{|_{\sBig(}}\n\send{array}\n$$\n$$\n\sudff{F} = \sf{d} \sudf{A} + {\sscriptsize \sfrac{1}{2}} \sbig[ \sudf{A}, \sudf{A} \sbig]\n$$

This hasn't been around long enough for any questions asked to be frequent, but I'll try to anticipate some.\n\n!!Who?\nThe site is principally authored by me, [[Garrett Lisi]]. I may open it up for collaboration in the future.\n\nAs 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.\n\n!!What?\nIt's sort of a "choose your own adventure" book in theoretical physics &mdash; 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 &mdash; 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.\n\n!!Why?\nI 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 &mdash; 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.\n\n!!How?\nThe 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/]].\n\nTo use it... try things. Click on buttons and see what happens, you'll figure it out.\n\n!!What about money for food?\n\nMy research is supported entirely by private contributions. Support is always appreciated, is tax deductible, and may be contributed by contacting me or [[Theiss Research|http://www.theissresearch.org/]], a 501(c)(3) corporation.\n\n!!Where\nThis 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 &mdash; it hates that.\n\n!!When\nNow.\n\n----\n<<slider chkSliderAbout 'About (slider)' 'More questions and answers »' 'Click to see more questions and answers'>>\n

!!How did you come up with the title, "Deferential Geometry"?\nMy 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 &mdash; good physics uses math, and this site is about using only the math needed by physics, differential geometry in particular. There shouldn't be any mathematical tangents here without physics ideas motivating them, the geometry is deferential to the physics &mdash; a beautiful idea, and a bad pun.

Modified BF action, using $\sff{\sod{B}} = \sff{B} + \sfff{\sod{B}} \s,$:\n\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F} + \snf{\sPhi} ( \sf{H}{}_1, \sf{H}{}_2, \sff{B} ) \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \sff{B} \sff{F} + {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \snf{e} \sfr{1}{16 \spi G} \sph^2 \sbig( R - \sfr{3}{2} \sph^2 \sbig) + \sfr{1}{4} \sff{F'} \sff{*F'} \sbig>\n\send{eqnarray}\n\nCosmological constant from the Higgs VEV: $\squad \sLa = \sfr{3}{4} \sph^2$\n\nImplies frame VEV is de Sitter: $\squad \sff{R} = \sfr{\sLa}{6} \sf{e} \sf{e} \sqquad R = 4 \sLa$\n\nVacuum expectation value of the curvature vanishes: $\squad \sudff{F} = 0$\n<<note HideTags>>

<<note HideTags>>$$\n\sudff{F} = \sf{d} \sudf{A} + \sudf{A} \sudf{A} = \sbig( \sf{d} \sf{H} + \sf{H} \sf{H} \sbig) + \sbig( \sf{d} \sf{G} + \sf{G} \sf{G} \sbig) + \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig)\n$$\nModified BF action for everything, using $\sff{\sod{B}} = \sff{B} + \sfff{\sod{B}} \s,$:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F} + \snf{\sPhi} ( \sf{H}, \sf{G}, \sff{B} ) \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig)\n+ \sff{B} \sff{F} - {\sscriptsize \sfrac{1}{4}} \sff{B_s} \sff{B_s} \sga + \sff{B_{m,h,G}} \sff{*B_{m,h,G}} \sbig>\n\send{eqnarray}\nFermionic part, using [[anti-ghost|BRST technique]] [[Grassmann|Grassmann number]] 3-form, $\sfff{\sod{B}} = \snf{e} \sod{\sps} \sve{e} \s,$:\n\sbegin{eqnarray}\nS_f &=& \sint \sbig< \sfff{\sod{B}} \sbig( \sf{d} \sud{\sps} + \sf{H} \sud{\sps} + \sud{\sps} \sf{G} \sbig) \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sps} \sve{e} \sbig( \sf{d} \sud{\sps} + {\sscriptsize \sfrac{1}{2}} \sf{\som} \sud{\sps} + {\sscriptsize \sfrac{1}{4}} \sf{e} \sph \sud{\sps} + \sf{B} \sud{\sps} + \sf{W} \sud{\sps} + \sud{\sps} \sf{G} \sbig) \sbig> \s\s\n&=& \sint \snf{d^4 x} |e| \s, \sbig< \sod{\sps} \sga^\smu (e_\smu)^i \sbig( \spa_i \sud{\sps} + {\sscriptsize \sfrac{1}{4}} \som_i^{\sp{i} \smu \snu} \sga_{\smu \snu} \sud{\sps} + B_i \sud{\sps} + W_i \sud{\sps} - \sud{\sps} G_i \sbig) + \sod{\sps} \s, \sph \s, \sud{\sps} \sbig>\n\send{eqnarray}

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author: [[Garrett Lisi]]\narxiv: http://arxiv.org/abs/0711.0770\nlocally: [[AESToE|papers/AESToE.pdf]]\nabstract:\nAll 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]].\n\nA [[talk for ILQGS 07]] on 11/13/07 was presented on this paper.\n\nInternet discussion of the paper, in chronological order: \n*[[Backreaction|http://backreaction.blogspot.com/2007/11/theoretically-simple-exception-of.html]]\n*[[Physics Forums|http://www.physicsforums.com/showthread.php?t=196498]]\n*[[The Reference Frame|http://motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html]]\n*[[Hidden Variables|http://blog.domenicdenicola.com/post/2007/11/Criteria-for-a-Theory-of-Everything.aspx]]\n*[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=617]]\n*[[Arcadian Functor|https://www.blogger.com/comment.g?blogID=28857369&postID=5548882952979522971]]\n*[[Freedom of Science|http://globalpioneering.com/wp02/an-exceptionally-simple-theory-of-everything/]]\n*[[Theoreman Egregium|http://egregium.wordpress.com/2007/11/10/physics-needs-independent-thinkers/]]\n*[[Science Forums|http://www.scienceforums.net/forum/showthread.php?t=29522]]\n\nAnd previously:\n*[[This Week's Finds 253]]

<<note HideTags>>Start with $E8$ principal bundle connection and its curvature,\n$$\n\sf{A} = \sf{H} + \sf{\sPs} \sqquad \squad\n\sff{F} = (\sf{d} \sf{H} + \sf{H} \sf{H} + \sf{\sPs} \sf{\sPs})\n+ (\sf{d} \sf{\sPs} + \sf{H} \sf{\sPs} + \sf{\sPs} \sf{H})\n$$\nAction such that $\sf{\sPs}$ part is pure gauge,\n$$\nS = \sint \sbig< \sff{B} \sff{F}\n+ {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig>\n$$\nBRST: Replace $\sf{\sPs}$ part with ghosts, $\sud{\sPs}$, in extended connection, \n$$\n\sudf{A} = \sf{H} + \sud{\sPs} \sqquad \squad\n\sudff{F} = \sbig( \sf{d} \sf{H} + \sf{H} \sf{H} \sbig) + \sbig( \sf{d} \sud{\sPs} + [ \sf{H}, \sud{\sPs} ] \sbig)\n= \sff{F}{}_H + \sf{D} \sud{\sPs}\n$$\nEffective action for gauge fields, ghosts, and anti-ghosts:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{\sod{B}} \sudff{F}\n+ {\sscriptsize \sfrac{\spi G}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig> \s\s\n&=& \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs}\n+ \snf{e} \sfr{1}{16 \spi G} \sph^2 \sbig( R - \sfr{3}{2} \sph^2 \sbig) + \sfr{1}{4} \sff{F'} \sff{*F'} \sbig>\n\send{eqnarray}

<<note HideTags>>$\sde \snf{L} = 0$ under [[gauge transformation]]: &nbsp;&nbsp; $\sde \sf{A} = - \sf{\sna} C = -\sf{d} C - \sbig[ \sf{A}, C \sbig]$\nAccount for gauge part of $\sf{A}$ by introducing [[Grassmann|Grassmann number]] valued ''ghosts'', $\sud{C} \sin \sud{\srm Lie}(G)_g$, ''anti-ghosts'', $\snf{\sod{B}}$, ''partners'', $\snf{\sla}$, and [[BRST transformation|BRST technique]]:\n$$\n\sbegin{array}{rclcrcl}\n\sud{\sde} \sf{A} &=& - \sf{\sna} \sud{C} & \s;\s;\s;\s;\s;\s;\s;\s;\s; & \sud{\sde} \sud{C} &=& - \sha \sbig[ \sud{C}, \sud{C} \sbig] \s\s\n\sud{\sde} \sff{B} &=& \sbig[ \sff{B}, \sud{C} \sbig] & \s;\s;\s;\s;\s;\s;\s;\s;\s; & \sud{\sde} \snf{\sod{B}} &=& \snf{\sla} \s\s\n\sud{\sde} \snf{\sla} &=& 0 & \s;\s;\s;\s;\s;\s;\s;\s;\s; & & &\n\send{array}\n$$\nThis satisfies $\sud{\sde} \snf{L} = 0_{\sphantom{\sbig(}}$and $\sud{\sde} \sud{\sde} = 0$.\nChoose a ''BRST potential'', $\snf{\sod{\sPs}} = \sbig< \snf{\sod{B}} \sf{A} \sbig>$, to get new Lagrangian:\n$$\n\snf{L'} = \snf{L} + \sud{\sde} \snf{\sod{\sPs}} = \snf{L} + \sbig< \snf{\sla} \sf{A_g} \sbig> + \sbig< \snf{\sod{B}} \sf{\sna} \sud{C} \sbig>\n$$\nBRST partners act as Lagrange multipliers; ''effective Lagrangian'':\n$$\n\snf{L^{\srm eff}} = \snf{L}[\sff{B'},\sf{A'}] + \sbig< \snf{\sod{B}} \sf{\sna'} \sud{C} \sbig>\n$$

//This is a speculative description of the [[BRST technique]] based on conversations with [[Michael Edwards]]//\n\nStart with a [[connection]] 1-form, $\sf{\som}$, defined over the entire space of a [[fiber bundle]], and some fiber bundle section, $\ssi$. The connection field over the base manifold is the pullback of the connection along the section,\n$$\n\sf{A} = \ssi^* \sf{\som}\n$$\nA BRST transformation may be a way of describing how $\sf{A}$ 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,\n$$\n\sve{\sva} = \sva^A(x) \sve{\sxi_A}(p)\n$$\non the entire space, with $\sve{\sxi_A}$ 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 $\sva^A(x)= \sva C^A(x)$. The BRST transformation then is\n$$\n\sf{\sde A} = - \ssi^* L_{\sve{\sva}} \sf{\som} = \sva s \sf{A} = - \sva (\sf{d} C + \sf{A} \stimes C)\n$$\nin which $C=C^A T_A$.\n//Hmm, this seems to give the change in A from flowing $\som$ under $\ssi$...// \n\n\nAnother 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.\n\n\nNah, none of this is going to work right. Have to work in the space of connections.\n\nvariational bicomplex\n\nRefs:\n*M. Ghiotti\n**[[Gauge fixing and BRST formalism in non-Abelian gauge theories|papers/Ghiotti - Gauge fixing and BRST formalism in non-Abelian gauge theories.pdf]]\n***Excellent new thesis.\n*G. Catren and J. Devoto\n**[[Extended Connection in Yang-Mills Theory|http://arxiv.org/abs/0710.5698]]\n*Bonora and Cotta-Ramusino\n**[[Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations|papers/1103922136.pdf]]\n***This is one of the first, and probably the best, description of ghosts as 1-forms in the space of connections.\n***${\scal G}$ is group of vertical [[automorphism]]s of $E$, equals group of [[gauge transformation]]s.\n*Stora and Kastler\n**[[A Differential Geometric Setting for BRS Transformations and Anomalies|papers/A Differential Geometric Setting for BRS Transformations and Anomalies.pdf]]\n***detailed exposition.\n***gauge transformation bundle\n***old (hard to read scanned text) but good. lengthy.\n***same gauge trasf as Viallet (below)\n*Viallet\n**[[The Geometry of the Space of Fields in Yang-Mills theory|papers/The Geometry of the Space of Fields in Yang-Mills theory.pdf]]\n***space of fields as bundle, physical fields as base\n***strange definition for group automorphism, which disagrees with mine and Wikipedia's.\n***gauge transf are ''equivariant'' automorphisms, $f(p)=p\sph(p)$, satisfying $f(ph)=f(p)h$ and hence $\sph(ph)=h^- \sph(p) h$.\n*J.W. van Holten\n**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]\n***excellent elementary practical intro\n***ghost to M-C form mapping not an identification\n**[[The BRST Complex and the Cohomology of Compact Lie Algebras|papers/The BRST Complex and the Cohomology of Compact Lie Algebras.pdf]]\n***BRST analysis analogous to [[Hodge decomposition]]\n***this paper's content is included in the paper above\n*http://en.wikipedia.org/wiki/BRST_Quantization\n*[[Principal Bundles, Connections and BRST Cohomology|papers/9408003.pdf]]\n**(//read this now//)\n**BRST cohomology in the space of connections\n**mathematically dense, but I'm hacking it so far\n**[[lots of geometric brst papers from spires|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+c+cmpha,87,589&SKIP=0]]\n*Jeffrey A. Harvey\n**[[TASI 2004 Lectures on Anomalies|papers/0509097.pdf]]\n***for particle physicists\n***ghosts are 1-forms in the space of gauge transformations\n***BRST operator is exterior derivative in this space\n*Barnich, Brandt, and Henneaux\n**[[Local BRST cohomology in gauge theories|papers/0002245.pdf]]\n***these guys are the big shots in the field, but I don't like this antifield approach yet.\n*[[Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory|papers/9705123.pdf]]\n**pretty good and succinct intros, plus treats BF\n**agrees with Viallet's def of equivariant automorphism.\n*Moritsch, Sorella, et al\n**[[Algebraic characterization of gauge anamolies on a nontrivial bundle|papers/9611168.pdf]]\n***nice algebraic treatment, with generalized connection\n**[[Algebraic characterization of the Wess-Zumino consistency conditions in gauge theory|papers/9302136.pdf]]\n***Sorrella's paper introducing $\sde$. Seems to work with jets without calling them that.\n**[[Algebraic structure of gravity in Ashtekar variables|papers/9409046.pdf]]\n***Blaga, using $\sde$.\n*Jim Stasheff\n**[[The (secret?) homological algebra of the Batalin-Vilkovisky approach|papers/9712157.pdf]]\n***abstract mathematical overview (jets) of relations between physics and math structures\n***ghost = Chevelley-Eilenberg generator\n***anti-ghost = Tate generator\n***anti-field = Koszul generator\n*Yang, Lee\n**[[Lie algebra cohomology and group structure of gauge theories|papers/9503204.pdf]]\n***maybe or maybe not useful\n*Kelnhofer\n**[[On the Geometrical Structure of Covariant Anomalies in Yang-Mills Theory|paper/9302012.pdf]]\n***damn this is a (unavoidable) mess\n***universal bundle\n*Thomas Schucker\n**see his book on Amazon for introductory material:\n***[[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]]\n**[[The Cohomological Construction of Stora's Solutions|papers/The Cohomological Construction of Stora's Solutions.pdf]]\n*Picken: http://www.iop.org/EJ/abstract/0305-4470/19/5/001\n**//(scan this)//\n*J. P. Zwart\n**[[BRST Reduction and Quantization of Constrained Hamiltonian Systems|papers/zwart98brst.pdf]]\n*Witten\n**[[Topological Quantum Field Theory|papers/1104161738.pdf]]\n**differential forms on the space of connections\n*Laurent Baulieu\n**[[On the Cohomological Structure of Gauge Theories|papers/On the Cohomological Structure of Gauge Theories.pdf]]\n***adds two Grassmann coordinates to spacetime\n*Rudolf Schmid\n**[[Local Cohomology in Gauge Theories BRST Tansformations and Anomalies|papers/localbrst.pdf]]\n***mathematically abstract, but geometric\n***ack, [[jet]]s. \n***connects ghosts to [[Maurer-Cartan form]]\n***pretty tough going, maybe delete this ref\n**[[A Few BRST Bicomplexes|papers/nankai.pdf]]\n*discussion with [[Michael Edwards]] on\n**[[Not Even Wrong|http://www.math.columbia.edu/~woit/wordpress/?p=436]]\n*Ian Anderson\n**[[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.\n\nA restricted BF Lagrangian,\n$$\n\snf{L} = \sli \snf{B} \sff{F} + \snf{\sPhi}(\sf{A},\snf{B}) \sri\n$$\ninvariant, $\sdelta_{G} \snf{L} = 0$, under some subset of the gauge transformation, $G \sin {\srm Lie}(H) \ssubset {\srm Lie}(G)$, is amenable to the BRST technique. A ''ghost field'', $\sud{C} = \sud{C^A} T_A \sin \sud{\srm Lie}(H)$, is introduced with Grassmann coefficients multiplying [[Lie algebra]] elements, along with an anti-Grassmann valued $(n-1)$-form ''antighost field'', $\snf{\sod{B}} = \snf{\sod{B}{}^A} T_A$, and a real valued $(n-1)$-form ''BRST partner field'', $\snf{\slambda} = \snf{\slambda^A} T_A$. This new system is equipped with a global ''BRST transformation'' -- a ''supersymmetry rotation'' between real and Grassmann valued variables,\n$$\n\sbegin{array}{rclcrcl}\n\sud{\sde} \sf{A} &=& -\sf{\snabla} \sud{C} & \s;\s;\s; & \sud{\sde} \sud{C} &=& -\sha \slb \sud{C}, \sud{C} \srb \s\s\n\sud{\sde} \snf{B} &=& \slb \snf{B}, \sud{C} \srb & \s;\s;\s; & \sud{\sde} \snf{\sod{B}} &=& \snf{\sla} \s\s\n\sud{\sde} \snf{\sla} &=& 0 & & & &\n\send{array}\n$$\nthat is nilpotent, $\sud{\sde} \sud{\sde} = 0$, and leaves the Lagrangian invariant (''BRST [[closed]]''), $\sud{\sde} \snf{L} = 0$. Physical observables are in the [[cohomology]] of this ''BRST operator'', $\sud{\sde}$. Dynamics are introduced for the ghosts by adding a ''BRST [[exact]]'' term to get a ''BRST extended Lagrangian'',\n$$\n\snf{L'} = \snf{L} + \sud{\sde} \snf{\sod{\sPsi}}\n$$\nwith some ''BRST potential'', $\snf{\sod{\sPsi}}$, chosen. For example, choosing\n$$\n\snf{\sod{\sPsi}} = \sli \snf{\sod{B}} \sf{A} \sri\n$$\ngives\n$$\n\sud{\sde} \snf{\sod{\sPsi}} = \sli \snf{\sla} \sf{A} \sri + \sli \snf{\sod{B}} \sf{\snabla} \sud{C} \sri\n$$\nThe BRST partner field, $\snf{\sla}$, acts as a Lagrange multiplier constraining the gauge freedom of the connection, so the ''gauge fixed connection'' is $\sf{A} = \sf{A'}$, with $\snf{\sla} \sf{A'} = 0$. The resulting ''effective Lagrangian'' is\n$$\n\snf{L^{\srm eff}} = \sli \snf{B'} \sff{F'} + \snf{\sPhi}(\sf{A'},\snf{B'}) \sri \n+ \sli \snf{\sod{B}} \sf{\snabla'} \sud{C} \sri\n$$\n\nThis form of the Lagrangian suggests the introduction of a ''BRST extended connection'',\n$$\n\sudf{A} = \sf{A'} + \sud{C}\n$$\nwith ''BRST extended curvature'',\n$$\n\sudff{F} = \sf{d} \sudf{A} + \sha \slb \sudf{A} , \sudf{A} \srb = \sff{F'} + \sf{\snabla'} \sud{C} + \sha \slb \sud{C} , \sud{C} \srb\n$$\nallowing the effective Larangian to be written as\n$$\n\snf{L^{\srm eff}} = \sli \snf{\sod{B'}} \sudff{F} + \snf{\sPhi}(\sf{A'},\snf{B'}) \sri\n$$\nwith $\snf{\sod{B'}} = \snf{B'} + \snf{\sod{B}}$.\n\nRef:\n*J.W. van Holten\n**[[Aspects of BRST Quantization|papers/JHolten_BRST.pdf]]

<<note HideTags>>$$\sbegin{array}{rcl}\n\sf{H} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W} =\n{\sscriptsize\n\slb \sbegin{array}{cccc} \n\sha \sf{\som_L} \s!+\s! i \sf{W^3} & i \sf{W^1} \s!+\s! \sf{W^2} & - \sfr{1}{4} \sf{e_R} \sph_0^* & \sfr{1}{4} \sf{e_R} \sph_+ \s\s\ni \sf{W^1} \s!-\s! \sf{W^2} & \sha \sf{\som_L} \s!-\s! i \sf{W^3} & \sp{-} \sfr{1}{4} \sf{e_R} \sph_+^* & \sfr{1}{4} \sf{e_R} \sph_0 \s\s\n- \sfr{1}{4} \sf{e_L} \sph_0 & \sfr{1}{4} \sf{e_L} \sph_+ & \sha \sf{\som_R} \s!+\s! i \sf{B} & \s\s\n\sp{-} \sfr{1}{4} \sf{e_L} \sph_+^* & \sfr{1}{4} \sf{e_L} \sph_0^* & & \sha \sf{\som_R} \s!-\s! i \sf{B}\n\send{array} \srb_{\sp{(}}\n} \s\s\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \sha h_a^{\sp{a} \sal\sbe} \sga_{\sal\sbe} \s;\s; \sin \s;\s; \sf{so}(1,7) = \sf{Cl}^2(1,7) \ssubset \sf{\smathbb{C}}(8\stimes8)\n\send{array}$$\n@@display:block;text-align:center;[[Clifford bivector|Clifford algebra]] parts:@@$$\n\sbegin{array}{rcl}\n\sf{\som} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \sha \som_a^{\sp{a} \smu \snu} \sga_{\smu \snu}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\sleftarrow \stext{spin connection} \s\s\n\sf{e} \sph \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \slp e_a \srp^\smu \sph^\sph \sga_{\smu \sph}\n\s, \sleft\s{\n\sbegin{array}{rcl}\n\sf{e} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{dx^a} \slp e_a \srp^\smu \sga_\smu\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\n\sleftarrow \stext{frame (vierbein)} \s\s\n\sph \s!\s!&\s!\s!=\s!\s!&\s!\s! \sph^\sph \sga_\sph\n\s, \sleft\s{\n\sbegin{array}{rcl}\n\sph_+ \s!\s!&\s!\s!=\s!\s!&\s!\s! (-\sph^5 \s!+\s! i \sph^6) \s\s\n\sph_0 \s!\s!&\s!\s!=\s!\s!&\s!\s! (\sph^7 \s!+\s! i \sph^8)\n\send{array}\n\sright\s}\n\sbegin{array}{c}\n\s;\s;\s;\s;\n\sleftarrow \stext{Higgs} \s\s\n\sph \sph = -M^2\n\send{array}\n\send{array}\n\srd\n\send{array}\n$$\n$$\n\sbegin{array}{rcl}\n\sf{B} \s!\s!&\s!\s!=\s!\s!&\s!\s! - \s! \sf{dx^a} \sha B_a \sbig( \sga_{56} - \sga_{78} \sbig) \n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s,\n\sleftarrow \s; \sdownarrow \stext{electroweak gauge fields} \s\s\n\sf{W} \s!\s!&\s!\s!=\s!\s!&\s!\s! - \s! \sha \sf{W^1} \sbig( \sga_{67} + \sga_{58} \sbig)\n- \sha \sf{W^2} \sbig(-\sga_{57} + \sga_{68} \sbig)\n- \sha \sf{W^3} \sbig( \sga_{56} + \sga_{78} \sbig) \n\s;\s;\s;\s;\s;\s;\s;\s, \s\s\n\send{array}\n$$\n@@display:block;text-align:center;[[indices]]: $\s;\s;\s;\s; 0 \sle a,b \sle 3 \s;\s;\s;\s;\s;\s; 0 \sle \smu,\snu \sle 3 \s;\s;\s;\s;\s;\s; 5 \sle \sph,\sps \sle 8 $@@

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>\n</center></html>@@\n

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Soft Coral_620.jpg" width="827" height="620"></embed>\n</center></html>@@\n

Creating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n{{{\nCreating bulleted lists is simple.\n* Just add an asterisk\n* at the beginning of a line.\n** If you want to create sub-bullets\n** start the line with two asterisks\n*** And if you want yet another level\n*** use three asterisks\n* You can also do [[Numbered Lists]]\n}}}

\n\nRef:\n*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]]

[[arxiv|http://arxiv.org/abs/0706.0217]]\nS. Raby, A. Wingerter\nAbstract: 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). \n\n*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\n*Location: Marseille\n*CV: http://www.cpt.univ-mrs.fr/%7Erovelli/vita.pdf\n*arXiv papers: http://arxiv.org/find/gr-qc/1/au:+Rovelli_C/0/1/0/all/0/1\n\nSelected work:\n*[[Quantum Gravity|http://www.cpt.univ-mrs.fr/%7Erovelli/book.pdf]]\n*[[Graviton propagator in loop quantum gravity|http://arxiv.org/abs/gr-qc/0604044]]\n**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 $H \ssubset G$; 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 $E_H$.\n\nThe Cartan H-bundle is mapped into a section (a [[submanifold]]), $E'_H$, of the Ehressmann Cartan geometry, $E_G$, by the reference section of the [[Cartan homogeneous space bundle]],\n\sbegin{eqnarray}\n\ssi'^S &:& E_H \sto E_G \s\s\n\ssi'^S(x,y) &=& (x,x_{s\ssi}(x),y)\n\send{eqnarray}\nThe [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Ehresmann Cartan H-connection form'' over $E_H$,\n\sbegin{eqnarray}\n\sf{{\scal C}_H}(x,y) &=& \ssi'^{S*} \sf{\scal C} = \slp \sf{C^J}(x) L^I{}_J(x_{s\ssi}(x),y) + \ssi'^{S*} \sf{\sxi_R^I}(x_{s\ssi}(x),y) \srp T_I \s\s\n&=& g^-(x_{s\ssi}(x),y) \s, \sf{C}(x) \s, g(x_{s\ssi}(x),y) + g^-(x_{s\ssi}(x),y) \s, \sf{d} \s, g(x_{s\ssi}(x),y) \s\s\n&=& h^-(y) \sBig( r^-(x_{s\ssi}(x)) \s, \sf{C}(x) \s, r(x_{s\ssi}(x)) + r^-(x_{s\ssi}(x)) \s, \sf{d} \s, r(x_{s\ssi}(x)) \sBig) h(y) + h^-(y) \s, \sf{d} h(y) \n\send{eqnarray}\nin which the [[coset representative section|homogeneous space]], $r:G/H \sto G$, is used to write $g(x_s,y)=r(x_s) \s, h(y)$. Choosing the homogeneous space bundle zero reference section, $r(x_{s\ssi_0}(x)) = r(0) = 1$, this gives\n$$\n\sf{{\scal C}_H}(x,y) = h^-(y) \s, \sf{C}(x) \s, h(y) + h^-(y) \s, \sf{d} \s, h(y)\n$$\nPulling this back along the [[canonical reference section|Ehresmann principal bundle connection]] gives the [[Cartan connection|Cartan geometry]], $\ssi_0^{H*} \sf{{\scal C}_H} = \sf{C}$, over $M$.\n\nIf $H$ 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'',\n\sbegin{eqnarray}\n\sf{{\scal C}_H} &=& \sf{{\scal E}_H} + \sf{{\scal A}_H} \s\s \n\sf{{\scal E}_H} &=& h^-(y) \s, \sf{e}(x) \s, h(y) \sin \sf{\srm Lie}(G/H) \s\s\n\sf{{\scal A}_H} &=& h^-(y) \s, \sf{A}(x) \s, h(y) + h^-(y) \s, \sf{d} \s, h(y) \sin \sf{\srm Lie}(H)\n\send{eqnarray}\nThe Ehresmann H-connection form, $\sf{{\scal A}_H}(x,y)$, over $E_H$ is an [[Ehresmann principal bundle connection]] form for the bundle.\n\nWhen the Ehresmann Cartan H-connection form equals the [[Maurer-Cartan form]], $\sf{{\scal C}_H} = \sf{\scal I}$, 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 $\sf{{\scal C}_H}$ deviating from $\sf{\scal I}$ to give the new [[frame]] 1-forms, $\sf{{\scal C}_H^J} = \sf{E^J}$, of the [[Cartan tangent bundle geometry]] over what was $G$.

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, $H \ssubset G$, usually assumed to be [[reductive]] in $G$. The wavy ''Cartan geometry base manifold'', $M$, is ''modeled'' on the [[homogeneous space]], $M \ssim S=G/H$, and has the same dimension, $n_S = (n_G - n_H)$. The ''Cartan connection'' over $M$,\n$$\n\sf{C}(x) = \sf{e} + \sf{A} \sin \sf{\srm Lie}(G)\n$$\nis a [[Lieform]] modeled on the [[Maurer-Cartan frame|homogeneous space]], $\sf{C} \ssim \sf{I} = r^- \sf{d} r(x)$, and splits (for $H$ reductive in $G$) into the ''Cartan frame'', $\sf{e}(x) = \sf{e^A} K_A \sin \sf{\srm Lie}(G/H)$, and ''Cartan H-connection'', $\sf{A}(x) = \sf{A^P} H_P \sin \sf{\srm Lie}(H)$, which (unlike their homogeneous space counterparts) may vary freely. \n\nThe ''Cartan [[curvature]]'' of the connection is\n\sbegin{eqnarray}\n\sff{F}(x) &=& \sf{d} \sf{C} + \sha \slb \sf{C}, \sf{C} \srb \s\s\n&=& \sf{d} \sf{e} + \sf{d} \sf{A} + \sha \slb \sf{e}, \sf{e} \srb + \slb \sf{A}, \sf{e} \srb + \sha \slb \sf{A}, \sf{A} \srb \s\s\n&=& \sff{F^A} K_A + \sff{F^P} H_P \n\send{eqnarray}\nwhich (like the [[homogeneous space curvature|homogeneous space tangent bundle geometry]]) splits into\n\sbegin{eqnarray}\n\sff{F^A} &=& \sf{d} \sf{e^A} + \sf{A^P} \sf{e^B} C_{PB}{}^A + \sha \sf{e^C} \sf{e^B} C_{CB}{}^A \s\s \n\sff{F^P} &=& \sff{F_H^P} + \sha \sf{e^C} \sf{e^D} C_{CD}{}^P\n\send{eqnarray}\nwith the ''curvature of the Cartan H-connection'' defined by:\n$$\n\sff{F_H^P} = \sf{d} \sf{A^P} + \sha \sf{A^Q} \sf{A^R} C_{QR}{}^P\n$$\nNote that $\sha \sf{e^C} \sf{e^B} C_{CB}{}^A = 0$ if $G/H$ is a [[symmetric space]].\n\nThere 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]].\n\nRefs:\n*http://en.wikipedia.org/wiki/Cartan_connection\n*[[Differential Geometry of Cartan Connections|papers/9412232.pdf]]\n**by [[Peter Michor]] and Alekseevsky (note $G \ssubset H$)\n*[[The Works of Charles Ehresmann on Connections: From Cartan Connections to Connections on Fibre Bundles|papers/CMMarle.pdf]]\n**Ehresmann version of Cartan, nicely explained. See p9 for main def.\n**http://www.math.jussieu.fr/~marle/\n*[[MacDowell-Mansouri Gravity and Cartan Geometry|papers/0611154.pdf]]\n**a new paper by Derek Wise\n*[[Natural Operations on the Bundle of Cartan Connections|papers/Natural Operations on the Bundle of Cartan Connections.pdf]]\n*[[The Existance of Cartan Connections and Geometrizable Principal Bundles|papers/0206136.pdf]]\n**a very concise and interesting mathematical treatment.\n

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 $n_M \sneq n_S$ 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.\n\nThe 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]], $\ssi^S : M \sto E_S$, of the Cartan homogeneous space bundle determines the ''points of tangency'' -- the points, $x_{s\ssi}(x)$, 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'', $\ssi_0^S(x) = (x,0)$.\n\nThe 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]],\n$$\n\ssi'^H(x,x_s) = (x,x_s,y_\ssi(x))\n$$\nThe [[Ehresmann Cartan connection]] form [[pulls back|pullback]] along this map to give the ''Cartan homogeneous space connection form'' over $E_S$,\n\sbegin{eqnarray}\n\sf{{\scal C}_S}(x,x_s) &=& \ssi'^{H*} \sf{\scal C} = \slp \sf{C^J}(x) L^I{}_J(x_s,y_\ssi(x)) + \ssi'^{H*} \sf{\sxi_R^I}(x_s,y_\ssi(x)) \srp T_I \s\s\n&=& g^-(x_s,y_\ssi(x)) \s, \sf{C}(x) \s, g(x_s,y_\ssi(x)) + g^-(x_s,y_\ssi(x)) \s, \sf{d} \s, g(x_s,y_\ssi(x)) \s\s\n&=& h^-(y_\ssi(x)) \sBig( r^-(x_s) \s, \sf{C}(x) \s, r(x_s) + r^-(x_s) \s, \sf{d} \s, r(x_s) \sBig) h(y_\ssi(x)) + h^-(y_\ssi(x)) \s, \sf{d} h(y_\ssi(x)) \n\send{eqnarray}\nin which the [[coset representative section|homogeneous space]], $r:S \sto G$, is used to write $g(x_s,y)=r(x_s) \s, h(y)$. Choosing the canonical H-bundle reference section, $h(y_{\ssi_0}(x)) = h(0) = 1$, this gives\n$$\n\sf{{\scal C}_S}(x,x_s) = r^-(x_s) \s, \sf{C}(x) \s, r(x_s) + r^-(x_s) \s, \sf{d} \s, r(x_s)\n$$\nPulling this back along the zero point reference section gives the [[Cartan connection|Cartan geometry]], $\ssi_0^{S*} \sf{{\scal C}_S} = \sf{C}$, over $M$.

<<note HideTags>>Mutually [[commuting|commutator]] set of $r$ [[Lie algebra]] generators:\n$$\n\sleft\s{ T_1, T_2, ..., T_r \sright\s} \s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; [ T_i, T_j ] = 0\n$$\n[[Cartan subalgebra|Lie algebra structure]]: $\s;\s;\s; C=c^i T_i \s;\s; \sin {\srm Lie(G)} \sp{{}_{(}}$\n[[Eigenvalues|eigen]], $\sal^a$, and [[eigenvectors|eigen]], $V_a \sin {\srm Lie(G)}$, using the Lie bracket:\n$$\n[ C , V_a ] = \sal^a V_a = \ssum_i c^i \sal_i^a V_a\n$$\nUnique eigenvalue for each of the $(n-r)$ eigenvectors, corresponding to $(n-r)$ ''roots'', $\sal_i^a$, in $r$ dimensional vector space.\n\nCartan subalgebra of the standard model and gravity:\n$$\nC = {\sscriptsize \sfrac{1}{2}} \som^{01} \sga_{01} + {\sscriptsize \sfrac{1}{2}} \som^{12} \sga_{12} + W^3 i \sSi_3 + B i Y + G^3 i \sla_3 + G^8 i \sla_8 \n$$\nEigenvectors are elementary particles, roots are their charges:\n$$\n\sal(e_L) = ( \spm {\sscriptsize \sfrac{1}{2}}, \smp {\sscriptsize \sfrac{1}{2}}, -1, -1, 0, 0 ) \sp{{}_{\sBig(}}\n$$

The [[Riemann curvature]] for the [[Cartan tangent bundle geometry]] is calculated from the [[Cartan tangent bundle spin connection]],\n$$\n\sff{R}^J{}_I = \sf{d} \sf{W}^J{}_I + \sf{W}^J{}_K \sf{W}^K{}_I\n$$\nWe'll tackle this in pieces. Using a [[left-right rotator]] identity,\n$$\n\sf{d} \slp L^h \srp^J{}_I = \sf{e_H^P} C_P{}^J{}_K \slp L^h \srp^K{}_I\n$$\nthe [[exterior derivative]]s are:\n\sbegin{eqnarray}\n\sf{d} \sf{W}^B{}_A &=& \sf{d} \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E - \sf{e_H^P} C_P{}^B{}_A \srp \s\s\n&=& \slp \sf{d} \sf{\snu}^E{}_F \srp \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \sf{e_H^P} C_P{}^B{}_D \slp L^h \srp^D{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \sf{e_H^P} C_P{}^F{}_C \slp L^h \srp^C{}_A \s\s\n&-& \sha \slp \sf{d} \sf{A^Q} + \slp \sf{d} \sf{e_H^P} \srp \slp L^h \srp_P{}^Q - \sf{e_H^P} \sf{e_H^R} C_{RP}{}^T \slp L^h \srp_T{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E \s\s\n&+& \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp\n\slp \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \sf{e_H^R} C_R{}^B{}_C \slp L^h \srp^{CD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \slp L^h \srp^{BD} \sf{e_H^R} C_R{}_{AC} \slp L^h \srp^{CE} \srp \s\s\n&-& \sf{d} \sf{e_H^P} C_P{}^B{}_A \s\s\n\s\s\n\sf{d} \sf{W}^B{}_R &=& \sf{d} \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \srp \s\s\n&=& \sha \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \sf{e_H^P} C_P{}^B{}_C \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \sf{e_H^P} C_{PRS} \slp L^h \srp^{SQ} \s\s\n\s\s\n\sf{d} \sf{W}^Q{}_R &=& - \sha \sf{d} \slp \sf{A^S} \s, \slp L^h \srp^P{}_S + \sf{e_H^P} \srp C_P{}^Q{}_R \s\s\n&=& - \sha \slp \slp \sf{d} \sf{A^S} \srp \slp L^h \srp^P{}_S\n- \sf{A^S} \sf{e_H^U} C_U{}^P{}_T \slp L^h \srp^T{}_S\n+ \sf{d} \sf{e_H^P} \srp C_P{}^Q{}_R \n\send{eqnarray}\nThe pieces quadratic in the spin connection are:\n\sbegin{eqnarray}\n\sf{W}^B{}_K \sf{W}^K{}_A &=& \sf{W}^B{}_C \sf{W}^C{}_A + \sf{W}^B{}_P \sf{W}^P{}_A \s\s\n&=&\n\slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp \s\s\n&\stimes&\n\slp \sf{\snu}^G{}_H \slp L^h \srp^C{}_G \slp L^h \srp_A{}^H - \sha \slp \sf{A^R} + \sf{e_H^S} \slp L^h \srp_S{}^R \srp F^H_{HGR} \slp L^h \srp^{CH} \slp L^h \srp_A{}^G - \sf{e_H^Q} C_Q{}^C{}_A \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n\slp \sha \sf{e^C} F^H_{CFR} \slp L^h \srp_A{}^F \slp L^h \srp^{PR} \srp \s\s\n\s\s\n\sf{W}^B{}_K \sf{W}^K{}_R &=& \sf{W}^B{}_C \sf{W}^C{}_R + \sf{W}^B{}_P \sf{W}^P{}_R \s\s\n&=& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp\n\slp \sha \sf{e^G} F^H_{GHS} \slp L^h \srp^{CH} \slp L^h \srp_R{}^S \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n \sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^P{}_R \s\s\n\s\s\n\sf{W}^Q{}_K \sf{W}^K{}_R &=& \sf{W}^Q{}_C \sf{W}^C{}_R + \sf{W}^Q{}_P \sf{W}^P{}_R \s\s\n&=& - \slp \sha \sf{e^A} F^H_{AFR} \slp L^h \srp_C{}^F \slp L^h \srp^{QR} \srp\n\slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \srp \s\s\n&+&\n\sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^Q{}_P\n\sha \slp \sf{A^U} \s, \slp L^h \srp^V{}_U + \sf{e_H^V} \srp C_V{}^P{}_R\n\send{eqnarray}\nCombining these gives the curvature,\n\sbegin{eqnarray}\n\sff{R}^B{}_A &=& \slp \sf{d} \sf{\snu}^E{}_F \srp \slp L^h \srp^B{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \sf{e_H^P} C_P{}^B{}_D \slp L^h \srp^D{}_E \slp L^h \srp_A{}^F - \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \sf{e_H^P} C_P{}^F{}_C \slp L^h \srp^C{}_A \s\s\n&-& \sha \slp \sf{d} \sf{A^Q} + \slp \sf{d} \sf{e_H^P} \srp \slp L^h \srp_P{}^Q - \sf{e_H^P} \sf{e_H^R} C_{RP}{}^T \slp L^h \srp_T{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_A{}^E \s\s\n&+& \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp\n\slp \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \sf{e_H^R} C_R{}^B{}_C \slp L^h \srp^{CD} \slp L^h \srp_A{}^E\n+ F^H_{DEQ} \slp L^h \srp^{BD} \sf{e_H^R} C_R{}_{AC} \slp L^h \srp^{CE} \srp \s\s\n&-& \sf{d} \sf{e_H^P} C_P{}^B{}_A \s\s\n&+& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp \s\s\n&\stimes&\n\slp \sf{\snu}^G{}_H \slp L^h \srp^C{}_G \slp L^h \srp_A{}^H - \sha \slp \sf{A^R} + \sf{e_H^S} \slp L^h \srp_S{}^R \srp F^H_{HGR} \slp L^h \srp^{CH} \slp L^h \srp_A{}^G - \sf{e_H^Q} C_Q{}^C{}_A \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n\slp \sha \sf{e^C} F^H_{CFR} \slp L^h \srp_A{}^F \slp L^h \srp^{PR} \srp \s\s\n\s\s\n\sff{R}^B{}_R &=& \sha \slp \sf{d} F^H_{DEQ} \srp \slp L^h \srp^{BE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \sf{e_H^P} C_P{}^B{}_C \slp L^h \srp^{CE} \slp L^h \srp_R{}^Q \n- \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \sf{e_H^P} C_{PRS} \slp L^h \srp^{SQ} \s\s\n&+& \slp \sf{\snu}^E{}_F \slp L^h \srp^B{}_E \slp L^h \srp_C{}^F - \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp^{BD} \slp L^h \srp_C{}^E - \sf{e_H^P} C_P{}^B{}_C \srp\n\slp \sha \sf{e^G} F^H_{GHS} \slp L^h \srp^{CH} \slp L^h \srp_R{}^S \srp \s\s\n&-& \slp \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp^{BE} \slp L^h \srp_P{}^Q \srp\n \sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^P{}_R \s\s\n\s\s\n\sff{R}^Q{}_R &=& - \sha \slp \slp \sf{d} \sf{A^S} \srp \slp L^h \srp^P{}_S\n- \sf{A^S} \sf{e_H^U} C_U{}^P{}_T \slp L^h \srp^T{}_S\n+ \sf{d} \sf{e_H^P} \srp C_P{}^Q{}_R \s\s\n&+&\n\sha \slp \sf{A^S} \s, \slp L^h \srp^T{}_S + \sf{e_H^T} \srp C_T{}^Q{}_P\n\sha \slp \sf{A^U} \s, \slp L^h \srp^V{}_U + \sf{e_H^V} \srp C_V{}^P{}_R\n\send{eqnarray}\n\nFrom this ugly mess, the [[Ricci curvature]], $\sf{R}{}_I = \sve{E_J} \sff{R}^J{}_I$, is\n\sbegin{eqnarray}\n\sf{R}{}_B &=& \sve{E_A} \sff{R}^A{}_B + \sve{E_R} \sff{R}^R{}_B \s\s\n\sf{R}{}_R &=& \sve{E_B} \sff{R}^B{}_R + \sve{E_Q} \sff{R}^Q{}_R \s\s\n\send{eqnarray}\n\n//ack, I've got to work on something else for awhile.//\n\n[[curvature scalar]]\n\nCheck that it matches [[reductive Lie group tangent bundle geometry]] as special case.\n

A ''Cartan tangent bundle geometry'' is a [[reductive Lie group tangent bundle geometry]] that has gone a little wavy. The [[frame]] 1-forms, $\sf{E^J}$, over what was the Lie group manifold, $E_H \ssim G$, split in adapted coordinates as\n\sbegin{eqnarray}\n\sf{E^A}(x,y) &=& \sf{e^B}(x) \s, \slp L^h\srp^A{}_B(y) \s\s\n\sf{E^P}(x,y) &=& \sf{A^Q}(x) \s, \slp L^h \srp^P{}_Q(y) + \sf{e_H^P}(y)\n\send{eqnarray}\nin which $\sf{e^B}$ and $\sf{A^Q}$ are the [[Cartan frame|Cartan geometry]] forms and [[Cartan H-connection|Cartan geometry]] forms, $\slp L^h \srp^A{}_B = \slp H^A, h^- H_B h(y) \srp$ is the [[left-right rotator]] over $H$, and $\sf{e_H^P}$ are the frame 1-forms over $H$. These frame 1-forms are components of the Ehresmann Cartan H-connection form, $\sf{E^J} = \sf{{\scal C}_H^J}$, over the [[Cartan H-bundle]], $E_H$, with $\sf{E^A} = \sf{{\scal E}_H^A}$ and $\sf{E^P} = \sf{{\scal A}_H^P}$. 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]], $\sf{W}{}^J{}_K$, 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]],\n$$\n\sff{T^J} = 0 = \sf{d} \sf{E^J} + \sf{W}{}^J{}_K \sf{E^K}\n$$\nwhich may be solved for the ''Cartan tangent bundle spin connection'', $\sf{W}{}^J{}_K$. To construct the solution, we first compute the [[exterior derivative]] of the [[frame]] 1-forms,\n\sbegin{eqnarray}\n\sf{d} \sf{E^A} &=& \slp \sf{d} \sf{e^B} \srp \slp L^h\srp^A{}_B - \sf{e^B} \sf{d} \slp L^h\srp^A{}_B \s\s\n&=& \slp \sf{d} \sf{e^B} \srp \slp L^h\srp^A{}_B - \sf{e^B} \sf{e_H^P} \slp L^h \srp^D{}_B C_{DP}{}^A \s\s\n\sf{d} \sf{E^P} &=& \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp^P{}_Q - \sf{A^Q} \sf{d} \slp L^h \srp^P{}_Q + \sf{d} \sf{e_H^P} \s\s\n&=& \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp^P{}_Q - \sf{A^Q} \sf{e_H^R} \slp L^h \srp^T{}_Q C_{TR}{}^P - \sha \sf{e_H^Q} \sf{e_H^R} C_{QR}{}^P\n\send{eqnarray}\nusing the [[left-right rotator]], $\slp L^h\srp^I{}_J = \slp T^I, h^- T_J h \srp$, and the [[Maurer-Cartan equation|Maurer-Cartan form]] over $H$. Next we write down the [[orthonormal basis vectors|frame]] (satisfying $\sve{E_J} \sf{E^K} = \sde_J^K$),\n\sbegin{eqnarray}\n\sve{E_A}(x,y) &=& \slp L^h \srp_A{}^B \s, \sve{e_B}(x) - \slp L^h \srp_A{}^C \slp \sve{e_C} \sf{A^Q} \srp \slp L^h \srp^P{}_Q \s, \sve{e^H_P} \s\s\n\sve{E_P}(x,y) &=& \sve{e^H_P}(y)\n\send{eqnarray}\nand compute the [[anholonomy|Cartan's equation]] coefficients, $f_{IJ}{}^K = \sve{E_J} \sve{E_I} \slp \sf{d} \sf{E^K} \srp$, getting\n\sbegin{eqnarray}\nf_{AB}{}^C &=& \sve{e_E} \sve{e_D} \slp \sf{d} \sf{e^F} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^C{}_F \n+ 2 \slp - \slp L^h \srp_{\slb B \srd}{}^E \s, \sve{e_E} \slp L^h \srp_{\sld A \srb}{}^G \slp \sve{e_G} \sf{A^Q} \srp \slp L^h \srp^R{}_Q \s, \sve{e^H_R} \srp \slp - \sf{e^F} \sf{e_H^P} \slp L^h \srp^D{}_F C_{DP}{}^C \srp \s\s\n&=& f^M_{DE}{}^F \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp^C{}_F \n- 2 \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^R{}_Q \slp L^h \srp_{\slb A \srd}{}^E \s, C_{\sld B \srb R}{}^C \s\s\nf_{AQ}{}^C &=& - C_{AQ}{}^C \s\s\nf_{AB}{}^R &=& \sve{e_E} \sve{e_D} \slp \sf{d} \sf{A^Q} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^R{}_Q\n- \slp L^h \srp_A{}^E \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^P{}_Q \slp L^h \srp_B{}^D \slp \sve{e_D} \sf{A^S} \srp \slp L^h \srp^T{}_S C_{TP}{}^R \s\s\n&=& \sve{e_E} \sve{e_D} \slp \sff{F_H^Q} \srp \slp L^h \srp_B{}^E \slp L^h \srp_A{}^D \slp L^h \srp^R{}_Q \s\s\n&=& F^H_{DE}{}^Q \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp^R{}_Q \s\s\nf_{AQ}{}^R &=& 0 \s\s\nf_{PQ}{}^C &=& 0 \s\s\nf_{PQ}{}^R &=& - C_{PQ}{}^R\n\send{eqnarray}\nin which $f^M_{DE}{}^F(x)$ is the anholonomy for $\sf{e^A}$ and $\sff{F_H^Q}(x) = \sf{d} \sf{A^Q} + \sha \sf{A^P} \sf{A^R} C_{PR}{}^Q$ is the [[curvature]] for $\sf{A^Q}$. Using these, the solution to Cartan's equation, $W_{IJK} = \sha \slp f_{IJK} - f_{JKI} + f_{KIJ} \srp$, gives the Cartan tangent bundle spin connection coefficients,\n\sbegin{eqnarray}\nW_{ABC} &=& \snu_{DEF} \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F - \slp L^h \srp_A{}^E \slp \sve{e_E} \sf{A^Q} \srp \slp L^h \srp^R{}_Q C_{B R C} \s\s\nW_{ABR} &=& \sha f_{ABR} = \sha F^H_{DEQ} \slp L^h \srp_A{}^D \slp L^h \srp_B{}^E \slp L^h \srp_R{}^Q \s\s\nW_{AQR} &=& 0 \s\s\nW_{PBC} &=& - C_{PBC} - \sha f_{BCP} = - C_{PBC} - \sha F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E \slp L^h \srp_P{}^Q \s\s\nW_{PQC} &=& 0 \s\s\nW_{PQR} &=& - \sha C_{PQR}\n\send{eqnarray}\nwith $\snu_{DEF}(x)$ the coefficients of the torsionless spin connection for $\sf{e^A}$. From these, the ''Cartan tangent bundle spin connection'', $\sf{W}{}_{JK} = \sf{E^I} W_{IJK}$, is\n\sbegin{eqnarray}\n\sf{W}{}_{BC} &=& \sf{E^A} W_{ABC} + \sf{E^P} W_{PBC} \s\s\n&=& \sf{e^D} \slp \snu_{DEF} \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F - \slp \sve{e_D} \sf{A^Q} \srp \slp L^h \srp^R{}_Q C_{B R C} \srp \n+ \slp \sf{A^R} \s, \slp L^h \srp^P{}_R + \sf{e_H^P} \srp \slp - C_{PBC} - \sha F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E \slp L^h \srp_P{}^Q \srp \s\s\n&=& \sf{\snu}{}_{EF} \slp L^h \srp_B{}^E \slp L^h \srp_C{}^F \n- \sha \slp \sf{A^Q} + \sf{e_H^P} \slp L^h \srp_P{}^Q \srp F^H_{DEQ} \slp L^h \srp_B{}^D \slp L^h \srp_C{}^E - \sf{e_H^P} C_{PBC} \s\s\n\sf{W}{}_{BR} &=& \sf{E^A} W_{ABR} = \sha \sf{e^D} F^H_{DEQ} \slp L^h \srp_B{}^E \slp L^h \srp_R{}^Q \s\s\n\sf{W}{}_{QR} &=& \sf{E^P} W_{PQR} = - \sha \slp \sf{A^S} \s, \slp L^h \srp^P{}_S + \sf{e_H^P} \srp C_{PQR}\n\send{eqnarray}\nThis 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,\n$$\n0 = \sf{d} \sf{e} + \sf{\som} \stimes \sf{e} \n$$\nor, equivalently,\n$$\n0 = \sf{d} \sf{e^\sal} + \sf{\som}^\sal{}_\sbe \sf{e^\sbe} \n$$\nThis equation may be solved in closed form for the spin or [[tangent bundle connection]] coefficients. To generalize, lets solve\n$$\n\sf{\som} \stimes \sf{e} = -\sff{f}\n$$\nfor $\sf{\som}$ in terms of the frame and an arbitrary Clifford vector valued 2-form, $\sff{f}$. In components, using the [[index bracket]], this is\n$$\n\som_{\slb i \srd}{}^{\sal \sbe} \slp e_{\sld j \srb} \srp_\sbe = -\sha f_{ij}{}^\sal\n$$\nUsing 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\n$$\n\som_{\slb \sbe \sga \srb \sal} = \sha f_{\sbe \sga \sal}\n$$\nBy once again juggling spin connection indices, we see from this that\n$$\n\som_{\slp \sbe \sga \srp \sal} = \som_{\slb \sal \sbe \srb \sga} + \som_{\slb \sal \sga \srb \sbe} = f_{\sal \slp \sbe \sga \srp}\n$$\nAdding these last two expressions gives the explicit solution for the spin connection coefficients:\n$$\n\som_{\sal \sbe \sga} = \som_{\slb \sal \sbe \srb \sga} + \som_{\slp \sal \sbe \srp \sga} = \sha \slp f_{\sal \sbe \sga} - f_{\sbe \sga \sal} + f_{\sga \sal \sbe} \srp\n$$\nPutting these indices in their more familiar positions is done using the frame and [[Minkowski metric]]: $\som_{i}{}^{\sde \sep} = \slp e_i \srp^\sal \set^{\sde \sbe} \set^{\sep \sga} \som_{\sal \sbe \sga}$. Cartan's equation is solved by simply plugging $\sff{f}=\sf{d} \sf{e}$ into the above equation -- in coefficients,\n$$\nf_{\sal \sbe \sga} = \slp e_\sal \srp^i \slp e_\sbe \srp^j \set_{\sga \sde} \slp \spa_i \slp e_j \srp^\sde - \spa_j \slp e_i \srp^\sde \srp\n= \sve{e_\sbe} \sve{e_\sal} \slp \sf{d} \sf{e}{}_\sga \srp\n$$\n\nNote that this last tensor, the ''anholonomy'', may also be expressed using the [[Lie bracket|Lie derivative]] of the orthonormal basis vectors,\n$$\n\slb \sve{e_\sal} , \sve{e_\sbe} \srb_L = 2 \slb \slp e_{\slb \sal \srd} \srp^j \spa_j \slp e_{\sld \sbe \srb} \srp^i \srb \sve{\spa_i}\n= - 2 \slb \slp e_{\slb \sal \srd} \srp^j \slp e_{\sld \sbe \srb} \srp^k \slp e_\sga \srp^i \spa_j \slp e_k \srp^\sga \srb \sve{\spa_i}\n= - f_{\sal \sbe}{}^\sga \sve{e_\sga} \n$$\n\nIt is possible to express the solution to Cartan's equation in a particularly pretty, index free way using [[Clifform algebra]]:\n$$\n\sf{\som} = - \sve{e} \stimes \sff{f} + \sfr{1}{4} \slp \sve{e} \stimes \sve{e} \srp \slp \sf{e} \scdot \sff{f} \srp\n$$\n(A cute, if not particularly useful expression.)

[[Lie algebra]]\nhttp://en.wikipedia.org/wiki/Casimir_invariant\n\nrelated to [[Laplacian]]\n

For a $2 \stimes 2$ square matrix or $2$ or $3$ dimensional [[Clifford algebra]] element, $A$, using the [[trace]] and products gives\n$$\n0 = A^2 - \sli A \sri A + \sha \slp \sli A \sri^2 - \sli A^2 \sri \srp\n$$\nFor a $3 \stimes 3$ square matrix, $A$,\n$$\n0 = A^3 - \sli A \sri A^2 + \sha \slp \sli A \sri^2 - \sli A^2 \sri \srp A - \sfr{1}{6} \slp \sli A \sri^3 - 3 \sli A^2 \sri \sli A \sri + 2 \sli A^3 \sri \srp\n$$\nThis generalizes to formula for larger matrices,\nhttp://arxiv.org/hep-th/0701116

A ''Chern-Simons form'', $\snf{\som_p}$, is a grade $p$ [[differential form]] defined (for odd $p$) to satisfy\n$$\n\sf{d} \snf{\som_p} = Tr\slp \sff{F}^{\sfr{p+1}{2}} \srp\n$$\nin which $\sff{F}=\sf{d} \sf{A} + \sf{A} \sf{A}$ is the [[curvature]] for some [[principal bundle]] [[connection]], $\sf{A}$. The first few Chern-Simons forms are\n\sbegin{eqnarray}\n\sf{\som_1} &=& Tr\slp \sf{A} \srp \s\s\n\snf{\som_3} &=& Tr\slp \sff{F} \sf{A} - \sfr{1}{3} \sf{A} \sf{A} \sf{A} \srp \s\s\n\snf{\som_5} &=& Tr\slp \sff{F} \sff{F} \sf{A} - \sfr{1}{2} \sff{F} \sf{A} \sf{A} \sf{A} + \sfr{1}{10} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \srp \s\s\n\snf{\som_7} &=& Tr\slp \sff{F} \sff{F} \sff{F} \sf{A} + ? \sff{F} \sff{F} \sf{A} \sf{A} \sf{A} + ? \sff{F} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} + ? \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \sf{A} \srp\n\send{eqnarray}\n\nThe [[integral|integration]] of a Chern-Simons p-form over a $p$ dimensional [[manifold]] is a homotopy invariant called the ''Chern number'',\n$$\nc_p = \sint \snf{\som_p}\n$$\ncorresponding to the topology of the manifold. For a $(p+1)$ dimensional manifold with a boundary,\n$$\n\sint Tr\slp \sff{F}^{\sfr{p+1}{2}} \srp = \sint \sf{d} \snf{\som_p} = \sint_\spa \snf{\som_p} = c_p \n$$\n\nAlso of potential interest is the relationship to the ''Pfaffian'',\n$$\n\sff{F}^{\sfr{p+1}{2}} = Pf\slp F \srp \snf{d^{p+1}x}\n$$\nwhere $Pf(F) = \ssqrt{\sll F \srl}$\nref:\nhttp://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\n$$\nD_{i}T^{k}{}_{j}=\spartial_{i}T^{k}{}_{j}+\sGamma^{k}{}_{im}T^{m}{}_{j}-\sGamma^{m}{}_{ij}T^{k}{}_{m}\n$$\nwith the ''Christoffel symbols'', $\sGa^k{}_{ij}$, defined as tangent bundle connection coefficients, \n$$\n\sna_i \sve{\spa_j} = \sGa^k{}_{ij} \sve{\spa_k} \n$$\nThe Christoffel symbols are determined from the assumptions that the [[torsion]] vanishes,\n$$\n\sGa^k{}_{\slb ij \srb} = 0\n$$\nand that the covariant derivative is ''[[metric]] compatible'',\n$$\n0 = D_i g_{jk} = \spa_i g_{jk} - \sGamma^{m}{}_{ij} g_{mk} - \sGamma^{m}{}_{ik} g_{jm}\n$$\nIt is then computed explicitly in terms of the metric, metric inverse, and its partial derivatives as\n$$\n\sGa^k{}_{ij} = \sha g^{km} \slp \spa_j g_{mi} + \spa_i g_{jm} - \spa_m g_{ij} \srp\n$$\nComputing the Christoffel symbols from vanishing torsion and metric compatibility is equivalent to calculating the [[spin connection]] from [[Cartan's equation]].\n\nThe 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]],\n$$\n\sna_i \sve{e_\sal} = \slp \spa_i \slp e_\sal \srp^j + \slp e_\sal \srp^k \sGa^j{}_{ik} \srp \sve{\spa_j} = w_{i}{}^\sbe{}_\sal \sve{e_\sbe}\n= w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j \sve{\spa_j}\n$$\nto get\n$$\n\sGa^j{}_{ik} = \slp e_k \srp^\sal \slp w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j - \spa_i \slp e_\sal \srp^j \srp = \slp e_\sbe \srp^j \slp w_{i}{}^\sbe{}_\sal \slp e_k \srp^\sal + \spa_i \slp e_k \srp^\sbe \srp\n$$\nThis last expression may be used to easily determine how the Christoffel symbols, which do not constitute a tensor, transform under a [[coordinate change]] to\n\sbegin{eqnarray}\n\sGa^n{}_{ml} &=& \sfr{\spa x^k}{\spa y^l} \slp e_k \srp^\sal \sfr{\spa x^i}{\spa y^m} \slp w_{i}{}^\sbe{}_\sal \slp e_\sbe \srp^j \sfr{\spa y^n}{\spa x^j} - \spa_i \slp \slp e_\sal \srp^j \sfr{\spa y^n}{\spa x^j} \srp \srp \s\s\n&=& \sfr{\spa y^n}{\spa x^j} \sfr{\spa x^i}{\spa y^m} \sfr{\spa x^k}{\spa y^l} \sGa^j{}_{ik} - \sfr{\spa x^i}{\spa y^m} \sfr{\spa x^j}{\spa y^l} \sfr{\spa^2 y^n}{\spa x^i \spa x^j} \n\send{eqnarray}\nFrom 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, $\sGa^k{}_{\slb ij \srb} = 0$. It is sometimes argued, along the lines of the [[equivalence principle|frame]], that such a choice should be possible and hence torsion should vanish. \n\nUsing the Christoffel symbols is quite old fashioned, but sometimes practical. Things may be fancied up a bit by defining the ''Christoffel 1-form''s, $\sf{\sGa^k{}_j} = \sf{dx^i} \sGa^k{}_{ij}$, and using the [[vector-form algebra]] and [[partial derivative]] to get $\sf{\sna} \sve{\spa_j} = \sf{\sGa^k{}_j} \sve{\spa_k}$ and\n$$\n\sf{\sna} \sve{e_\sal} = \sf{\sna} \slp e_\sal \srp^k \sve{\spa_k} = \sf{\spa} \sve{e_\sal} + \slp e_\sal \srp^k \sf{\sGa^j{}_k} \sve{\spa_j} = \sf{\som^\sbe{}_\sal} \sve{e_\sbe}\n$$\n

The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(1,3), is built from 4 anti-commuting, [[Clifford basis vectors]], $\sga_\smu$, with positive time [[Minkowski norm|Minkowski metric]],\n$$\n\sga_\smu \scdot \sga_\snu = \sha \slp \sga_\smu \sga_\snu + \sga_\snu \sga_\smu \srp = \set_{\smu \snu}\n$$\nThe full algebra has $2^4 = 16$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sga_\smu$ | $1$ | $4$ |vector |\n| $\sga_{\smu \snu}$ | $2$ | $6$ |bivector |\n| $\sga_{\smu \snu \ska } = \sfr{1}{3!} \sep_{\smu \snu \ska \sla} \sga^\sla \sga $ | $3$ | $4$ |trivector |\n| $\sga_{\smu \snu \ska \sla} = \sep_{\smu \snu \ska \sla} \sga$ | 4 | $1$ |4-vector, pseudoscalar |\nThe ''spacetime [[pseudoscalar]]'', $\sga = \sga_0 \sga_1 \sga_2 \sga_3$, satisfies $\sga \sga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\stimes4$ 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:\n\sbegin{eqnarray}\n\sga_{01} &=& \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sga_{02} &=& \ssi^P_3 \sotimes \ssi^P_2 \s\s\n\sga_{03} &=& \ssi^P_3 \sotimes \ssi^P_3 \s\s\n\sga_{12} &=& -i 1 \sotimes \ssi^P_3 \s\s\n\sga_{13} &=& +i 1 \sotimes \ssi^P_2 \s\s\n\sga_{23} &=& -i 1 \sotimes \ssi^P_1\n\send{eqnarray}\nAny ''Cl(1,3) bivector'' may thus be represented as\n\sbegin{eqnarray}\nB &=& \sha B^{\smu \snu} \sga_{\smu \snu} =\n\slb \sbegin{array}{cc}\nB_L & 0 \s\s\n0 & B_R\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\nB^{0 \sva} \ssi^P_\sva - i \sha B^{\sva \sze} \sep_{\sva \sze \sta} \ssi^P_\sta & 0 \s\s\n0 & - B^{0 \sva} \ssi^P_\sva - i \sha B^{\sva \sze} \sep_{\sva \sze \sta} \ssi^P_\sta\n\send{array} \srb \s\s\n&=&\n\slb \sbegin{array}{cccc}\nB^{03}- i B^{12} & B^{01}+B^{13}-i B^{02}- i B^{23} & 0 & 0 \s\s\nB^{01}- B^{13}+i B^{02}-i B^{23} & -B^{03}+i B^{12} & 0 & 0 \s\s\n0 & 0 & -B^{03}- i B^{12} & -B^{01}+B^{13}+i B^{02}- i B^{23} \s\s\n0 & 0 & -B^{01}- B^{13}-i B^{02}-i B^{23} & B^{03}+i B^{12}\n\send{array} \srb\n\send{eqnarray}\nin which $B_{L/R}$ are the ''left and right [[chiral]] bivector parts'', projected out by the [[left/right chirality projector]], and $\sep_{\sva \sze \sta}$ is the three dimensional [[permutation symbol]]. These $2\stimes2$ matrices satisfy $B_L^\sdagger = - B_R$, using Hermitian conjugation. 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.

The $n=16$ dimensional [[Clifford algebra]], $Cl(16,0)$, is built from 16 anti-commuting, positive norm [[Clifford basis vectors]], $\sga^{\slp16\srp}_\sal$. The full algebra has $2^{16} = 65,536$ [[Clifford basis elements]]. This algebra has many [[Clifford matrix representation]]s in real or complex $256\stimes256$ matrices. One particularly nice representation, built using the [[Kronecker product]] of [[Cl(8)]] elements, is\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_\sal &=& \sGa_\sal \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp} &=& \sGa \sotimes \sGa_\sal\n\send{eqnarray}\nwith $1 \sle \sal \sle 8$. (Note that this rep is not [[chiral]].) These $16$ ''Cl(16) basis vectors'' may be multiplied to get the $120$ ''Cl(16) basis bivectors'',\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_{\sal \sbe} &=& \sGa_{\sal \sbe} \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp \slp\sbe+8\srp} &=& 1 \sotimes \sGa_{\sal \sbe} \s\s\n\sga^{\slp16\srp}_{\sal \slp\sbe+8\srp} = -\sga^{\slp16\srp}_{\slp\sbe+8\srp \sal} &=& \slp \sGa_\sal \sGa \srp \sotimes \sGa_{\sbe}\n\send{eqnarray}\nThe [[pseudoscalar]] in this rep, $\sga^{\slp16\srp} = \sGa \sotimes \sGa$, satisfies $\sga^{\slp16\srp} \sga^{\slp16\srp} = 1$ and anti-commutes with odd-graded elements.\n\nA 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 $\sGa_8$, and using it to build the ''chiral Cl(16) basis vectors'':\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_\sal &=& \sGa_\sal \sotimes 1 \s\s\n\sga^{\slp16\srp}_8 &=& \sGa_8 \sotimes \sGa \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp} &=& \sGa_8 \sotimes \sGa_\sal \s\s\n\sga^{\slp16\srp}_{(16)} &=& \sGa_8 \sotimes \sGa_8\n\send{eqnarray}\nwith $1 \sle \sal \sle 7$. The pseudoscalar in this rep is\n$$\n\sga^{\slp16\srp} = ( \sGa \sotimes \sGa ) ( 1 \sotimes \sGa ) = \sGa \sotimes 1\n$$\nThe (1st level) chirality projector is $P_{\spm} = \sha \slp 1 \smp \sga^{\slp16\srp} \srp = \sha \slp 1 \smp \sGa \srp \sotimes 1$. The basis vectors may be used to build the ''chiral Cl(16) basis bivectors'',\n\sbegin{eqnarray}\n\sga^{\slp16\srp}_{\sal \sbe} &=& \sGa_{\sal \sbe} \sotimes 1 \s\s\n\sga^{\slp16\srp}_{\sal 8} &=& \sGa_{\sal 8} \sotimes \sGa \s\s\n\sga^{\slp16\srp}_{\sal \slp\sbe+8\srp} &=& \sGa_{\sal 8} \sotimes \sGa_\sbe \s\s\n\sga^{\slp16\srp}_{\sal (16)} &=& \sGa_{\sal 8} \sotimes \sGa_8 \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp \slp\sbe+8\srp} &=& 1 \sotimes \sGa_{\sal \sbe} \s\s\n\sga^{\slp16\srp}_{\slp\sal+8\srp (16)} &=& 1 \sotimes \sGa_{\sal 8} \s\s\n\sga^{\slp16\srp}_{8 \slp\sbe+8\srp} &=& 1 \sotimes \sGa \sGa_\sbe \s\s\n\sga^{\slp16\srp}_{8 (16)} &=& 1 \sotimes \sGa \sGa_8\n\send{eqnarray}\n

The three dimensional [[Clifford algebra]], Cl(3,0), is generated by three [[Clifford basis vectors]], $\ssi_\sio$. These basis vectors have a matrix representation as the three [[Pauli matrices]], $\ssi_\sio=\ssi_\sio^P$. 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 product identities]], is (row header times column header equals entry):\n| | !$1$ | !$\ssi_1$ | !$\ssi_2$ | !$\ssi_3$ | !$\ssi_{12}$ | !$\ssi_{13}$ | !$\ssi_{23}$ | !$\ssi$ |\n| !$1$ | $1$ | $\ssi_1$ | $\ssi_2$ | $\ssi_3$ | $\ssi_{12}$ | $\ssi_{13}$ | $\ssi_{23}$ | $\ssi$ |\n| !$\ssi_1$ | $\ssi_1$ | $1$ |bgcolor(#a0ffa0): $\ssi_{12}$ |bgcolor(#a0ffa0): $\ssi_{13}$ | $\ssi_2$ | $\ssi_3$ | $\ssi$ |bgcolor(#a0ffa0): $\ssi_{23}$ |\n| !$\ssi_2$ | $\ssi_2$ |bgcolor(#a0ffa0): $-\ssi_{12}$ | $1$ |bgcolor(#a0ffa0): $\ssi_{23}$ | $-\ssi_1$ | $-\ssi$ | $\ssi_3$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |\n| !$\ssi_3$ | $\ssi_3$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |bgcolor(#a0ffa0): $-\ssi_{23}$ | $1$ | $\ssi$ | $-\ssi_1$ | $-\ssi_2$ |bgcolor(#a0ffa0): $\ssi_{12}$ |\n| !$\ssi_{12}$ | $\ssi_{12}$ | $-\ssi_2$ | $\ssi_1$ | $\ssi$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\ssi_{23}$ |bgcolor(#88ccff): $\ssi_{13}$ | $-\ssi_3$ |\n| !$\ssi_{13}$ | $\ssi_{13}$ | $-\ssi_3$ | $-\ssi$ | $\ssi_1$ |bgcolor(#88ccff): $\ssi_{23}$ |bgcolor(#88ccff): $-1$ |bgcolor(#88ccff): $-\ssi_{12}$ | $\ssi_2$ |\n| !$\ssi_{23}$ | $\ssi_{23}$ | $\ssi$ | $-\ssi_3$ | $\ssi_2$ |bgcolor(#88ccff): $-\ssi_{13}$ |bgcolor(#88ccff): $\ssi_{12}$ |bgcolor(#88ccff): $-1$ | $-\ssi_1$ |\n| !$\ssi$ | $\ssi$ |bgcolor(#a0ffa0): $\ssi_{23}$ |bgcolor(#a0ffa0): $-\ssi_{13}$ |bgcolor(#a0ffa0): $\ssi_{12}$ | $-\ssi_3$ | $\ssi_2$ | $-\ssi_1$ | $-1$ |\nThe blue square shows the bivector subalgebra. This bivector subalgebra is the [[three dimensional special unitary group Lie algebra|su(2)]], with the identification $T_A = \ssi \ssi_A = \sepsilon_{ABC} \ssi_{BC} = i \ssi_A^P$ giving the three $su(2)$ generators,\n$$\n\sbegin{array}{ccc}\nT_1 = i \ssigma_{1}^{P} = \ssi_{23}\n&\nT_2 = i \ssigma_{2}^{P} = -\ssi_{13}\n&\nT_3 = i \ssigma_{3}^{P} = \ssi_{12}\n\send{array}\n$$\nwhich 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]], $\ssi=\ssi_1 \ssi_2 \ssi_3$, squares to $-1$ and has the matrix representation $\ssi = i$.\n\nThe ''three dimensional Clifford algebra of negative signature'', $Cl(0,3)$, is obtained by using $\ssi'_\sio = i \ssi_\sio$ as the basis vectors.

The $n=4$ dimensional ''[[spacetime]] [[Clifford algebra]]'', Cl(3,1), is built from 4 anti-commuting, [[Clifford basis vectors]], $\sga_\smu$, with negative time [[Minkowski norm|Minkowski metric]],\n$$\n\sga_\smu \scdot \sga_\snu = \sha \slp \sga_\smu \sga_\snu + \sga_\snu \sga_\smu \srp = \set_{\smu \snu}\n$$\nThe full algebra has $2^4 = 16$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sga_\smu$ | $1$ | $4$ |vector |\n| $\sga_{\smu \snu}$ | $2$ | $6$ |bivector |\n| $\sga_{\smu \snu \ska } = \sfr{1}{3!} \sep_{\smu \snu \ska \sla} \sga^\sla \sga $ | $3$ | $4$ |trivector |\n| $\sga_{\smu \snu \ska \sla} = \sep_{\smu \snu \ska \sla} \sga$ | 4 | $1$ |4-vector, pseudoscalar |\nThe ''spacetime [[pseudoscalar]]'', $\sga = \sga_0 \sga_1 \sga_2 \sga_3$, satisfies $\sga \sga = -1$ and anti-commutes with odd-graded elements. This algebra has several nice [[Clifford matrix representation]]s in real or complex $4\stimes4$ matrices -- the [[Dirac matrices]].

The $n=8$ dimensional [[Clifford algebra]], ''Cl(5,3)'', is built from 5 positive norm and 3 negative norm [[Clifford basis vectors]], $\sGa_\sal$. It is the same as [[Cl(8)]] except for the signature.\n\nThis algebra has many [[Clifford matrix representation]]s in real or complex $16\stimes16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is\n\sbegin{eqnarray}\n\sGa_1 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_2 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& i \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_0 = \sGa_4 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_8 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\ngiving $\sGa = - i \s, \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0$.\n\nStandard model using this Cl(5,3) rep:\nHas correct Higgs.\nAh, no good -- would give negative [[cosmological constant]].\n$$\n{\sscriptsize\n\slb \sbegin{array}{cccccccc}\n\sha \sf{\som_L} \s!+\s! i \sf{W^3} & i \sf{W^1} \s!+\s! \sf{W^2} & - \sfr{1}{4} \sf{e_R} \sph_0^* & \sfr{1}{4} \sf{e_R} \sph_+ &\n\sud{\snu_L} & \sud{u_L^r} & \sud{u_L^b} & \sud{u_L^g} \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} & \sha \sf{\som_L} \s!-\s! i \sf{W^3} & \sfr{1}{4} \sf{e_R} \sph_+^* & \sfr{1}{4} \sf{e_R} \sph_0 &\n\sud{e_L} & \sud{d_L^r} & \sud{d_L^b} & \sud{d_L^g} \s\s\n\n\sfr{1}{4} \sf{e_L} \sph_0 & -\sfr{1}{4} \sf{e_L} \sph_+ & \sha \sf{\som_R} \s!+\s! i \sf{B} & &\n\sud{\snu_R} & \sud{u_R^r} & \sud{u_R^b} & \sud{u_R^g} \s\s\n\n-\sfr{1}{4} \sf{e_L} \sph_+^* & -\sfr{1}{4} \sf{e_L} \sph_0^* & & \sha \sf{\som_R} \s!-\s! i \sf{B} &\n\sud{e_R} & \sud{d_R^r} & \sud{d_R^b} & \sud{d_R^g} \s\s\n\n& & & & i \sf{B} & & & \s\s\n& & & & & \sfr{-i}{3} \sf{B} \s!+\s! i \sf{G^{3+8}} & i\sf{G^1} \s!-\s! \sf{G^2} & i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n& & & & & i\sf{G^1} \s!+\s! \sf{G^2} & \sfr{-i}{3} \sf{B} \s!-\s! i \sf{G^{3+8}} & i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n& & & & & i\sf{G^4} \s!+\s! \sf{G^5} & i\sf{G^6} \s!+\s! \sf{G^7} & \sfr{-i}{3} \sf{B} \s!-\s!\s! \sfr{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n}\n$$

The $n=8$ dimensional [[Clifford algebra]], $Cl(8,0)$, is built from 8 anti-commuting, positive norm [[Clifford basis vectors]], $\sGa_\sal$. The full algebra has $2^8 = 256$ [[Clifford basis elements]],\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | $1$ |scalar |\n| $\sGa_\sal$ | $1$ | $8$ |vector |\n| $\sGa_{\sal \sbe}$ | $2$ | $28$ |bivector |\n| $\sGa_{\sal \sbe \sga}$ | $3$ | $56$ |trivector |\n| $\sGa_{\sal \sbe \sga \sde}$ | $4$ | $70$ |4-vector |\n| $\sGa_{\sal \sbe \sga \sde \sep}$ | $5$ | $56$ |5-vector |\n| $\sGa_{\sal \sdots \sbe}$ | $6$ | $28$ |6-vector |\n| $\sGa_{\sal \sdots \sbe}$ | $7$ | $8$ |7-vector |\n| $\sGa_{\sal \sdots \sbe} = \sep_{\sal \sdots \sbe} \sGa$ | $8$ | $1$ |8-vector, psuedoscalar |\nThe [[pseudoscalar]], $\sGa$, satisfies $\sGa \sGa = 1$ and anti-commutes with odd-graded elements. This algebra has many [[Clifford matrix representation]]s in real or complex $16\stimes16$ matrices. One particularly nice, [[chiral]], complex representation, built using the [[Kronecker product]] of [[Pauli matrices]], is\n\sbegin{eqnarray}\n\sGa_1 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_2 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_4 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_0 = \sGa_8 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\ngiving $\sGa = \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0$. These may all be expressed in a $16\stimes16$ matrix (using $2\stimes2$ sub-matrices) as\n\sbegin{eqnarray}\n& & \sGa_\spi + z \sGa_4 + a \sGa_5 + b \sGa_6 + c \sGa_7 + \sGa_8 = \s\s\n& & \slb\n\sbegin{array}{cccccccc}\n& & & & 1-i\ssi^p_\spi & & -z-ic & -b-ia \s\s\n& & & & & 1-i\ssi^p_\spi & b-ia & -z+ic \s\s\n& & & & z-ic & -b-ia & 1+i\ssi^p_\spi & \s\s\n& & & & b-ia & z+ic & & 1+i\ssi^p_\spi \s\s\n1+i\ssi^p_\spi & & z+ic & b+ia & & & & \s\s\n& 1+i\ssi^p_\spi & -b+ia & z-ic & & & & \s\s\n-z+ic & b+ia & 1-i\ssi^p_\spi & & & & & \s\s\n-b+ia & -z-ic & & 1-i\ssi^p_\spi & & & &\n\send{array}\n\srb\n\send{eqnarray}\n\nA nice chiral, real representation of $Cl(8,0)$ is\n\sbegin{eqnarray}\n\sGa_1 &=& \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_2 &=& \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_3 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_4 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_5 &=& \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \s\s\n\sGa_6 &=& \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sGa_7 &=& \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sGa_8 &=& \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0\n\send{eqnarray}\n\nThe ''chirality operator for Cl(8)'' is $P^{\slp8\srp}_\spm = \sha \slp 1 \spm \sGa \srp$.

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:\n\sbegin{eqnarray}\n\sGa_{01} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_{02} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_{03} &=& i \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{12} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{13} &=& -i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \s\s\n\sGa_{23} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n&\s,& \s\s\n\sGa_{04} &=& i \ssi^P_3 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \sotimes \ssi^P_0 \s\s\n\sGa_{05} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \s\s\n\sGa_{06} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{07} &=& i \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{14} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \s\s\n\sGa_{15} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \s\s\n\sGa_{16} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sGa_{17} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sGa_{24} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sGa_{25} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sGa_{26} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sGa_{27} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_2 \s\s\n\sGa_{34} &=& -i \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \s\s\n\sGa_{35} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_1 \sotimes \ssi^P_3 \s\s\n\sGa_{36} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \s\s\n\sGa_{37} &=& i \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_3 \sotimes \ssi^P_3 \s\s\n&\s,& \s\s\n\sGa_{45} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_1 \sotimes \ssi^P_0 \s\s\n\sGa_{46} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{47} &=& -i \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{56} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_3 \sotimes \ssi^P_0 \s\s\n\sGa_{57} &=& -i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_2 \sotimes \ssi^P_0 \s\s\n\sGa_{67} &=& i \ssi^P_0 \sotimes \ssi^P_0 \sotimes \ssi^P_1 \sotimes \ssi^P_0\n\send{eqnarray}\n

The ''Clifford adjoint'' transformation of a [[Clifford element]], $A$, by a [[Clifford group]] element, $U$, is\n\s[ A' = U A U^- \s]\nThe ''Clifford inner [[automorphism]]'', a.k.a. //''similarity transformation''//, is the Clifford adjoint transformation of the [[Clifford basis vectors]], \n\s[ \sga'_\sal = U \sga_\sal U^- \s]\nThis subsequently produces the Clifford adjoint transformation of all Clifford elements built from these basis vectors, since\n$$\n\sga'_{\sal \sdots \sbe} = \sga'_\sal \sdots \sga'_\sbe = U \sga_\sal U^- \sdots U \sga_\sbe U^- = U \sga_{\sal \sdots \sbe} U^-\n$$\nIt is an automorphism because it is a map, specified by $U \sin Cl^*$, from the [[Clifford algebra]] into itself.\n\nThe adjoint transformation does not necessarily preserve the [[grade|Clifford grade]] of elements. It does, however, preserve scalars, $\sli A' B' \sri = \sli UAU^- UBU^- \sri = \sli A B \sri$. The [[fundamental Clifford identity|Clifford basis vectors]], $\sga_\sal \scdot \sga_\sbe = \set_{\sal \sbe}$, is preserved by the Clifford automorphism, $\sga'_\sal \scdot \sga'_\sbe = \set_{\sal \sbe}$, preserving the structure of the Clifford algebra and the decomposition of [[Clifford element]]s even though the transformed basis "vectors", $\sga'_\sal$, may no longer be grade 1 with respect to the old basis.\n\nFor Clifford group elements near the identity, $U \ssimeq 1 + \sha C$, the Clifford adjoint is approximately\n\s[ A' = U A U^- \ssimeq \slp 1 + \sha C \srp A \slp 1 - \sha C \srp \ssimeq A + C \stimes A \s]\nwith 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 [[Lie 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]], $\sga_\sal$. 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,\n\sbegin{eqnarray}\nAB &=& A \scdot B + A \stimes B\s\s\nA \scdot B &=& \sha \slp AB+BA \srp\s\s\nA \stimes B &=& \sha \slp AB-BA \srp\n\send{eqnarray}\nThe product is associative and distributive,\n\sbegin{eqnarray}\nA \slp B C \srp = \slp A B \srp C\s\s\nA \slp B + C \srp = A B + A C\n\send{eqnarray}\nAnd, just as for matrices, [[almost all elements|Clifford group]] have an [[inverse]], $AA^-=1$.\n\nA Clifford algebra is a graded "geometric algebra" in that the elements of [[Clifford grade]] $0,1,2,3,\sdots$ 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 product identities]].

Any two [[Clifford basis elements]] are orthogonal under the [[scalar part|Clifford grade]] operator. Taking the scalar part of two multiplied basis elements of grade $r$ gives the orthogonality relation,\n\s[ \sli \sga_{\sal \sdots \sbe} \sga^{\sga \sdots \sde} \sri = r! \sde^\sga_{\slb \sbe \srd} \sdots \sde^\sde_{\sld \sal \srb} \s]\nin 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.\n\nThe 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\n\s[ \sli A B \sri = \sfr{1}{4} A^{\sal \sbe} B_{\sga \sde} \sli \sga_{\sal \sbe} \sga^{\sga \sde} \sri = \sfr{1}{4} A^{\sal \sbe} B_{\sga \sde} 2 \sde^\sga_{\slb \sbe \srd} \sde^\sde_{\sld \sal \srb} = \sha A^{\sal \sbe} B_{\sbe \sal} \s]\n\nThe scalar part operator, and the orthogonality relations, is 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, ${n \schoose q} = \sfr{n!}{q!(n-q)!}$, equal to the number of their ordered combinations,\n| !Element(s) | !Grade | !Multiplicity |!Names |\n| $1$ | $0$ | ${n \schoose 0} = 1$ |scalar, real number |\n| $\sga_\sal$ | $1$ | ${n \schoose 1} = n$ |vector, [[Clifford basis vectors]]|\n| $\sga_{\sal \sbe} = \sga_{\slb \sal \sbe \srb} = \sga_\sal \sga_\sbe = \slb \sga_\sal,\sga_\sbe \srb_A$ | 2 | ${n \schoose 2} = \sha n (n-1)$ |bivector, 2-vector |\n| $\sga_{\sal \sbe \sga} = \sga_{\slb \sal \sbe \sga \srb} = \sga_\sal \sga_\sbe \sga_\sga = \slb \sga_\sal,\sga_\sbe,\sga_\sga \srb_A$ | 3 | ${n \schoose 3} = \sfr{1}{3!} n (n-1)(n-2)$ |trivector, 3-vector |\n| $\svdots$ | $\svdots$ | $\svdots$ |$\svdots$ |\n| $\sga_{\sal \sdots \sbe} = \sep_{\sal \sdots \sbe} \sga$ | $n$ | ${n \schoose n} = 1$ |n-vector |\nEach of these $\ssum_{k=0}^n {n \schoose k} =2^n$ Clifford basis elements is a [[Lie algebra]] generator, with structure coefficients corresponding to [[Clifford basis product identities]]. The Clifford basis elements also satisfy [[Clifford basis element orthogonality]].\n\nUsing [[Clifford dual]]ity, it is often convenient to express high grade basis elements in terms of the Clifford [[pseudoscalar]],\n$$\sga = \sga_0 \sga_1 \sdots \sga_{n-1}$$\nand the [[permutation symbol]]. In this way, the basis r-vectors can be written as\n$$\sga_{\sal \sdots \sbe} = \sfr{1}{\slp n-r \srp!} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \sga$$\nFor example, the $n$ basis (n-1)-vectors are\n$$\sga_{\sal \sdots \sbe} = \sfr{1}{\slp n-1 \srp!} \sep_{\sal \sdots \sbe \sga} \sga^\sga \sga$$\nThis reduces the number of indices necessary to represent high grade [[Clifford element]]s.\n\nThe $2^n$ basis elements can also be written via a generalized index as $\sga_A$, with $A$ enumerating\neach different antisymmetric combination of the usual Clifford indices.

The ''Clifford basis product 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,\n\s[ \sga_\sal \sga_\sbe = \sga_\sal \scdot \sga_\sbe + \sga_\sal \stimes \sga_\sbe = \set_{\sal \sbe} + \sga_{\sal \sbe} \s]\nor going in reverse &mdash; 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,\n\sbegin{eqnarray}\n\sga_\sal \stimes \sga_\sbe &=& \sga_{\sal \sbe}\s\s\n\sga_\sal \stimes \sga_{\sbe \sga} &=& \set_{\sal \sbe} \sga_{\sga} - \set_{\sal \sga} \sga_{\sbe}\s\s\n\sga_{\sal \sbe} \stimes \sga_{\sga \sde} &=& - \set_{\sal \sga} \sga_{\sbe \sde} + \set_{\sal \sde} \sga_{\sbe \sga} + \set_{\sbe \sga} \sga_{\sal \sde} - \set_{\sbe \sde} \sga_{\sal \sga}\s\s\n\sga_\sal \stimes \sga_{\sbe \sga \sde} &=& \sga_{\sal \sbe \sga \sde} \s\s\n&\svdots& \n\send{eqnarray}\nEqually useful identities arise for the symmetric product,\n\sbegin{eqnarray}\n\sga_\sal \scdot \sga_\sbe &=& \set_{\sal \sbe}\s\s\n\sga_\sal \scdot \sga_{\sbe \sga} &=& \sga_{\sal \sbe \sga}\s\s\n\sga_{\sal \sbe} \scdot \sga_{\sga \sde} &=& \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde}\s\s\n\sga_\sal \scdot \sga_{\sbe \sga \sde} &=& \set_{\sal\sbe} \sga_{\sga\sde} - \set_{\sal\sga} \sga_{\sbe\sde} + \set_{\sal\sde} \sga_{\sbe\sga} \s\s\n &\svdots& \n\send{eqnarray}\nContinuing the series, the product of two basis elements of [[grade|Clifford grade]]s $p$ and $q$, such as\n\sbegin{eqnarray}\n\sga_{\sal \sbe} \sga_{\sga \sde} &=& \sga_{\sal \sbe} \scdot \sga_{\sga \sde} + \sga_{\sal \sbe} \stimes \sga_{\sga \sde}\s\s\n &=& \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \slp \set_{\sal \sga} \sga_{\sbe \sde} + \set_{\sal \sde} \sga_{\sbe \sga} + \set_{\sbe \sga} \sga_{\sal \sde} + \set_{\sbe \sde} \sga_{\sal \sga} \srp + \sga_{\sal \sbe \sga \sde}\n\send{eqnarray}\ngives a result of mixed grades $|p-q|$ through $p+q \sle n$ 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.\n\nThe products of even or odd graded elements are\n| !grade of $A$ | !grade of $B$ | !grade of $AB$ |\n| even | even | even |\n| odd | odd | even |\n| odd | even | odd |\nThe cross product of anything 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]], $\seta_{\sal \sbe}$, 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//'', $\sga_\sal$. These Clifford basis vectors provide a means for invariantly describing local geometric objects.\n\nLike [[coordinate basis 1-forms]], two unequal Clifford basis vectors anti-commute, and their product represents a geometric area, or ''bivector'', element such as $\sga_1 \sga_2 = - \sga_2 \sga_1$, representing a unit area element spanned by $\sga_1$ and $\sga_2$. The orthonormality of Clifford basis vectors is expressed by the ''fundamental Clifford identity'',\n\s[ \sga_\sal \scdot \sga_\sbe = \sha \slp \sga_\sal \sga_\sbe + \sga_\sbe \sga_\sal \srp = \set_{\sal \sbe} \s]\nwhich 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 $\sha n (n-1)$ bivector [[Clifford basis elements]],\n\s[ \sga_\sal \stimes \sga_\sbe = \sha \slb \sga_\sal, \sga_\sbe \srb = \sha \slp \sga_\sal \sga_\sbe - \sga_\sbe \sga_\sal \srp = \sga_\sal \sga_\sbe = \sga_{\slb \sal \sbe \srb} = \sga_{\sal \sbe} \s]

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]], $\sga_{\sal \sdots \sbe}$. 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,\n$$\n\sga_{\sal \sdots \sbe}^2 = U_{21} \sga_{\sal \sdots \sbe}^1 U_{21}^-\n$$\nwhich 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?)//\n\nFor a section, $C(x)$, transforming under the adjoint action [[gauge transformation]], $C \smapsto C'=U C U^-$, the [[covariant derivative]] is\n$$\n\sf{\sna} C = \sf{d} C + \sha \sf{A} C - \sha C \sf{A} = \sf{d} C + \sf{A} \stimes C\n$$\n(defined with a $\sha$ in it to keep things pretty) with the [[Clifford connection]], $\sf{A}$, applied using the [[cross product|Clifford algebra]].\n\nAny 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, $U(t) = Pe^{- \sha \sint_0^t \sf{A}}$, satisfying the [[path holonomy]] equation,\n$$\n\sfr{d}{dt} U(t) = - \sha \sve{v} \sf{A} U\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n\sbegin{eqnarray}\n\sf{\sna} \sf{\sna} C &=& \sf{d} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \sf{A} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp \sf{A} \s\s\n&=& \sha \slp \sf{d} \sf{A} \srp C - \sha \sf{A} \sf{d} C - \sha \slp \sf{d} C \srp \sf{A} - \sha C \sf{d} \sf{A} \n + \sha \sf{A} \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp + \sha \slp \sf{d} C + \sha \sf{A} C - \sha C \sf{A} \srp \sf{A} \s\s\n&=& \sff{F} \stimes C\n\send{eqnarray}\ngives the [[Clifford curvature|Clifford-Riemann curvature]],\n$$\n\sff{F} = \sf{d} \sf{A} + \sha \sf{A} \stimes \sf{A}\n$$\na Clifford valued 2-form. This expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\sha$ in the path holonomy equation).\n\nUnder a gauge transformation, $C(x) \smapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} C' &=& U \slp \sf{\sna} C \srp U^-\s\s\n\sf{d} \slp U C U^- \srp + \sha \sf{A'} U C U^- - \sha U C U^- \sf{A'} &=& U \slp \sf{d} C \srp U^- + \sha U \sf{A} C U^- - \sha U C \sf{A} U^-\n\send{eqnarray}\ngiving the transformation law for the connection,\n$$\n\sf{A'} = U \sf{A} U^- - 2 \slp \sf{d} U \srp U^- = U \sf{A} U^- + 2 U \slp \sf{d} U^- \srp \n$$\nAn infinitesimal transformation, $U \ssimeq 1 + \sha C$, changes the connection to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} C - \sha \sf{A} C + \sha C \sf{A} = \sf{A} - \sf{\sna} C\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{F'} = \sf{d} \sf{A'} + \sha \sf{A'} \stimes \sf{A'} = U \sff{F} U^- \ssimeq \sff{F} + C \stimes \sff{F}\n$$\n\nThe covariant derivative acting on a [[Clifform]] such as the curvature, transforming under the adjoint action, $\sff{F'} = U \sff{F} U^-$, is still \n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{A} \stimes \sff{F} \n$$\n\nThe [[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.

One may carry out several unary operations on [[Clifford element]]s.\n\nThe [[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, $v^- = \sfr{v}{v \scdot v}$.\n\nThe [[Clifford dual]] of an element, $A \sga^-$, is often a useful object.\n\nThe ''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, $\shat{A^r} = \slp -1 \srp^r A^r$, also expressible as $\shat{A} = A^e - A^o$.\n\nThe ''reversion operator'', a.k.a. ''//reverse//'', reverses the order of all vectors multiplied in an element, producing $\stilde{A^r} = \slp -1 \srp^{\sha r(r-1)} A^r$.\n\n''Clifford conjugation'' combines these last two, $\sbar{A} = \stilde{\shat{A^r}} = \slp -1 \srp^{\sha r(r+1)} A^r$.\n\nFor a set of [[Dirac matrices]] in which $\sga_0$ 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 $A^\sdagger = \sga_0 \stilde{A} \sga^0$. When written as a matrix, this gives the transpose of the complex conjugate, $A^\sdagger = A^{*T}$.\n\nOne often encounters the ''Dirac conjugate'', $\soverline{A} = A^\sdagger \sga_0 = \sga_0 \stilde{A}$, 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'', $\sf{A} \sin \sf{Cl}$, acting on basis elements via the [[cross product|antisymmetric bracket]],\n$$\n\sf{\sna} \sga_\sal = \sf{A} \stimes \sga_\sal \n$$\nwhich gives the ''Clifford covariant derivative'' acting on any Clifford valued field (Clifford bundle section),\n$$\n\sf{\sna} C = \sf{d} C + \sf{A} \stimes C \n$$\n \nNote that the covariant derivative for the Clifford bundle does not necessarily preserve [[Clifford grade]].

The ''Clifford curvature scalar'' is obtained by taking the [[scalar part|Clifford grade]] of the [[frame]] contracted twice with the [[Clifford bundle]] curvature,\n$$\nR = \sli \sve{e} \sve{e} \sff{F} \sri\n$$\nIf, 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]],\n$$\nR = \sli \sve{e} \sve{e} \sff{R} \sri = \sve{e} \scdot \sf{R} = \slp \sve{e} \stimes \sve{e} \srp \scdot \sff{R}\n$$\nand equals the [[curvature scalar]] written in terms of the [[spin connection]] and frame.\n\nThe Clifford curvature scalar also comes from the expression:\n\sbegin{eqnarray}\n\sfr{2}{\slp n-2 \srp!} \sli \slp \sf{e} \srp^{n-2} \sff{R} \sga^- \sri\n&=& \sfr{2}{\slp n-2 \srp!} \sf{e}^\sal \sdots \sf{e}^\sbe \sf{e}^\smu \sf{e}^\snu \sfr{1}{4} R_{\smu\snu}{}^{\srh\ssi} \sli \sga_{\sal \sdots \sbe} \sga_{\srh \ssi} \sga^- \sri \s\s\n&=& \sfr{2}{\slp n-2 \srp!} \snf{e} \sep^{\sal \sdots \sbe \smu \snu} \sfr{1}{4} R_{\smu\snu}{}^{\srh\ssi} \sep_{\sal \sdots \sbe \srh \ssi} \s\s\n&=& \snf{e} \sde_{\slb \srh \ssi \srb}^{\smu\snu} R_{\smu\snu}{}^{\srh \ssi}=\snf{e} R\n\send{eqnarray}\nusing the [[volume form]] and [[permutation identities]].

The ''Clifford dual'' of any [[Clifford element]], $A$, is obtained by right multiplying it by the inverse [[pseudoscalar]], $A \sga^-$. For a [[Clifford grade]] $r$ element, this gives a grade $(n-r)$ element,\n\s[ A^r \sga^- = \sfr{1}{r!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} \sga^- = \sfr{1}{r! \slp n - r \srp !} A^{\sal \sdots \sbe} \sli \sga_{\sal \sdots \sbe \sga \sdots \sde} \sga^- \sri \sga^{\sga \sdots \sde} = \sfr{1}{r! \slp n - r \srp !} A^{\sal \sdots \sbe} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \s]\nin which $\sep_{\sal \sdots \sbe \sga \sdots \sde}$ is the [[permutation symbol]], and [[indices]] are raised with the [[Minkowski metric]].\n\nThe Clifford dual transformation is analogous to the [[Hodge dual]].

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]],\n\sbegin{eqnarray}\nA &=& A^s + A^\sal \sga_\sal + \sha A^{\sal \sbe} \sga_{\sal \sbe} + \sfr{1}{3!} A^{\sal \sbe \sga} \sga_{\sal \sbe \sga} + \sdots + A^p \sga\s\s\n&=& A^0 + A^1 + A^2 + A^3 + \sdots + A^n\n\send{eqnarray}\n(Some people choose to limit the sums so they don't run over all index values &mdash; but this isn't done here.) Like the coefficients of [[differential form]]s, the Clifford element coefficients are [[antisymmetric|index bracket]] in their indices, $A^{\sal \sdots \sbe}=A^{\slb \sal \sdots \sbe \srb}$. Unlike differential forms, Clifford elements may be of mixed [[grade|Clifford grade]].\n\nA clifford element has a geometric interpretation as a collection of variously sized scalar, vector, oriented area set, ..., and n-volume objects.\n\nClifford elements have a faithful [[matrix representation|Clifford matrix representation]].\n\nThe high grade terms of Clifford elements may be written with fewer indices by using the [[pseudoscalar]],\n$$A^r = \sfr{1}{r!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = \sfr{1}{r!} A^{\sal \sdots \sbe} \sfr{1}{\slp n-r \srp!} \sep_{\sal \sdots \sbe \sga \sdots \sde} \sga^{\sga \sdots \sde} \sga\n= \sfr{1}{\slp n-r \srp!} \slp \sfr{1}{r!} A^{\sal \sdots \sbe} \sep_{\sal \sdots \sbe \sga \sdots \sde} \srp \sga^{\sga \sdots \sde} \sga\n= \sfr{1}{\slp n-r \srp!} A^r_{\sga \sdots \sde} \sga^{\sga \sdots \sde} \sga$$\nSo, for example, the pseudoscalar (n-vector, grade $n$) part is\n$$A^n = \sfr{1}{n!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = A^p \sga$$\nand the (n-1)-vector part is\n$$A^{n-1} = \sfr{1}{\slp n-1 \srp!} A^{\sal \sdots \sbe} \sga_{\sal \sdots \sbe} = A^{n-1}_\sal \sga^\sal \sga$$

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]],\n$$\n\sga'_\sal = U \sga_\sal U^-\n$$\nThis gauge transformation is an active transformation of bundle elements, and transforms any Clifford valued field (section), $\sPh$, to\n$$\n\sPh' = U \sPh U^-\n$$\nBy definition, the [[Clifford covariant derivative|Clifford connection]] of any Clifford valued field transforms under a gauge transformation such that,\n$$\n\sf{\sna'} \sPhi' = \slp \sf{\sna} \sPhi \srp'\n$$\nWriting out the covariant derivative operators in this equation using the [[Clifford connection]],\n\sbegin{eqnarray}\n\sf{\sna'} \slp U \sPhi U^- \srp &=& U \slp \sf{\sna} \sPhi \srp U^- \s\s\n\slp \sf{d} U \srp \sPhi U^- + U \slp \sf{d} \sPhi \srp U^- + U \sPhi \slp \sf{d} U^- \srp + \sf{A'} \stimes \slp U \sPhi U^-\srp &=& U \slp \sf{d} \sPhi + \sf{A} \stimes \sPhi \srp U^- \s\s\nU^- \slp \sf{d} U \srp \sPhi + \sPhi \slp \sf{d} U^- \srp U + \sha U^- \sf{A'} U \sPhi - \sha \sPhi U^- \sf{A'} U &=& \sha \sf{A} \sPhi - \sha \sPhi \sf{A}\n\send{eqnarray}\ngives the transformation law for the connection under a gauge transformation:\n$$\n\sf{A'} = U \sf{A} U^- - 2 \slp \sf{d} U \srp U^- \n$$\nFor an infinitesimal gauge transformation, $U \ssimeq 1 + \sha C$, the connection changes to\n$$\n\sf{A'} \ssimeq \sf{A} - \sf{d} C - \sha \sf{A} C + \sha C \sf{A} = \sf{A} - \sf{\sna} C\n$$\ngiving the change $\sde \sf{A} = - \sf{\sna} C$.

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,\n\s[ t = \sfr{1}{3!} t^{\sal \sbe \sga} \sga_{\sal \sbe \sga} \s]\nis a 3-vector, or ''trivector'', while\n\s[ w = w^s + \sha w^{\sal \sbe} \sga_{\sal \sbe} \s]\nis a multivector of grades 0 and 2.\n\nThe ''grade operator'', $\sli A \sri_q = A^q$, acts as a filter, passing only the grade $q$ parts of $A$. For example, the bivector part of $w$ is\n\s[ \sli w \sri_2 = \sha w^{\sal \sbe} \sga_{\sal \sbe} \sin \sli Cl \sri_2 = Cl^2 \s] \nThe grade operator may also be used to filter the even or odd graded parts of an element, such as $\sli w\sri_e = \sli w\sri_2$ and $\sli w\sri_o = 0$. Of special interest is the grade 0 operator, $\sli A\sri = \sli A\sri_0 = A^0 = A^s$, 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 number.

Combining [[Clifford basis product identities]] with the [[grade|Clifford grade]] operator gives a ''Clifford graded [[commutation|commutator]]'' relationship for two Clifford elements of grades $r$ and $s$,\n\s[ \sli A^r B^s \sri_q = \slp -1 \srp^\sep \sli B^s A^r \sri_q \s]\nwith\n\s[ \sep = \sha \slp q^2 + r^2 + s^2 - q - r - s \srp \s]\nThis relation implies that any two Clifford elements commute inside the scalar part operator, $\sli AB \sri = \sli BA \sri$.\n\nThe Clifford product of two elements of grades $r$ and $s$ can produce elements of various grades,\n\s[ A^r B^s = \sli A^r B^s \sri_{\sll r - s \srl} + \sli A^r B^s \sri_{\sll r - s \srl + 2} + \sdots + \sli A^r B^s \sri_{r + s} \s]

The ''Clifford group'' consists of [[Clifford algebra]] elements having an inverse,\n\s[ Cl^* = \sleft\s{ U \sin Cl \smid \sexists \s; U^- \sni U U^- = 1 \sright\s} \s]\nIt is the [[Lie group]] corresponding to [[exponentiation]] of the [[Clifford basis elements]],\n$$U = e^{B^A \sga_A}$$\nThe [[Lie algebra]] corresponding to the Clifford group is the Clifford algebra.

Each [[Clifford algebra]] has a faithful representation in the complex matrices, $GL(2^{[n/2]},\smathbb{C})$, with the Clifford product isomorphic to matrix multiplication. This corresponds to the traditional definition of [[Pauli matrices]] and [[Dirac matrices]] as the $\sgamma_{\salpha}$ 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.\n\nA 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]],\n$$\n\sbegin{array}{cc}\n\ssi_1 = \ssigma_{1}^{P} =\n\sleft[\sbegin{array}{cc}\n0 & 1\s\s\n1 & 0\n\send{array}\sright]\n&\n\ssi_2 = \ssigma_{2}^{P}=\sleft[\sbegin{array}{cc}\n0 & -i\s\s\ni & 0\send{array}\sright]\n\send{array}\n$$\nand multiplying to get the scalar and bivector,\n$$\n\sbegin{array}{cc}\n1 = \ssi_1 \ssi_1 =\n\sleft[\sbegin{array}{cc}\n1 & 0\s\s\n0 & 1\send{array}\sright]\n&\n\ssi_{12} = \ssi_1 \ssi_2 =\n\sleft[\sbegin{array}{cc}\ni & 0\s\s\n0 & -i\n\send{array}\sright]\n= i \ssi_3^P\n\send{array}\n$$\ncompleting the list of $Cl(2,0)$ [[Clifford basis elements]] represented as $2 \stimes 2$ complex matrices. To build larger Clifford algebras we can use the [[Kronecker product]] of any smaller Clifford algebras. For example, $Cl(2,2) = Cl(2,0) \sotimes Cl(2,0)$. The tricky part is finding a set of orthogonal, anticommuting, ''Clifford basis vector matrix representatives'' after doing the product, such as picking out:\n$$\n\sbegin{array}{cc}\n\sga_1 = \ssi_1 \sotimes 1 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 1 & 0\s\s\n0 & 0 & 0 & 1\s\s\n1 & 0 & 0 & 0\s\s\n0 & 1 & 0 & 0\n\send{array}\sright]\n&\n\sga_2 = \ssi_2 \sotimes \ssi_2 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 0 & -1\s\s\n0 & 0 & 1 & 0\s\s\n0 & 1 & 0 & 0\s\s\n-1 & 0 & 0 & 0\n\send{array}\sright]\n\s\s\n\sga_3 = \ssi_2 \sotimes \ssi_1 =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 0 & -i\s\s\n0 & 0 & -i & 0\s\s\n0 & i & 0 & 0\s\s\ni & 0 & 0 & 0\n\send{array}\sright]\n&\n\sga_4 = \ssi_2 \sotimes \ssi_{12} =\n\sleft[\sbegin{array}{cccc}\n0 & 0 & 1 & 0\s\s\n0 & 0 & 0 & -1\s\s\n-1 & 0 & 0 & 0\s\s\n0 & 1 & 0 & 0\n\send{array}\sright]\n\send{array}\n$$\nA matrix representation, such as above, allows any $Cl(2,2)$ element to be represented by a $4 \stimes 4$ complex matrix. For example,\n$$\na \s, \sga_{12} + b \s, \sga_{34} = a \s, \ssi_{12} \sotimes \ssi_2 + b \s, 1 \sotimes \ssi_2 =\n\sleft[\sbegin{array}{cccc}\n0 & a - i b & 0 & 0\s\s\n-a + i b & 0 & 0 & 0\s\s\n0 & 0 & 0 & -a - i b\s\s\n0 & 0 & a + i b & 0\n\send{array}\sright]\n$$\nTo get a different signature we can multiply any basis vector representaive by $i$, such as multiplying $\sga_3$ above by $i$ to get a ''real representation'' of $Cl(3,1)$ -- 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.\n\nEverything 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.\n\nRefs:\n*http://en.wikipedia.org/wiki/Representations_of_Clifford_algebras\n*Andrzej Trautman\n**[[Clifford Algebras and their Representations|papers/Clifford Algebras and their Representations.pdf]]\n***p20 describes construction of reps for arbitrarily high dimension

Iteration of the [[cross product|antisymmetric bracket]] produces the ''Clifford Jacobi identity'',\n\s[ A \stimes \slp B \stimes C \srp + B \stimes \slp C \stimes A \srp + C \stimes \slp A \stimes B \srp = 0 \s]\nand the ''cross product distributive rule'',\n\s[ A \stimes \slp B C \srp = \slp A \stimes B \srp C + B \slp A \stimes C \srp \s]\nA combination of [[Clifford algebra]] dot and cross products is\n\s[ A \scdot \slp B \stimes C \srp + A \stimes \slp B \scdot C \srp = \sha \slp ABC - CBA \srp = \slp A \scdot B \srp \stimes C + \slp A \stimes B \srp \scdot C \s]\n\nA string of cross products without parenthesis, $A \stimes B \stimes C$, is not well defined because $A \stimes \slp B \stimes C \srp \sne \slp A \stimes B \srp \stimes C$; but a string of dot products, $A \scdot B \scdot C$, 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.\n\nTo calculate Clifford products, it is best to use the [[Clifford basis product identities]]

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 product identities]] and antisymmetry of bivector indices,\n\sbegin{eqnarray}\nB \stimes v &=& \sha B^{\sal \sbe} v^\sga \sga_{\sal \sbe} \stimes \sga_\sga = B^{\sal \sbe} v^\sga \sga_{\slb \sal \srd} \set_{\sld \sbe \srb \sga} = B^{\sal \sbe} v_\sbe \sga_\sal \s\s\nv \scdot \slp B \stimes v \srp &=& v^\sde B^{\sal \sbe} v_\sbe \sga_\sde \scdot \sga_\sal = B^{\sal \sbe} v_\sal v_\sbe = 0\n \send{eqnarray}\nA small rotational transformation in the plane (or planes) of a bivector may be carried out by\n\s[ v' = v + \sfr{1}{N}B \stimes v \ssimeq \slp 1 + \sfr{1}{2N} B \srp v \slp 1 - \sfr{1}{2N} B \srp \s]\nfor a large parameter, $N$. A finite rotation comes from [[exponentiating|exponentiation]] the bivector,\n\s[ v' = \slim_{N \sto \sinfty} \slp 1 + \sfr{1}{2N} B \srp^N v \slp 1 - \sfr{1}{2N} B \srp^N = e^{\sha B} v e^{- \sha B} = U v U^- \s]\nFor example, rotating $v$ by an angle of $\sth$ in the $\sga_{12}$ plane gives\n\s[ v' = e^{\sha \sth \sga_{12}} v e^{- \sha \sth \sga_{12}} = \slp \scos \sha \sth + \sga_{12} \ssin \sha \sth \srp v \slp \scos \sha \sth - \sga_{12} \ssin \sha \sth \srp \s]\nSince $U^- = e^{- \sha B} = \stilde{U}$ is the [[reverse|Clifford conjugate]] and the [[inverse]] of $U = e^{\sha B}$, Clifford rotation is a special case of the [[Clifford adjoint]]. Such a $U$ is sometimes called a ''rotor''. Any [[Clifford element]] may be rotated by, $A' = U A U^-$ -- which preserves the [[Clifford grade]] of the element.\n\nA Clifford rotation may be readily translated into the standard matrix coefficient notation for a [[Lorentz rotation]] via\n\s[ \sga'_\sal = \sga_\sbe L^\sbe{}_\sal = U \sga_\sal U^- \s]\n\nThe group of Clifford rotations, $\smbox{Spin}{}^+$, in any [[spacetime]] is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. Boosts along any [[spatial|indices]] direction, $\snu = \snu^\spi \sga_\spi$, are Clifford rotations in the $\sga_0 \snu$ plane,\n\s[ v' = e^{\sha \sga_{0 \spi} \snu^\spi} v e^{- \sha \sga_{0 \srh} \snu^\srh} \s]\nFor example, a boost of $\snu$ along $\sga_3$ gives\n\s[ v' = e^{\sha \sga_{03} \snu} v e^{- \sha \sga_{03} \snu} = \slp \scosh \sha \snu + \sga_{03} \ssinh \sha \snu \srp v \slp \scosh \sha \snu - \sga_{03} \ssinh \sha \snu \srp \s]\n\nA simple rotor is defined as a rotor that can be written as the product of two vectors, $U_{s}=ab=e^{\sfrac{1}{2}i_{2}\stheta}$, in which $i_{2}$ is a unit bivector of a rotation plane and $\stheta$ is a rotation angle. A rotor may always be factored into a product of $\sleq\sfrac{n}{2}$ simple rotors. A standard decomposition uses the choice of a non-singular vector, $v$, to factor a rotor into $U=\spm U'U_{s}$, in which $U'$ is a rotor that leaves $v$ invariant, $U'v=vU'$, and $U_{s}$ is a simple rotor that rotates $v$. As an example, in a four dimensional Lorentzian spacetime, a rotor can be factored using the time-like frame vector, $\sgamma_{0}$, into the spatial rotation and Lorentz boost,\s[\nU=e^{\sfrac{1}{2}\sgamma\sgamma_{0}\sgamma_{\spi}\stheta^{\spi}}\s: e^{\sfrac{1}{2}\sgamma_{0}\sgamma_{\srh}\snu^{\srh}}\s]\nin which $\sth = \sgamma_{\spi}\stheta^{\spi}$ is the spatial rotation vector and $\snu = \sgamma_{\srh}\snu^{\srh}$ is the spatial boost vector used to construct the [[Cl(1,3) bivector]] corresponding to the Clifford rotation.

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]], $\sga_\sal$. The fiber at each base manifold point, $p$, is the space of grade 1 Clifford elements, $Cl^1 = \sli Cl \sri_1$. The transition functions for the basis elements over overlapping patches, $U_1$ and $U_2$, are given by [[Clifford rotation]]s,\n$$\n\sga_\sal^2 = U_{12} \sga_\sal^1 U_{12}^- = \slp t^{12} \srp_\sal{}^\sbe \sga_\sbe^1 = \slp L^{12} \srp^\sbe{}_\sal \sga_\sbe^1\n$$\nThrough equating the transition functions, $L^\sbe{}_\sal$, and using the [[frame]], $\sve{e_\sal} \sf{e} = \sga_\sal$, the Clifford vector bundle may be [[associated]], $\sga_\sal \sleftrightarrow \sve{e_\sal}$, to the [[tangent bundle]], with a corresponding equivalence between all their geometric structures. The structure group of the Clifford vector bundle, $\smbox{Spin}{}^+$, is a double cover of the [[special orthochronous Lorentz group|Lorentz group]]. A Clifford vector field, $v = v(x) = v^\sal(x) \sga_\sal$, over the manifold is a section of the bundle, and gives a Clifford vector at each manifold point.\n\n[[Clifford grade]] $p$ fields are sections of the ''Clifford p-vector bundle'', $Cl^p M$, which has the $\sfrac{n!}{\sleft(n-p\sright)!p!}$ grade $p$ [[Clifford basis elements]], $\sga_{\sla \sdots \sbe}$, as basis. The combined collection of these Clifford vector product bundles is the ''graded Clifford bundle'', $Cl^g M = \sbigoplus_{p=0}^{n} Cl^p M$, having dimension $2^{n}$. The transition functions for the graded Clifford bundle are also [[Clifford rotation]]s,\n$$\sga_{\sal \sdots \sbe}^2 = U_{12} \sga_{\sal \sdots \sbe}^1 U_{12}^-$$\nwhich preserve the grade of the basis elements. The graded Clifford bundle fiber, $Cl$, is the same as for the [[Clifford bundle]] &mdash; 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]].\n\nFor a section, $C(x)$, transforming under the Clifford rotation [[gauge transformation]], $C \smapsto C'=U C U^-$, the [[covariant derivative]] is\n$$\n\sf{\sna} C = \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} = \sf{d} C + \sf{\som} \stimes C\n$$\n(defined with a $\sha$ in it to keep things pretty) with the [[spin connection]], $\sf{\som}$, applied using the [[cross product|Clifford algebra]].\n\nAny 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, $U(t) = Pe^{- \sha \sint_0^t \sf{\som}}$, satisfying the [[path holonomy]] equation,\n$$\n\sfr{d}{dt} U(t) = - \sha \sve{v} \sf{\som} U\n$$\n\nApplying the covariant derivative twice (being careful with the signs of 1-forms hopping sides),\n\sbegin{eqnarray}\n\sf{\sna} \sf{\sna} C &=& \sf{d} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \sf{\som} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp \sf{\som} \s\s\n&=& \sha \slp \sf{d} \sf{\som} \srp C - \sha \sf{\som} \sf{d} C - \sha \slp \sf{d} C \srp \sf{\som} - \sha C \sf{d} \sf{\som} \n + \sha \sf{\som} \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp + \sha \slp \sf{d} C + \sha \sf{\som} C - \sha C \sf{\som} \srp \sf{\som} \s\s\n&=& \sff{R} \stimes C\n\send{eqnarray}\ngives (after using one of the [[Clifford basis product identities]]) the [[Clifford-Riemann curvature]],\n$$\n\sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som}\n$$\nThis expression for the curvature may alternatively be obtained from the [[holonomy]] (minding the new factor of $\sha$ in the path holonomy equation).\n\nUnder a gauge transformation, $C(x) \smapsto C'(x) = U(x) C(x) U^-(x)$, the covariant derivative changes to\n\sbegin{eqnarray}\n\sf{\sna'} C' &=& U \slp \sf{\sna} C \srp U^-\s\s\n\sf{d} \slp U C U^- \srp + \sha \sf{\som'} U C U^- - \sha U C U^- \sf{\som'} &=& U \slp \sf{d} C \srp U^- + \sha U \sf{\som} C U^- - \sha U C \sf{\som} U^-\n\send{eqnarray}\ngiving the transformation law for the spin connection,\n$$\n\sf{\som'} = U \sf{\som} U^- - 2 \slp \sf{d} U \srp U^- = U \sf{\som} U^- + 2 U \slp \sf{d} U^- \srp \n$$\nAn infinitesimal transformation, $U \ssimeq 1 + \sha B$, in which $B$ is a Clifford bivector, changes the spin connection to\n$$\n\sf{\som'} \ssimeq \sf{\som} - \sf{d} B - \sha \sf{\som} B + \sha B \sf{\som} = \sf{\som} - \sf{\sna} B\n$$\nThe curvature consequently transforms under a gauge transformation to\n$$\n\sff{R'} = \sf{d} \sf{\som'} + \sha \sf{\som'} \stimes \sf{\som'} = U \sff{R} U^- \ssimeq \sff{R} + B \stimes \sff{R}\n$$\nThese expressions equate to those for a [[tangent bundle gauge transformation]].\n\nThe covariant derivative acting on a [[Clifform]] such as the curvature, transforming under a Clifford rotation, $\sff{F'} = U \sff{F} U^-$, is still \n$$\n\sf{\sna} \sff{F} = \sf{d} \sff{F} + \sf{\som} \stimes \sff{F} \n$$\n\nClifford 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,\n$$\n\sf{R} = \sve{e} \stimes \sff{F}\n$$\nIf, specifically, we are working with the [[Clifford vector bundle]], the Clifford-Ricci curvature is then a Clifford vector valued 1-form,\n$$\n\sf{R} = \sf{dx^i} R_i{}^\sal \sga_\sal = \sve{e} \stimes \sff{R} = \sve{e} \stimes \slp \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som} \srp\n$$\nwith coefficients equal to those of the [[Ricci curvature]], $R_i{}^\sal = \slp e_\sbe \srp^j R_{ji}{}^{\sbe \sal} = \set^{\sal \sbe} R_{i \sbe}$.

The ''Clifford curvature'' is a [[Clifform]] describing the [[curvature]] of a [[Clifford bundle]],\n$$\n\sff{F} = \sf{d} \sf{A} + \sha \sf{A} \stimes \sf{A}\n$$\nIf, 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]],\n$$\n\sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \stimes \sf{\som} = \sf{dx^i} \sf{dx^j} \sfr{1}{4} R_{ij}{}^{\sal \sbe} \sga_{\sal \sbe}\n$$\n$$\nR_{ij}{}^{\sal \sbe} = 2 \spa_{\slb i \srd} \som_{\sld j \srb}{}^{\sal \sbe} + 2 \som_{\slb i \srd}{}^\sal{}_\sga \som_{\sld j \srb}{}^{\sga \sbe}\n$$\nwith coefficients equal to those of the [[Riemann curvature]], $R_{ij}{}^{\sal\sbe}$, when the [[tangent bundle connection]] and spin connection coefficients are identified, $\sf{w^{\sal\sbe}}=\sf{\som^{\sal\sbe}}$.

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\n$$\n\snf{A} = \sf{dx^i} \sdots \sf{dx^k} \sfr{1}{p!} \slp A_{i \sdots k}{}^s + A_{i \sdots k}{}^\sal \sga_\sal + \sha A_{i \sdots k}{}^{\sal \sbe} \sga_{\sal \sbe} + \sfr{1}{3!} A_{i \sdots k}{}^{\sal \sbe \sga} \sga_{\sal \sbe \sga} + \sdots + A_{i \sdots k}{}^p \sga \srp\n$$\nFor example, a bivector 2-form is written (using the coordinate or [[frame]] basis forms) as\n$$\n\sff{R} = \sff{R^2} = \sf{dx^i} \sf{dx^j} \sfr{1}{4} R_{ij}{}^{\sal \sbe} \sga_{\sal \sbe}\n= \sf{e^\sga} \sf{e^\sde} \sfr{1}{4} R_{\sga \sde}{}^{\sal \sbe} \sga_{\sal \sbe}\n$$\nThe 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.\n\nThe 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 $\scdot$, $\stimes$, $[,]$, 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,\n\sbegin{eqnarray}\n\svv{L} \scdot \sff{R} &=& \slp \sve{\spa_i} \sve{\spa_j} \sfr{1}{4} L^{i j \sal \sbe} \sga_{\sal \sbe} \srp \scdot \slp \sf{dx^k} \sf{dx^m} \sfr{1}{4} R_{km}{}^{\sga \sde} \sga_{\sga \sde} \srp\s\s\n&=& \slp \sve{\spa_i} \sve{\spa_j} \slp \sf{dx^k} \sf{dx^m} \srp \srp \sfr{1}{16} L^{i j \sal \sbe} R_{km}{}^{\sga \sde} \slp \sga_{\sal \sbe} \scdot \sga_{\sga \sde} \srp\s\s\n&=& \slp - 2 \sde_i^{\slb k \srd} \sdelta_j^{\sld m \srb} \srp \sfr{1}{16} L^{i j \sal \sbe} R_{km}{}^{\sga \sde} \slp \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde} \srp\s\s\n&=& - \sfr{1}{8} L^{i j \sal \sbe} R_{ij}{}^{\sga \sde} \slp \slp \set_{\sal \sde} \set_{\sbe \sga} - \set_{\sal \sga} \set_{\sbe \sde} \srp + \sga_{\sal \sbe \sga \sde} \srp\s\s\n&=& \sfr{1}{4} L^{i j \sal \sbe} R_{ij \sal \sbe} - \sfr{1}{8} L^{i j \sal \sbe} R_{ij}{}^{\sga \sde} \sga_{\sal \sbe \sga \sde}\s\s\n\send{eqnarray}\nusing vector-form algebra and [[Clifford basis product identities]].\n\nClifform product identities can be inferred from the identities of the two respective algebras. For example, since 1-forms anti-commute,\n\s[ \sf{A} \sti \sf{B} = \sha \slp \sf{A} \sf{B} + \sf{B} \sf{A} \srp = \sf{B} \sti \sf{A} \s]\nSome useful identities can be computed using the [[frame]]. For example, for any Clifford vector valued 2-form, $\sff{f}$,\n\sbegin{eqnarray}\n\sve{e} \sti \sff{f} & = & -\slp \sve{e} \sti \sve{e} \srp \slp \sf{e} \scdot \sff{f} \srp + \sve{e} \sti \slp \slp \sve{e} \sti \sff{f} \srp \sti \sf{e} \srp\s\s\n\sff{f} & = & \slp \sve{e} \sti \sff{f} \srp \sti \sf{e} - \sve{e} \scdot \slp \sf{e} \scdot \sff{f} \srp\s\s\n\slp n-2 \srp \sff{f} & = & \sve{e} \sti \slp \sf{e} \sti \sff{f} \srp - \sf{e} \scdot \slp \sve{e} \scdot \sff{f} \srp\n\send{eqnarray} \n(//add identities as needed//)

The [[Coleman-Mandula theorem|http://prola.aps.org/abstract/PR/v159/i5/p1251_1]] states:\n<<<\nLet 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 Poincaré 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 Poincaré group.\n<<<\n\nThe 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 Poincaré 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 Poincaré group. At low energies the deviation from the Poincaré group is infinitesimally small, and the Coleman-Mandula theorem applies to a good approximation, with gravity separate from the other symmetries.\n\nRef:\n*R. Percacci\n**[[Mixing internal and spacetime transformations: some examples and counterexamples|http://arxiv.org/abs/0803.0303]]\n*K. Cahill\n**[[On the unification of the gravitational and electronuclear forces|papers/Cahill - On the unification of the gravitational and electronuclear forces.pdf]]\n*** Phys. Rev. D 26, 1916 - 1922 (1982).\n*T. Love\n**The Geometry of Grand Unification\n***Int. J. Th. Phys., 801 (1984).\n*F. Nesti and R. Percacci\n**[[Gravi-Weak Unification|http://arxiv.org/abs/0706.3307]]\n*S. Alexander\n**[[Isogravity: Toward an Electroweak and Gravitational Unification|http://arxiv.org/abs/0706.4481]]

/***\nAuthors: Eric Shulman & Bradley Meck\nversion: 2007.30.03\nsource: http://www.tiddlytools.com/\n***/\n/*{{{*/\nconfig.commands.collapseNote = {\ntext: "-",\ntooltip: "Collapse this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nif(e.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(e.getAttribute("template") != t ){\ne.setAttribute("oldTemplate",e.getAttribute("template"));\nstory.displayNote(null,title,t);\n}\n}\n}\n}\n\nconfig.commands.expandNote = {\ntext: " | ",\ntooltip: "Expand this note",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.displayNote(null,title,e.getAttribute("oldTemplate"));\n}\n}\n\nconfig.macros.collapseAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"-","Collapse all notes",function(){\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE])\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nstory.displayNote(null,title,t);\n})})\n}\n}\n\nconfig.macros.expandAll = {\nhandler: function(place,macroName,params,wikifier,paramString,note){\ncreateTiddlyButton(place,"expand all","Expand all notes",function(){\nstory.forEachNote(function(title,note){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif(note.getAttribute("template") == t) story.displayNote(null,title,note.getAttribute("oldTemplate"));\n})})\n}\n}\n\nconfig.commands.collapseOthers = {\ntext: "Ø",\ntooltip: "Expand this note and collapse all others",\nhandler: function(event,src,title)\n{\nvar e = story.findContainingNote(src);\nstory.forEachNote(function(title,note){\nif(note.getAttribute("template") != config.noteTemplates[DEFAULT_EDIT_TEMPLATE]){\nvar t = (readOnly&&store.noteExists("WebCollapsedTemplate"))?"WebCollapsedTemplate":"CollapsedTemplate";\nif (e==note) t=e.getAttribute("oldTemplate");\n//////////\n// ELS 2006.02.22 - removed this line. if t==null, then the *current* view template, not the default "ViewTemplate", will be used.\n// if (!t||!t.length) t=!readOnly?"ViewTemplate":"WebViewTemplate";\n//////////\nstory.displayNote(null,title,t);\n}\n})\n}\n}\n/*}}}*/

<div>\n<span class='toolbar' macro='toolbar +editNote expandNote collapseOthers closeOthers -closeNote'></span>\n<span class='title' macro='view title'></span>\n</div>

This site is powered by [[TiddlyWiki|http://www.tiddlywiki.com]] <<version>>\n!I installed these plugins (need t(T)iddler -> n(N)ote find and replace):\n*[[InlineJavascriptPlugin]]\n**used for the [[DisplayControl]]\n**and for [[HideTags]] (used for slides)\n*[[TextAreaPlugin]] -- disabled for now\n**deselect the edit contents, and adds ctr-f,ctrl-g,cmd-v search/replace to editing.\n*[[jsMathPlugin]]\n**this processes the [[LaTeX]]. The AJAX part had problems, so I put the jsmath load into the source directly.\n**inserted custom LaTeX/jsmath command abbreviations into plugin.\n*[[CollapsePlugin]]\n**[[CollapsedTemplate]]\n*[[RearrangeNotesPlugin]]\n*[[ListTaggedPlugin]]\n**used for folder/tag listings\n*[[AllTagsExceptPlugin]]\n**advanced checkbox to see system tags\n*[[CopyNotePlugin]]\n*[[DisableWikiLinksPlugin]]\n**remove checkbox so it's always on\n**this is very tricky in combination with [[jsMathPlugin]] and \sss pytw problem\n*[[FaviconPlugin]]\n*[[ReferencesPlugin]]\n*[[RecentPlugin]]\n**set to show last 2\ncheck to make sure I didn't install any<<tag plugin>>and forget to list it here. Try using the [[PluginManager]].\n\n!I changed these notes to configure operation and appearance:\n*These control the content of several boxes:\n**[[SiteTitle]]\n**[[SiteSubtitle]]\n**[[SiteUrl]]\n**[[DefaultNotes]]\n**[[MainMenu]]\n**[[SideBarOptions]]\n**[[OptionsPanel]]\n***[[SideBarOptionsText]]\n**[[AdvancedOptions]]\n**[[SideBarTabs]]\n***[[TabContents]]\n***[[TabTimeline]]\n***[[TabAll]] - nope, for some reason this has the text built in. :(\n***[[TabTags]]\n**[[DisplayControl]]\n*These are css layout templates:\n**[[PageTemplate]]\n**[[ViewTemplate]]\n**[[EditTemplate]]\n**[[CollapsedTemplate]]\n*And these change the system and css options:\n**[[SystemConfig]]\n**[[StyleSheet]]\n***Trouble with [[MyColors]] conflicting with [[ColorPalette]]\n**[[StyleSheetPrint]]\nThe default config files are invisible and listed as [[ShadowNotes]]. These:\n*[[StyleSheetLayout]]\n*[[StyleSheetColors]]\nare 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]]. \n\n!Evil raw html/javascript TW source code tweakage\n*edit cookie options, since setting them in [[SystemConfig]] overrides user cookies\n*maybe add ctrl-w accessKey -- just fooling around\n*comment out a couple of displayMessage s\n*switch line order in {{{config.macros.search.handler}}} for search button after search field\n*comment out tag prompt line in {{{config.macros.tags.handler}}}\n*Insert this just after body. (This starts jsmath)\n**{{{<scriipt src="jsMath/jsMath.js"></scriipt>}}}\n*add B's logging script call\n**make index executable so log script will run\n\n!edited {{{jsMath/easy/load.js}}}\n*changed default font scaling to {{{scale: 110}}} and {{{warn: 0}}}\n*remove doubleclick show\n*reduced vertical margins by adding {{{margin-top: 0.5em; margin-bottom: 0.5em;}}}\n*hide jsMath button\n\n!And finally\nI did a global find/replace of "t(T)iddler" -> "n(N)ote" in the base file or directory via editor or the noteify shell script. Make sure to do this when more plugins are installed, so they'll work.\n\nThen, save a bare copy, without folders or editing tips, and a minimal copy, with them. Then try to [[ImportNotes]].

http://arxiv.org/abs/gr-qc/0603062\n*Concise treatment of Hamiltonian formulation of GR with a conformal factor.\n*uses metric instead of frame

[[Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity|papers/9403058.pdf]]\nAuthors: Sean M. Carroll, George B. Field\n\nWe 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. \n

/***\nAuthors: Eric Shulman\nversion: 2.1.2\nsource: http://www.tiddlytools.com/\nadds a "copy" option to duplicate a note\n***/\n/*{{{*/\nversion.extensions.copyNote= {major: 2, minor: 1, revision: 2, date: new Date(2007,5,17)};\nconfig.commands.copyNote = {\n text: '\sxA9',\n hideReadOnly: true,\n tooltip: 'Make a copy of this note',\n prefix: "Copy of ",\n handler: function(event,src,title) {\n var text=store.getNoteText(title); // get text from note (or shadow)\n var tags=[]; var tid=store.getNote(title); if (tid) tags=tid.getTags();\n var textfield=story.getNoteField(title,"text");\n if (textfield&&textfield.getAttribute("edit")=="text") var text=textfield.value; // edit mode, use field value\n var tagsfield=story.getNoteField(title,"tags");\n if (tagsfield&&tagsfield.getAttribute("edit")=="tags") var tags=tagsfield.value; // edit mode, use field value\n var newTitle = this.prefix + title;\n story.displayNote(null,newTitle,DEFAULT_EDIT_TEMPLATE);\n story.getNoteField(newTitle,"text").value=text;\n story.getNoteField(newTitle,"tags").value=tags;\n story.focusNote(newTitle,"title");\n return false;\n }\n};\n/*}}}*/

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/TED08/images/Coral_reef_s620.JPG" width="827" height="620"></embed></center></html>@@\n

<<note HideTags>>$$\n\sbegin{array}{rcll}\n\sff{F} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{d} \sf{H} + \sf{H} \sf{H}\n\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s;\s; \sf{H} = \sha \sf{\som} + \sfr{1}{4}\sf{e}\sph + \sf{B} + \sf{W}\n\s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sBig( \sha ( \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} ) + \sfr{1}{16} M^2 \sf{e} \sf{e} \sBig)_{\sp{(}}\n\s!&\s!\s! \sleftarrow \stext{spacetime} \s; \sga_{\smu\snu} \s\s\n\n&&\s!\s!\s! + \sBig( \sfr{1}{4} \sbig( \sf{d} \sf{e} + \sha [ \sf{\som}, \sf{e} ] \sbig) \sph - \sfr{1}{4} \sf{e} \sbig( \sf{d} \sph + [ \sf{B} \s!+\s! \sf{W}, \sph ] \sbig) \sBig)_{\sp{(}}\n\s!&\s!\s! \sleftarrow \stext{mixed} \s; \sga_{\smu\sph} \s\s\n\n&&\s!\s!\s! + \sBig( \sf{d} \sf{B} + \sf{d} \sf{W} + \sf{W} \sf{W} \sBig)_{\sp{\sbig(}}\n\s!&\s!\s! \sleftarrow \stext{higher} \s; \sga_{\sph\sps} \s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sha \sbig( \sff{R} + \sfr{1}{8} M^2 \sf{e} \sf{e} \sbig)\n+ \sfr{1}{4} \sbig( \sff{T} \sph - \sf{e} \sf{D} \sph \sbig)\n+ \sbig( \sff{F_B} + \sff{F_W} \sbig) \s\s\n\n\s!\s!&\s!\s!=\s!\s!&\s!\s! \sff{F_s} + \sff{F_m} + \sff{F_h}\n\send{array}\n$$\nModified BF action over 4D base [[manifold]]:\n\sbegin{eqnarray}\nS &=& \sint \sbig< \sff{B} \s, \sff{F} + \sPh(\sf{H},\sff{B}) \sbig>\n= \sint \sbig< \sff{B} \s, \sff{F} - {\sscriptsize \sfrac{1}{4}} \sff{B_s} \sff{B_s} \sga + \sff{B_m} \sff{*B_m} + \sff{B_h} \sff{*B_h} \sbig> \s\s\n&=& \sint \sbig< \sff{F_s} \s, \sff{F_s} \sga^- + {\sscriptsize \sfrac{1}{4}} \sff{F_m} \sff{*F_m} + {\sscriptsize \sfrac{1}{4}} \sff{F_h} \sff{*F_h} \sbig>\n\send{eqnarray}

\nNew paper. How to go from a higher dimensional gauge theory, with Chern Simons or Born Infeld action, to Einstein gravity in 4D:\n*[[D=4 Einstein gravity from higher D CS and BI gravity and an alternative to dimensional reduction|papers/0703034.pdf]]

Welcome

The kinetic [[Lagrangian]] term for a [[Dirac spinor]] field in curved [[spacetime]] is\n$$\nL = \sPsi^\sdagger \sga_0 \sve{e} \slp \sf{\spa} + \sha \sf{\som} \srp \sPsi \n$$\n//(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\n\sbegin{eqnarray}\nL &=& \slb \sPsi_L^\sdagger \s;\s; \sPsi_R^\sdagger \srb\n\slb \sbegin{array}{cc}\n0 & 1 \s\s\n1 & 0\n\send{array} \srb\n\slb \sbegin{array}{cc}\n0 & \sve{e}_L \s\s\n\sve{e}_R & 0\n\send{array} \srb\n\slp \sf{\spa} + \n\sha\n\slb \sbegin{array}{cc}\n\sf{\som}{}_L & 0 \s\s\n0 & \sf{\som}{}_R\n\send{array} \srb\n\srp\n\slb \sbegin{array}{c}\n\sPsi_L \s\s\n\sPsi_R\n\send{array} \srb\n \s\s\n&=& \sPsi_L^\sdagger \sve{e}_R \slp \sf{\spa} + \sha \sf{\som}{}_L \srp \sPsi_L + \sPsi_R^\sdagger \sve{e}_L \slp \sf{\spa} + \sha \sf{\som}{}_R \srp \sPsi_R\n\send{eqnarray}

The ''Dirac matrices'' provide a $4\stimes4$ [[Clifford matrix representation]] of [[Cl(1,3)]] or [[Cl(3,1)]]. There are several standard choices, built from the [[Kronecker product]] of [[Pauli matrices]]:\n\nThe ([[chiral]]) ''Weyl representation'' of the Dirac matrices of Cl(1,3) is:\n\sbegin{eqnarray}\n\sga_0 &=& \ssi^P_1 \sotimes 1 \s\s\n\sga_1 &=& -i \ssi^P_2 \sotimes \ssi^P_1 \s\s\n\sga_2 &=& -i \ssi^P_2 \sotimes \ssi^P_2 \s\s\n\sga_3 &=& -i \ssi^P_2 \sotimes \ssi^P_3\n\send{eqnarray}\ngiving a complex rep for ''Cl(1,3) vectors'',\n\sbegin{eqnarray}\nv &=& v^\smu \sga_\smu =\n\slb \sbegin{array}{cc}\n0 & v_R \s\s\nv_L & 0\n\send{array} \srb\n=\n\slb \sbegin{array}{cc}\n0 & v^0 - v^\sva \ssi^P_\sva \s\s\nv^0 + v^\sva \ssi^P_\sva & 0\n\send{array} \srb\n\s\s\n&=& \n\slb \sbegin{array}{cccc}\n0 & 0 & v^0-v^3 & -v^1+iv^2 \s\s\n0 & 0 & -v^1-iv^2 & v^0+v^3 \s\s\nv^0+v^3 & v^1-iv^2 & 0 & 0 \s\s\nv^1+iv^2 & v^0-v^3 & 0 & 0\n\send{array} \srb\n\send{eqnarray}\nand [[spacetime pseudoscalar|Cl(1,3)]], $\sga = i \ssi^P_3 \sotimes 1$. The $v_{L/R}$ are ''left and right chiral vector parts'' -- $2\stimes2$ Hermitian matrices projected out by the [[left/right chirality projector]]. (//They satisfy...//) Note that a vector is completely determined by one of its chiral parts.\n\n''Dirac representation'' of CL(1,3),\n\sbegin{eqnarray}\n\sga_0 &=& \ssi^P_3 \sotimes 1 \s\s\n\sga_\sva &=& -i \ssi^P_2 \sotimes \ssi^P_\sva\n\send{eqnarray}\n\n(real) ''Majorana representation'' of Cl(3,1),\n\sbegin{eqnarray}\n\sga_0 &=& i \ssi^P_1 \sotimes \ssi^P_2 \s\s\n\sga_1 &=& \ssi^P_1 \sotimes \ssi^P_3 \s\s\n\sga_2 &=& \ssi^P_3 \sotimes \ssi^P_1 \s\s\n\sga_3 &=& \ssi^P_1 \sotimes \ssi^P_1\n\send{eqnarray}\nMultiplying these matrices by $i$ switches them between representations of Cl(1,3) and Cl(3,1).\n\nRef:\nhttp://en.wikipedia.org/wiki/Dirac_matrices

If $\sPsi$ is a [[spinor]] field and $\sf{A} \sin \sf{\srm Lie}(G)$ a [[principal bundle]] connection in a representation matched to the spinor, the [[covariant derivative]] of the spinor field is\n$$\n\sf{\sna} \sPsi = \slp \sf{d} + \sf{A} \srp \sPsi\n$$\nNote that $\sf{A}$ includes the [[spin connection]], $\sf{\som}$ (as the connection for the [[Clifford vector bundle]] subbundle of the full principal bundle) and usually other parts, which will be written as $\sf{G}$, so\n$$\n\sf{A} = \sha \sf{\som} + \sf{G}\n$$\nIf we write the [[frame]] over the base manifold as $\sve{e} = \sga^\smu \sve{e_\smu}$, the ''Dirac operator'' acting on the spinor is defined as\n$$\n\sna \sPsi = \sve{e} \sf{\sna} \sPsi = \sga^\smu \slp e_\smu \srp^i \slp \spa_i + \sfr{1}{4} \som_i{}^{\snu \srh} \sga_{\snu \srh} + G_i{}^B T_B \srp \sPsi\n$$\nusing the [[vector-form algebra]].

A ''Dirac [[spinor]]'', $\sPsi$, 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,\n$$\n\sPsi = \sPsi_L + \sPsi_R =\n\slb \sbegin{array}{c}\n\sps_L \s\s\n0\n\send {array} \srb\n+\n\slb \sbegin{array}{c}\n0 \s\s\n\sps_R\n\send {array} \srb\n=\n\slb \sbegin{array}{c}\n\sps_L \s\s\n\sps_R\n\send {array} \srb\n=\n\slb \sbegin{array}{c}\n\sps_L^\swedge \s\s\n\sps_L^\svee \s\s\n\sps_R^\swedge \s\s\n\sps_R^\svee\n\send {array} \srb\n$$\nThese parts are the ''left handed Weyl spinor'',\n$$\n\sps_L = \slb \sbegin{array}{c}\n\sps_L^\swedge \s\s\n\sps_L^\svee\n\send {array} \srb\n$$\nand ''right handed Weyl spinor'', $\sps_R$ -- each represented by a column of 2 complex (or complex [[Grassmann|Grassmann number]]) numbers -- the ''spin up'' and ''spin down'' components. These weyl spinors may be projected out,\n$$\n\sPsi_{L/R} = P_{L/R} \sPsi\n$$\nby the [[left/right chirality projector]],\n$$\nP_{L/R} = \sha \slp 1 \smp i \sga \srp\n$$\n(In this equation, the four component column with two zero entries is equated to a two component column.)\n

/***\nAuthors: Eric Shulman\nversion: 1.0.0\nsource: http://www.tiddlytools.com/\nThis plugin allows you to disable TiddlyWiki's automatic WikiWord linking behavior, so that WikiWords embedded in note content will be rendered as regular text, instead of being automatically converted to note links. To create a note link when automatic linking is disabled, you must enclose the link text within {{{[[}}} and {{{]]}}}.\n!!!!!Code\n***/\n//{{{\nversion.extensions.disableWikiLinks= {major: 1, minor: 0, revision: 0, date: new Date(2005,12,9)};\n\n// G changed to have this on, without checkbox\nconfig.options.chkDisableWikiLinks= true;\n\n// find the formatter for wikiLink and replace handler with 'pass-thru' rendering\nfor (var i=0; i<config.formatters.length && config.formatters[i].name!="wikiLink"; i++);\nconfig.formatters[i].coreHandler=config.formatters[i].handler;\nconfig.formatters[i].handler=function(w) {\n // if not enabled, just do standard WikiWord link formatting\n if (!config.options.chkDisableWikiLinks) return this.coreHandler(w);\n // supress any leading "~" (if present)\n var skip=(w.matchText.substr(0,1)==config.textPrimitives.unWikiLink)?1:0;\n w.outputText(w.output,w.matchStart+skip,w.nextMatch)\n}\n//}}}

<<note HideTags>>What is done:\n*All [[gauge fields|connection]], [[gravity|spacetime]], and Higgs in ''one'' [[connection]], with fermions as [[BRST ghosts|BRST technique]].\n\nTo do:\n*Will particle assignments work with [[E8]]? (Get the CKMPMNS matrix?)\n*Why is the action what it is? (How does symmetry breaking happen?)\n*Is a four dimensional base [[manifold]] emergent?\n*How does this theory get quantized? (LQG methods should apply.)\n**Natural explanation for QM as a bonus?\n\nWhat this theory will mean, if it all works:\n*Gravitational [[frame]] and Higgs are intimately related.\n*Naturally combines standard model with gravity -- so it's a [[T.O.E.|theory of everything]]\n**(It's also a U.F.T., but I don't like to call it that.)\n*Our universe is a very pretty shape!\n\n@@display:block;text-align:center;Gar@Lisi.com\nhttp://deferentialgeometry.org $\sp{{}_{(}}$@@

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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]].

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]].

<<note 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:\n$$\n\sudf{A} = \sf{H} + \sf{G} + \sud{\sPs}{}_I + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} = \n$$\n$$\n\stext{something like}_{\sp{\sbig(}}\n$$\n$$\n{\ssmall\n\sbegin{array}{c}\n\n\s!\s!\s! \slb \sbegin{array}{cccc}\n\sfrac{1}{2} \sf{\som_L} \s!+\s! i \sf{W^3} \s!&\s! i \sf{W^1} \s!+\s! \sf{W^2} \s!&\s! - \s! \sfrac{1}{4} \sf{e_R} \sph_0^* \s!& \sfrac{1}{4} \sf{e_R} \sph_+ \s! \s\s\n\ni \sf{W^1} \s!-\s! \sf{W^2} \s!&\s! \sfrac{1}{2} \sf{\som_L} \s!-\s! i \sf{W^3} \s!&\s! \sp{-} \sfrac{1}{4} \sf{e_R} \sph_+^* \s!& \sfrac{1}{4} \sf{e_R} \sph_0 \s! \s\s\n\n-\sfrac{1}{4} \sf{e_L} \sph_0 & \sfrac{1}{4} \sf{e_L} \sph_+ & \s!\s!\s!\s! \sfrac{1}{2} \sf{\som_R} \s!+\s! i \sf{B} \s!\s! \s!& & \s! \s\s\n\n\sp{-}\sfrac{1}{4} \sf{e_L} \sph_+^* & \sfrac{1}{4} \sf{e_L} \sph_0^* & &\s! \s!\s! \sfrac{1}{2} \sf{\som_R} \s!-\s! i \sf{B} \s!\s!\s!\s!\s!\n\send{array} \srb\n\n\s!\s!+\s!\s!\n\slb \sbegin{array}{cccc}\ni \sf{B} \s!\s! & & & \s\s\n&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!+\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^1} \s!-\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! i\sf{G^4} \s!-\s! \sf{G^5} \s\s\n&\s!\s!\s! i\sf{G^1} \s!+\s! \sf{G^2} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s! i \sf{G^{3+8}} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!-\s! \sf{G^7} \s\s\n&\s!\s!\s! i\sf{G^4} \s!+\s! \sf{G^5} \s!\s!\s!&\s!\s!\s! i\sf{G^6} \s!+\s! \sf{G^7} \s!\s!\s!&\s!\s!\s! \sfrac{-i}{3} \s! \sf{B} \s!-\s!\s! \sfrac{2i}{\ssqrt{3}}\sf{G^8}\n\send{array} \srb\n\s\s\n\s; \s\s\n+\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^e_L & \sud{u}{}_L^r & \sud{u}{}_L^g & \sud{u}_L^b \s\s\n\sud{e}{}_L & \sud{d}{}_L^r & \sud{d}{}_L^g & \sud{d}{}_L^b \s\s\n\sud{\snu}{}^e_R & \sud{u}{}_R^r & \sud{u}{}_R^g & \sud{u}{}_R^b \s\s\n\sud{e}{}_R & \sud{d}{}_R^r & \sud{d}{}_R^g & \sud{d}{}_R^b\n\send{array} \srb\n\s;+\s;\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^\smu_L & \sud{c}{}_L^r & \sud{c}{}_L^g & \sud{c}_L^b \s\s\n\sud{\smu}{}_L & \sud{s}{}_L^r & \sud{s}{}_L^g & \sud{s}{}_L^b \s\s\n\sud{\snu}{}^\smu_R & \sud{c}{}_R^r & \sud{c}{}_R^g & \sud{c}{}_R^b \s\s\n\sud{\smu}{}_R & \sud{s}{}_R^r & \sud{s}{}_R^g & \sud{s}{}_R^b\n\send{array} \srb\n\s;+\s;\n\slb \sbegin{array}{cccc}\n\sud{\snu}{}^\sta_L & \sud{t}{}_L^r & \sud{t}{}_L^g & \sud{t}_L^b \s\s\n\sud{\sta}{}_L & \sud{b}{}_L^r & \sud{b}{}_L^g & \sud{b}{}_L^b \s\s\n\sud{\snu}{}^\sta_R & \sud{t}{}_R^r & \sud{t}{}_R^g & \sud{t}{}_R^b \s\s\n\sud{\sta}{}_R & \sud{b}{}_R^r & \sud{b}{}_R^g & \sud{b}{}_R^b\n\send{array} \srb_{\sp{(}}\n\send{array}\n}\n$$\n

*Quantization\n**Coupling constants run.\n***Large $\sLa$ compatible with UV fixed point.\n**Just a connection -- amenable to LQG, spin foams, etc.\n*Understand triality-generation relationship better\n**Possible collapse or mixing to graviweak $SL(2,\smathbb{C})$.\n**The role of $\sf{w}+\sf{x}\sPh$ and symmetry breaking.\n**Getting the CKMPMNS matrix would be nice.\n*Why is the action what it is?\n**Pulling $\sf{e}$ out and putting it into $\sff{F} \sff{*F}$ and $\sfff{\sod{B}}$ seems weird.\n***Why $\sf{e}\sph$ simple?\n***Four dimensional base manifold emergent?\n\nWhat this theory will mean, if it all works:\n*Combines standard model with gravity -- with LQG, it's a T.o.E.\n*Our universe is very pretty.\n\n@@display:block;text-align:center; http://deferentialgeometry.org &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Garrett Lisi@@\n<<note HideTags>>

Everything in an $E8$ principal bundle connection,\n$$\n\sudf{A} \sin \sudf{e8}\n$$\nPeriodic table of interactions (Feynman vertices) from curvature,\n$$\n\sudff{F} = \sf{d} \sudf{A} + {\sscriptsize \sfrac{1}{2}} \sbig[ \sudf{A}, \sudf{A} \sbig]\n$$\ndescribed by the $E8$ root polytope. Three generations through triality,\n$$\nT \s, e = \smu \sqquad T \s, \smu = \sta \sqquad T \s, \sta = e\n$$\nPati-Salam $SU(2)_L \stimes SU(2)_R \stimes SU(4)$ GUT and MM gravity together,\n$$\nS = \sint \sbig< \sff{\sod{B}} \sudff{F}\n+ {\sscriptsize \sfrac{\spi}{4}} \sff{B}{}_G \sff{B}{}_G \sga + \sff{B'} \sff{*B'} \sbig>\n$$\nNo free parameters -- masses from Higgs VEV's,\n$$\ng_1 = \ssqrt{\sfr{3}{5}} \sqquad g_2=1 \sqquad g_3=1 \sqquad \sLa=\sfr{3}{4}\sph^2 \sqquad \sph_0 , \sph_1, \sPh \sdots \n$$\nEverything is pure geometry, and it's very beautiful.\n<<note HideTags>>

$$\n\sudf{A} = \sf{H}{}_1 + \sf{H}{}_2 + \sud{\sPs}{}_{I} + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} \squad \sin \s;\s; \sudf{e8} \svp{|_{\sbig(}}\n$$\n$$\n\sbegin{array}{rclcl}\n\sf{H}{}_1 \s!\s!&\s!\s!=\s!\s!&\s!\s! {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph + \sf{W} + \sf{B}{}_1 & \sin & \sf{so}(7,1) \s\s[-.1em]\n&& \sf{\som} & \sin & \sf{so}(3,1) \s\s[-.1em]\n&& \sf{e} \sph = (\sf{e}{}_1+\sf{e}{}_2+\sf{e}{}_3+\sf{e}{}_4)\stimes(\sph_{+/0}+\sph_{-/1}) & \sin & \sf{4} \stimes (2+\sbar{2}) \s\s\n&& \sf{W} + \sf{B}{}_1 & \sin & \sf{su}(2) + \sf{su}(2) \s\s\n\sf{H}{}_2 \s!\s!&\s!\s!=\s!\s!&\s!\s! \sf{w} + \sf{B}{}_2 + \sf{x} \sPh + \sf{g} & \sin & \sf{so}(8) \s\s[-.2em]\n&& \sf{w} + \sf{B}{}_2 & \sin & \sf{u}(1) + \sf{u}(1) \s\s[-.3em]\n&& \sf{x} \sPh = (\sf{x}{}_{1}+\sf{x}{}_{2}+\sf{x}{}_{3})\stimes(\sPh^{r/g/b} + {\sPh}{}^{\sbar{r}/\sbar{g}/\sbar{b}}) & \sin & \sf{3} \stimes (3+\sbar{3}) \s\s[-.1em]\n&& \sf{g} & \sin & \sf{su}(3) \s\s\n\sud{\sPsi}{}_{I} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_e + \sud{e} + \sud{u} + \sud{d} & \sin & 8_{S+} \s!\stimes 8_{S+} \s\s\n\sud{\sPsi}{}_{II} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_\smu + \sud{\smu} + \sud{c} + \sud{s} & \sin & 8_{V} \stimes 8_{V} \s\s \n\sud{\sPsi}{}_{III} \s!\s!&\s!\s!=\s!\s!&\s!\s! \sud{\snu}{}_\sta + \sud{\sta} + \sud{t} + \sud{b} & \sin & 8_{S-} \s!\stimes 8_{S-} \s\s\n\send{array}\n$$\n<<note HideTags>>

$$\n\sudff{F} = \sf{d} \sudf{A} + \sudf{A} \sudf{A}\n= \sff{F}{}_1+\sff{F}{}_2+ \sf{D} \sbig( \sud{\sPs}{}_{I} + \sud{\sPs}{}_{II} + \sud{\sPs}{}_{III} \sbig) \squad \sin \s;\s; \sudff{e8} \svp{|_{\sBig(}}\n$$\n$$\n\sbegin{array}{rlcl}\n\sff{F}{}_1 \s!\s!\s!\s!&=\n\sha \sbig( \sff{R} - \sfr{1}{8} \sf{e} \sf{e} \sph^2 \sbig)\n+ \sfr{1}{4} \sbig( \sff{T} \sph - \sf{e} \sf{D} \sph \sbig)\n+ \sbig( \sff{F}{}_{B_1} + \sff{F}{}_W \sbig) & \sin & \sf{so}(7,1) \s\s[.1em]\n& \sff{R} = \sf{d} \sf{\som} + \sha \sf{\som} \sf{\som} & \sin & \sf{so}(3,1) \s\s[.1em]\n& \sff{T} \sph \s!-\s! \sf{e} \sf{D} \sph = \sbig( \sf{d} \sf{e} \s!+\s! \sha [ \sf{\som}, \sf{e} ] \sbig) \sph - \sf{e} \sbig( \sf{d} \sph \s!+\s! [ \sf{B}{}_1 \s!+\s! \sf{W}, \sph ] \sbig) & \sin & \sf{4} \stimes (2+\sbar{2}) \s\s[.2em]\n& \sff{F}{}_{B_1} + \sff{F}{}_W = (\sf{d} \sf{B}{}_1 + \sf{B}{}_1 \sf{B}{}_1) + (\sf{d} \sf{W} + \sf{W} \sf{W}) & \sin & \sf{su}(2) \s!+\s! \sf{su}(2) \s\s[.4em]\n\sff{F}{}_2 \s!\s!\s!\s!&=\n\sbig( \sff{F}{}_{w} + \sff{F}{}_{B_2} + \sf{x}\sPh\sf{x}\sPh \sbig)\n+ \sbig( (\sf{D} \sf{x}) \sPh - \sf{x} \sf{D} \sPh \sbig)\n+\sff{F}{}_{g}\n& \sin & \sf{so}(8) \s\s[.1em]\n& \sff{F}{}_{w} + \sff{F}{}_{B_2} = \sf{d} \sf{w} + \sf{d} \sf{B}{}_2 & \sin & \sf{u}(1) + \sf{u}(1) \s\s[.1em]\n& (\sf{D} \sf{x}) \sPh \s!-\s! \sf{x} \sf{D} \sPh \s!=\s! \n\sbig( \sf{d} \sf{x} \s!+\s! [ \sf{w} \s!+\s! \sf{B}{}_2, \s! \sf{x} ] \sbig) \sPh \s!-\s! \sf{x} \sbig( \sf{d} \sPh \s!+\s! [ \sf{g}, \s! \sPh ] \sbig) \s!\s!\s!\n & \sin & \sf{3} \stimes (3+\sbar{3}) \s\s[0em]\n& \sff{F}{}_{g} = \sf{d} \sf{g} + \sf{g} \sf{g} & \sin & \sf{su}(3)\n\send{array}\n$$\n$$\n\sf{D} \sud{\sPsi} = \sbig( \sf{d} + {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph \sbig) \sud{\sPs}\n+ \sf{W} \sud{\sPs}{}_L + \sf{B}{}_1 \sud{\sPs}{}_R - \sud{\sPs} \sbig( \sf{w} + \sf{B}{}_2 + \sf{x} \sPh \sbig) - \sud{\sPs}{}_q \s, \sf{g}\n\svp{|^{\sBig(}}\n$$\n<<note HideTags>>

<<note HideTags>>Build new ${\srm Lie}(E8)$ generators from old ones:\n$$\n\sbegin{array}{rclclcll}\nH_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\sal\sbe} \s!\s!&\s!\s!=\s!&\s!\s! \sga^{(8)+}_{\sal\sbe} \sotimes 1 \s!\s!&\s!\s!\sin\s!&\s!\s! so(8)^+ \sotimes 1\n\s!\s!&\s!=\s, so(8)^H \s\s\nG_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\slp\sal+8\srp\slp\sbe+8\srp} \s!\s!&\s!\s!=\s!&\s!\s! P^{\slp8\srp}_+ \sotimes \sga^{(8)}_{\sal\sbe} \s!\s!&\s!\s!\sin\s!&\s!\s! 1 \sotimes so(8) \n\s!\s!&\s!=\s, so(8)^G \s\s\n\sPs^I_{\sal\sbe} \s!\s!&\s!=\s!&\s!\s! \sga^{\slp16\srp+}_{\sal\slp\sbe+8\srp} \s!\s!&\s!\s!=\s!&\s! \sga^{(8)+}_\sal \sotimes \sga^{(8)}_\sbe \s!\s!&\s!\s!\sin\s!&\s!\s! v^{(8)+} \sotimes v^{(8)}\n\s!\s!&\s!=\s, S^I \s\s\n\sPs^{II}_{ab} \s!\s!&\s!=\s!&\s!\s! Q^+_{16\slp a-1\srp+b} \s!\s!&\s!\s!=\s!&\s!\s! q^+_a \sotimes q^+_b \s!\s!&\s!\s!\sin\s!&\s!\s! S^{(8)+} \sotimes S^{(8)+}\n\s!\s!&\s!=\s, S^{II} \s\s\n\sPs^{III}_{ab} \s!\s!&\s!=\s!&\s!\s! Q^+_{16\slp a-1\srp+b+8} \s!\s!&\s!\s!=\s!&\s!\s! q^+_a \sotimes q^-_b \s!\s!&\s!\s!\sin\s!&\s!\s! S^{(8)+} \sotimes S^{(8)-}\n\s!\s!&\s!=\s, S^{III}\n\send{array}\n$$\n\nWith these basis generators, the ${\srm Lie}(E8)$ elements are:\n\sbegin{eqnarray}\nE &=& H + G + \sPs_I + \sPs_{II} + \sPs_{III} \s\s\n&=& \sha h^{\sal\sbe} H_{\sal\sbe} + \sha g^{\sal\sbe} G_{\sal\sbe} + \sps_I^{\sal\sbe} \sPs^I_{\sal\sbe} + \sps_{II}^{ab} \sPs^{II}_{ab} + \sps_{III}^{ab} \sPs^{III}_{ab} \s\s\n&\sin& so(8)^H + so(8)^G + S^I + S^{II} + S^{III}_{\sp{(}}\n\send{eqnarray}\n\n

<<note HideTags>>@@display:block;text-align:center;[img[images/png/e8 periodic table.png]]@@\n//"E8 is perhaps the most beautiful structure in all of mathematics, but it's very complex."// &nbsp;&nbsp;&nbsp;&nbsp;&nbsp; -- Hermann Nicolai

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour.mov" width="602" height="602" controller="false" autoplay="false" loop="false"></embed>\n<!-- <embed src="talks/Perimeter07/anim/e8tour (om up)/p1.png" width="608" height="609"></embed> -->\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n

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<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p562.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p642.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n

<<note HideTags>>@@display:block;text-align:center;<html><center>\n<embed src="talks/Perimeter07/anim/e8tour/p662.png" width="608" height="609"></embed>\n</center></html>E.S.T.o.E.: Each E8 vertex corresponds to an elementary particle field.@@\n

<<note HideTags>>@@display:block;text-align:center;${\srm Lie}(E8)$ has $(248-8)=240$ roots in 8D space -- vertices of $P4_{2,1}$:$\sp{{}_{\sbig(}}$\n<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.$\sp{{}{\sBig(}^{(}}$@@

<<note HideTags>>The ${\srm Lie}(E8)$ brackets between elements in the various parts:\n$$\n\sbegin{array}{cc}\n\sbegin{array}{rcl}\n\sbig[ H_1, H_2 \sbig] \s!\s!&\s!=\s!&\s!\s! H_1 H_2 - H_2 H_1 \s\s\n\sbig[ G_1, G_2 \sbig] \s!\s!&\s!=\s!&\s!\s! G_1 G_2 - G_2 G_1 \s\s\n&&\s\s\n\sbig[ H, \sPs_I \sbig] \s!\s!&\s!=\s!&\s!\s! H \s, \sPs_I \s\s\n\sbig[ H, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! H^+ \s, \sPs_{II} \s\s\n\sbig[ H, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! H^+ \s, \sPs_{III} \s\s\n&&\s\s\n\sbig[ G, \sPs_I \sbig] \s!\s!&\s!=\s!&\s!\s! \sPs_I \s, G \s\s\n\sbig[ G, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sPs_{II} \s, G^+ \s\s\n\sbig[ G, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sPs_{III} \s, G^-\n\send{array}\n&\n\sbegin{array}{rcl}\n\sbig[ \sPs^1_I, \sPs^2_I \sbig] \s!\s!&\s!=\s!&\s!\s! -2 \sbig( \sPs^1_I \s, {\sPs^2_I}^T \sbig)_H \s\s\n&& -2 \sbig( {\sPs^1_I}^T \sPs^2_I \sbig)_{G_{\sp{(}}} \s\s\n\s\s\n\sbig[ \sPs^1_{II}, \sPs^2_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs^1_{II} \sGa^+ {\sPs^2_{II}}^T \sbig)_H \s\s\n&&\s!\s! - \sbig( {\sPs^1_{II}}^T \sGa^+ \sPs^2_{II} \sbig)_{G_{\sp{(}}} \s\s\n\sbig[ \sPs^1_{III}, \sPs^2_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs^1_{II} \sGa^+ {\sPs^2_{II}}^T \sbig)_H \s\s\n&&\s!\s! - \sbig( {\sPs^1_{II}}^T \sGa^- \sPs^2_{II} \sbig)_G \s\s\n&&\s\s\n\sbig[ \sPs_I, \sPs_{II} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_I \sGa^{++} \sPs_{II} \sbig)_{III} \s\s\n\sbig[ \sPs_I, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_I \sGa^{+-} \sPs_{III} \sbig)_{II} \s\s\n\sbig[ \sPs_{II}, \sPs_{III} \sbig] \s!\s!&\s!=\s!&\s!\s! - \sbig( \sPs_{II} \sGa^{++} \sPs_{III} \sbig)_I\n\send{array}\n\send{array}\n$$\nNote: $H$ acts on $\sPs$'s from the left and $G$ acts from the right.$^{\sp{\sbig(}}_{\sp{(}}$

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The ''Ehresmann Cartan connection'', $\sf{\sve{\scal C}}$, 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\n\sbegin{eqnarray}\n\sf{\sve{\scal C}}(x, x_s, y) &=& \sf{C^J}(x) \s, \sve{\sxi^L_J}(x_s, y) + \sf{\sve{\scal I}} \s\s\n&=& \sf{C^J}(x) \s, L^I{}_J(x_s, y) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}} \n\send{eqnarray}\nin which $\sve{\sxi^L_J}$ and $\sve{\sxi^R_J} \ssim T_J$ are the [[left and right action vector fields|Lie group geometry]] for the fibers, $G_x$, the [[left-right rotator]] is\n$$\nL^I{}_J(x_s, y) = \sve{\sxi^L_J} \sf{\sxi_R^I} = \slp T^I, g^-(x_s,y) \s, T_J \s, g(x_s,y) \srp\n$$\nthe [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers) is\n$$\n\sf{\sve{\scal I}}(x_s,y) = \sf{\sxi_R^J} \sve{\sxi^R_J} = \sf{dx_s^a} \sve{\spa^s_a} + \sf{dy^p} \sve{\spa_p} = \sf{\sve{{\scal I}_{G/H}}} + \sf{\sve{{\scal I}_H}}\n$$\nand $\sf{C^J}(x)$ are the components of the [[Cartan connection|Cartan geometry]] over $M$. The Ehresmann Cartan connection is a projection, $\sf{\sve{\scal C}} \sf{\sve{\scal C}} = \sf{\sve{\scal C}}$, is [[right invariant]], $R_g^*\sf{\sve{\scal C}} = \sf{\sve{\scal C}}$, 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]], $\sf{\scal I}(x_s,y) = \sf{\sxi_R^J} T_J = g^- \sf{d} g$, over the total space to get the ''Ehresmann Cartan connection form'',\n\sbegin{eqnarray}\n\sf{\scal C}(x,x_s,y) &=& \sf{\sve{\scal C}} \sf{\scal I} = \sf{C^J}(x) \sve{\sxi^L_J} \sf{\sxi_R^I} T_I + \sf{\sve{\scal I}} \sf{\scal I} \s\s\n&=& \slp \sf{C^J} L^I{}_J(x_s,y) + \sf{\sxi_R^I} \srp T_I \s\s\n&=& g^-(x_s,y) \s, \sf{C}(x) \s, g(x_s,y) + g^-(x_s,y) \s, \sf{d} \s, g(x_s,y) \n\send{eqnarray}\nThis form [[pulls back|pullback]] along the canonical reference section, $\ssi_0^G$, to give the Cartan connection,\n$$\n\ssi_0^{G*} \sf{\scal C} = \sf{C}(x)\n$$\nand satisfies $R_g^* \sf{\scal C} = g^- \sf{\scal C} g$ under the right action.\n\nThe ''[[FuN curvature]] of the Ehresmann Cartan connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,x_s,y) &=& - \sha \slb \sf{\sve{\scal C}}, \sf{\sve{\scal C}} \srb_L \s\s\n&=& \slp \sf{d} \sf{C^K} + \sha \sf{C^I} \sf{C^J} C_{IJ}{}^K \srp \sve{\sxi^L_K}(x_s,y)\n\send{eqnarray}\nwhich is vector valued in the vertical subspace and right invariant, $R_g^* \sff{\sve{\scal F}} = \sff{\sve{\scal F}}$. The ''FuN curvature form of the Ehresmann Cartan connection'' is a $Lie(G)$ valued 2-form over $E_G$,\n\sbegin{eqnarray}\n\sff{\scal F} &=& \sff{\sve{\scal F}} \sf{\scal I} = \slp \sf{d} \sf{C^K} + \sha \sf{C^I} \sf{C^J} C_{IJ}{}^K \srp g^-(x_s,y) \s, T_K \s, g(x_s,y) \s\s\n&=& g^-(x_s,y) \slp \sf{d} \sf{C} + \sha \slb \sf{C}, \sf{C} \srb \srp g(x_s,y) \s\s\n&=& g^-(x_s,y) \slp \sf{d} \sf{C} + \sf{C} \sf{C} \srp g(x_s,y)\n\send{eqnarray}\nThis form pulls back along the canonical reference section to give the [[Cartan geometry]] curvature,\n$$\n\ssi_0^{G*} \sff{\scal F} = \sf{d} \sf{C} + \sf{C} \sf{C} = \sff{F}(x)\n$$\nand satisfies $R_g^* \sff{\scal F} = g^- \sff{\scal F} g$ under the right action.

When $H$ is [[reductive]] in $G$ (which is usually assumed) the [[Cartan connection|Cartan geometry]] splits as\n$$\n\sf{C}(x) = \sf{e}(x) + \sf{A}(x) = \sf{e^A} K_A + \sf{A^P} H_P\n$$\nthe [[Ehresmann Cartan connection]] can be made to follow this split,\n\sbegin{eqnarray}\n\sf{\sve{\scal C}}(x,x_s,y) &=& \sf{C^J}(x) \s, L^I{}_J(x_s, y) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}} \s\s\n&=& \sf{\sve{\scal E}} + \sf{\sve{\scal A}}\n\send{eqnarray}\nwith the ''Ehresmann Cartan frame'' and ''Ehresmann Cartan H-connection'' defined over patches of the total space, $E_G$, of the [[Ehresmann Cartan geometry]] as:\n\sbegin{eqnarray}\n\sf{\sve{\scal E}}(x, x_s, y) &=& \sf{e^A}(x) \s, (L^h)^I{}_K(y) \s, (L^r)^K{}_A(x_s) \s, \sve{\sxi^R_I}(x_s, y) \s\s\n\sf{\sve{\scal A}}(x, x_s, y) &=& \sf{A^P}(x) \s, (L^h)^I{}_K(y) \s, (L^r)^K{}_P(x_s) \s, \sve{\sxi^R_I}(x_s, y) + \sf{\sve{\scal I}}\n\send{eqnarray}\nwith 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]], $\sf{\scal C} = \sf{\sve{\scal C}} \sf{\scal I}$, also splits,\n\sbegin{eqnarray}\n\sf{\scal C}(x,x_s,y) &=& g^-(x_s,y) \s, \sf{C}(x) \s, g(x_s,y) + g^-(x_s,y) \s, \sf{d} \s, g(x_s,y) \s\s\n&=& \sf{\scal E} + \sf{\scal A}\n\send{eqnarray}\nwith the ''Ehresmann Cartan frame form'' and ''Ehresmann Cartan H-connection form'' defined over $E_G$ as:\n\sbegin{eqnarray}\n\sf{\scal E}(x, x_s, y) &=& \sf{\sve{\scal E}} \sf{\scal I} = \sf{{\scal E}^I} T_I = g^- \s, \sf{e} \s, g(x_s,y) \sin \sf{\srm Lie}(G) \s\s\n\sf{\scal A}(x, x_s, y) &=& \sf{\sve{\scal A}} \sf{\scal I} = \sf{{\scal A}^I} T_I = g^- \s, \sf{A} \s, g(x_s,y) + g^- \s, \sf{d} \s, g(x_s,y) \sin \sf{\srm Lie}(G)\n\send{eqnarray}\nwith $g(x_s,y) = r(x_s) \s, h(y)$.\n\nThis 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]]. \n

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]]), $\sf{\sve{\scal A}} = \sf{\sve{\scal C}}$, over a $(n_M + n_G)$ dimensional total space, $E_G \ssim M \stimes G$, 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, $H \ssubset G$, so the bundle produces two [[associated]] bundles, the [[Cartan H-bundle]], $E_H \ssim M \stimes H$, and the [[Cartan homogeneous space bundle]], $E_S \ssim M \stimes S$. The Ehresmann Cartan connection gives the ''Ehresmann Cartan connection form'', $\sf{\scal C} = \sf{\sve{\scal C}} \sf{\scal I} \sin \sf{\srm Lie}(G)$, which gives the associated [[Cartan H-bundle connection form|Cartan H-bundle]], $\sf{{\scal C}_H}$, over $E_H$ and [[Cartan homogeneous space connection form|Cartan homogeneous space bundle]], $\sf{{\scal C}_S}$, over $E_S$. A ''generalized Ehresmann Cartan geometry'' has $n_M \sneq n_S$.\n\nThere 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 $h(y) \sin H \ssubset G$, and the remaining $n_S$ homogeneous space coordinates, $x_s^a$, correspond to $x_s \sin S$. 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\n$$\np \ssim (x,x_s,y) \ssim (x,x_s,h(y)) \ssim (x,g(x_s,y))\n$$\nThe chosen [[coset representative section|homogeneous space]], $r : S \sto G$, allows points of $G$ to be specified in terms of points of $G/H$ and $H$ via the right action, \n$$\ng(x_s, y) = R_{h(y)} r(x_s) = r(x_s) \s, h(y)\n$$\nThe ''Cartan geometry [[reference section|Ehresmann gauge transformation]]'', $\ssi^G : M \sto E_G$, is then determined by the reference section, $\ssi^H : M \sto E_H$, of the Cartan H-bundle and the reference section, $\ssi^S : M \sto E_S$, of the Cartan homogeneous space bundle. With $\ssi^H(x) = {\sbig (} x,h(y_\ssi(x)) {\sbig )}$ and $\ssi^S(x) = (x, x_{s\ssi}(x))$ we have:\n$$\n\ssi^G(x) = {\sbig (} x, r(x_{s\ssi}(x)) \s, h(y_\ssi(x)) {\sbig )} \n$$\nThe [[canonical reference section|Ehresmann principal bundle connection]], $\ssi_0^G(x) = (x,1) \ssim (x,0,0)$, of $E_G$ corresponds to the canonical reference section, $\ssi_0^H(x) = (x,1) \ssim (x,0)$, of $E_H$ and the zero point reference section, $\ssi_0^S(x) = (x, 0)$, of $E_S$. \n\nThe 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, $\spi^G_H : E_G \sto E_H$ and $\spi^G_S : E_G \sto E_S$, are given by $\spi^G_H(x,x_s,y)=(x,y)$ and $\spi^G_S(x,x_s,y) = (x,x_s)$. There are also reference sections, $\ssi'^S : E_H \sto E_G$ and $\ssi'^H : E_S \sto E_G$, over these bases, determined by the reference sections over their partner bundle, $\ssi'^S(x,y)=(x,x_{s\ssi}(x),y)$ and $\ssi'^H(x,x_s)=(x,x_s,y_\ssi(x))$. The complete web of bundle maps is summarized by:\n$$\n\sbegin{array}{ccc}\nE_G & \smatrix{\slower8mu {\soverset{\ssi'^S}{\slongleftarrow}}\s\s \sraise8mu {\sunderset{\spi^G_H}{\slongrightarrow}}} & E_H\s\s\n{}^{\spi^G_S} \s! {\sbig \sdownarrow} {\sbig \suparrow} \s! {}_{\ssi'^H} & {}_{\spi_G} \s! \s! \s! \s! \ssearrow \s! \s! \snwarrow \s! \s! \s! \s! {}^{\ssi^G} & {}^{\spi_H} \s! {\sbig \sdownarrow} {\sbig \suparrow} \s! {}_{\ssi^H}\s\s\nE_S & \smatrix{\slower8mu {\soverset{\ssi^S}{\slongleftarrow}}\s\s \sraise8mu {\sunderset{\spi_S}{\slongrightarrow}}} & M\n\send{array}\n$$\nand we have\n\sbegin{eqnarray}\n\spi_G &=& \spi_H \scirc \spi^G_H = \spi_S \scirc \spi^G_S \s\s\n\ssi^G &=& \ssi'^H \scirc \ssi^S = \ssi'^S \scirc \ssi^H\n\send{eqnarray}\n\nThe 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]], $\sf{A}$, and [[covariant derivative]], $\sf{\sna}$, defined over the base manifold, $M$, or alternatively via an ''Ehresmann connection'', $\sf{\sve{\scal A}}$, defined over the total space, $E$, of the bundle. This [[vector valued form]] is a [[vector projection]], $\sf{\sve{\scal A}}\sf{\sve{\scal A}}=\sf{\sve{\scal A}}$, 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,\n$$\nT_p E = V_r + V_0\n$$\nThe range subspace of $\sf{\sve{\scal A}}$ is the ''vertical subspace'', $V_r=V_V$, and the collection of these vertical vector fields over $E$ is an involutive [[distribution]], $\sve{\sDe_V}=\sve{\sDe_r}$, of vectors tangent to the fibers of the bundle, $\spi_* \sve{\sDe_V} = 0$. In this way, the Ehresmann connection determines the fibers of the fiber bundle. The vector fields, $\sve{\sxi_A} \ssim T_A$, in $\sve{\sDe_V}$ are the flow fields of the group action on the fibers. They are in involution since\n$$\n\slb \sve{\sxi_A},\sve{\sxi_B} \srb_L = C_{AB}{}^C \sve{\sxi_C}\n$$\nThe kernel subspace of $\sf{\sve{\scal A}}$ is the ''horizontal subspace'', $V_0=V_H$, and the collection of these horizontal vector fields over $E$ form a ''horizontal distribution'', $\sve{\sDe_H}=\sve{\sDe_0}$, that may or may not be in involution (more on that further down). \n\nThe Ehresmann connection respects the symmetry of the structure group. If we take the group action on manifold points to be a right action $p \smapsto R_g p = pg$, the Ehresmann connection at different points along fibers are related by the [[pullback]],\n$$\nR_g^* \sf{\sve{\scal A}} (pg) = \sf{\sve{\scal A}} (p)\n$$\nIn 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\n$$\n\slp R_g^* \sf{dz^i} \srp {\scal A}_i{}^j (pg) \sve{\spa_j} = \sf{dz^i} {\scal A}_i{}^j (p) \slp R_{g*} \sve{\spa_j} \srp\n$$\nThe vertical and horizontal distributions satisfy\n\sbegin{eqnarray}\n\sve{\sDe_V} \sf{\sve{\scal A}} &=& \sve{\sDe_V} \s\s\n\sve{\sDe_H} \sf{\sve{\scal A}} &=& 0\n\send{eqnarray}\nand so must also respect the symmetry of the structure group,\n\sbegin{eqnarray}\n\sve{\sDe_V} (p g) &=& R_{g*} \sve{\sDe_V} (p) \s\s\n\sve{\sDe_H} (p g) &=& R_{g*} \sve{\sDe_H} (p)\n\send{eqnarray}\nso knowing the distributions at any point $p$ in $E$ implies the distributions at any other point on the fiber containing $p$.\n\nIff the [[FuN curvature]] of the Ehresmann connection vanishes,\n$$\n\sff{\sve{{\scal F}}} = - \sha \slb \sf{\sve{\scal A}},\sf{\sve{\scal A}} \srb_L = 0\n$$\nthe horizontal distribution is also in involution and may be integrated to get ''horizontal section''s. \n\nRefs:\n*http://philsci-archive.pitt.edu/archive/00002133/01/geometrie.pdf\n*http://www.mat.univie.ac.at/~michor/gaubook.pdf\n*http://www.mat.univie.ac.at/~michor/listpubl.html\n*http://www.emis.de/monographs/KSM/index.html

For any [[vector valued form]] field, $\snf{\sve{\scal K}}$, 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'',\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n$$\nOnce a choice of [[gauge|Ehresmann gauge transformation]] is made, the Ehresmann connection may be expressed in local coordinates as\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B}(x) \sve{\sxi_B}(y) + \sf{\sve{W}}(y)\n$$\nIf the vector valued form field is right invariant over the total space and may be written as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi_B}(y)\n$$\nthen, using a couple of [[FuN identities]], its Ehresmann covariant derivative is\n\sbegin{eqnarray}\n\sf{\scal D} \snf{\sve{\scal K}} &=& - \slb \sf{\sve{\scal A}}, \snf{\sve{\scal K}} \srb_L = - \slb \sf{A^B}(x) \sve{\sxi_B}(y), \snf{K^C}(x) \sve{\sxi_C}(y) \srb_L - \slb \sf{\sve{W}}(y), \snf{K^C}(x) \sve{\sxi_C}(y) \srb_L \s\s\n&=& - \sf{A^B} \snf{K^C} \slb \sve{\sxi_B}, \sve{\sxi_C} \srb_L - \sf{A^B} \slp {\scal L}_{\sve{\sxi_B}} \snf{K^C} \srp \sve{\sxi_C}\n+ \slp {\scal L}_{\sve{\sxi_C}} \sf{A^B} \srp \snf{K^C} \sve{\sxi_B}\n+ \slp \sf{d} \sf{A^B} \srp \sve{\sxi^B} \snf{K^C} \sve{\sxi_C}\n+ \slp \sve{\sxi^C} \sf{A^B} \srp \slp \sf{d} \snf{K^C} \srp \sve{\sxi_B} \s\s\n&-& \slp \sf{\sve{W}} \sf{\spa} \srp \snf{K^C} \sve{\sxi_C} + \slp -1 \srp^k \slp \snf{K^C} \sve{\sxi_C} \sf{\spa} \srp \sf{\sve{W}} + \slp \sf{\spa} \sf{\sve{W}} \srp \snf{K^C} \sve{\sxi_C} + \slp \sf{\spa} \snf{K^C} \sve{\sxi_C} \srp \sf{\sve{W}} \s\s\n&=& \slp \sf{d} \snf{K^C} \srp \sve{\sxi_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi_B}, \sve{\sxi_C} \srb_L\n\send{eqnarray}

An [[Ehresmann connection]] may be described in local coordinates by choosing a ''reference [[section|fiber bundle]]'', $\ssi_0$, 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\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{dx^a} A_a{}^B(x) \sxi_B{}^p(y) \sve{\spa_p} + \sf{dy^p} \sve{\spa_p} = \sf{A^B} \sve{\sxi_B} + \sf{\sve{\scal I}}\n$$\nIn which $\sve{\sxi_B}$ are the [[right (or left) invariant vector fields|Lie group geometry]] on the fibers. Another section ([[gauge|gauge transformation]]), $\ssi:M \srightarrow E$, may be chosen by flowing the original section along a [[diffeomorphism]] along the fibers, $\sph(x,y) = (x,y_\sph(x,y))$, to $\ssi = \sph \scirc \ssi_0$. The new section is described in the original coordinates by $y^p_\ssi(x)$. 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]],\n$$\n\sf{\sve{P_\ssi}} = \sf{dx^a} \sve{\spa_a} + \sf{dx^a} \sfr{\spa y_\ssi^p}{\spa x^a} \sve{\spa_p}\n$$\ncan be used to project to the TE valued 1-form on the section,\n$$\n\sf{\sve{{\scal A}_\ssi}} = \sf{\sve{P_\ssi}} \sf{\sve{\scal A}} = \sf{dx^a} \slp A_a{}^B(x) \sxi_B{}^p(y_\ssi) + \sfr{\spa y_\ssi^p}{\spa x^a} \srp \sve{\spa_p}\n$$\nThe 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]].\n\nAn alternative way of effecting a gauge transformation is to flow the Ehresmann connection by the diffeomorphism, $\sf{\sve{\scal A'}} = \sphi^*\sf{\sve{\scal A}}$, along the fibers while projecting it onto the original section, $\ssi_0$. This is called an ''active gauge transformation'', and gives the same result,\n$$\n\sf{\sve{{\scal A'}_{\ssi_0}}} = \sf{\sve{P_{\ssi_0}}} \sf{\sve{\scal A'}} = \sf{\sve{P_{\ssi_0}}} \sph^* \sf{\sve{\scal A}} = \sf{\sve{P_{\sph \scirc \ssi_0}}} \sf{\sve{\scal A}} = \sf{\sve{P_\ssi}} \sf{\sve{\scal A}} = \sf{\sve{\scal A_\ssi}}\n$$\n\nRef:\n*[[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]], $S = G/H$.\n\nPoints of the [[homogeneous space]], $x = [r(x)] \sin M = G/H$, are mapped to $G$, by the homogeneous reference section, $r: S \sto G$, and any point (element) in $G$, as a function of coordinates $x$ of $M$ and $y$ of $H$, may be specified by\n$$\ng(z) = g(x,y) = r(x) h(y) \sin G\n$$\nin which $h(y) \sin H$ operates on $r(x) \sin G$ 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]], $\sf{\scal I} = \sf{{\scal I}^J} T_J$, over $G$, is\n\sbegin{eqnarray}\n\sf{\scal I}(z) &=& g^- \sf{d} g = h^- \slp r^- \sf{d} r \srp h + h^- \sf{d} h \s\s\n&=& h^- \sf{I} h + h^- \sf{d} h \s\s\n&=& \sf{{\scal E}_S} + \sf{{\scal A}_S}\n\send{eqnarray}\nWhen $H$ is [[reductive]] in $G$ (which is assumed) the [[Maurer-Cartan frame|homogeneous space geometry]],\n$$\n\sf{I}(x) = r^- \sf{d} r = \sf{e_S} + \sf{A_S}\n$$\nsplits into the homogeneous space frame, $\sf{e_S}(x) = \sf{e_S^A} K_A \sin \sf{\srm Lie}(G/H)$, and homogeneous H-connection, $\sf{A_S}(x) = \sf{A_S^P} H_P \sin \sf{\srm Lie}(H)$. These correspond to the ''Ehresmann homogeneous space frame form'' and ''Ehresmann homogeneous H-connection form'' over $G$,\n\sbegin{eqnarray}\n\sf{{\scal E}_S} = \sf{{\scal E}_S^A} K_A &=& h^- \sf{e_S} h \sin \sf{\srm Lie}(G/H) \s\s\n\sf{{\scal A}_S} = \sf{{\scal A}_S^P} H_P &=& h^- \sf{A_S} h + h^- \sf{d} h \sin \sf{\srm Lie}(H)\n\send{eqnarray}\nTheir 1-form components are computed using the [[Killing form]],\n\sbegin{eqnarray}\n\sf{{\scal E}_S^A}(z) &=& \sf{e_S^B} \slp K^A, h^- K_B h \srp = \slp L^h \srp^A{}_B \s, \sf{e_S^B}(x) \s\s\n\sf{{\scal A}_S^P}(z) &=& \sf{A_S^Q} \slp H^P, h^- H_Q h \srp + \slp H^P, h^- \sf{d} h \srp = \slp L^h \srp^P{}_Q \s, \sf{A_S^Q}(x) + \sf{{\scal I}_H^P}(y)\n\send{eqnarray}\nwith the appearance of the [[left-right rotator]]s, $\slp L^h \srp^I{}_J(y)$, and the Maurer-Cartan form, $\sf{{\scal I}_H}$, for $H$. These are the same as the frame components, $\sf{e^A}(z) = \sf{{\scal E}_S^A}(z)$ and $\sf{e^P}(z) = \sf{{\scal A}_S^P}(z)$, 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, $T_I \ssim \sve{\sxi^R_I}$, allows us to write the [[Ehresmann-Maurer-Cartan vector valued form|Maurer-Cartan form]] as\n\sbegin{eqnarray}\n\sf{\sve{\scal I}} &=& \sf{\sve{{\scal E}_S}} + \sf{\sve{{\scal A}_S}} \s\s\n&=& \sf{{\scal E}_S^A} \sve{\sxi^R_A} + \sf{{\scal A}_S^P} \sve{\sxi^R_P} \s\s\n&=& \sf{e_S^B} \slp L^h \srp^A{}_B \s, \sve{\sxi^R_A} + \sf{A_S^Q} \slp L^h \srp^P{}_Q \s, \sve{\sxi^R_P} + \sf{\sve{{\scal I}_H}} \s\s\n&=& \sf{I^J} \slp L^r \srp_J{}^K \s, \sve{\sxi^L_K} + \sf{\sve{{\scal I}_H}}\n\send{eqnarray}\nwith the ''Ehresmann homogeneous space frame'', $\sf{\sve{{\scal E}_S}}$, and ''Ehresmann homogeneous H-connection'', $\sf{\sve{{\scal A}_S}}$, satisfying $\sf{\sve{{\scal E}_S}} \sf{\scal I} = \sf{{\scal E}_S}$ and $\sf{\sve{{\scal A}_S}} \sf{\scal I} = \sf{{\scal A}_S}$ using the Maurer-Cartan form, $\sf{\scal I} = \sf{\sxi_R^J}(z) T_J$, over $G$. The Ehresmann homogeneous H-connection, $\sf{\sve{{\scal A}_S}} = \sf{{\scal A}_S^P} \sve{\sxi^R_P}(z)$, is a [[Ehresmann principal bundle connection]] for an H-bundle and satisfies $\sf{\sve{{\scal A}_S}} \sf{\scal I_H} = \sf{{\scal A}_S}$, since $\sve{\sxi^R_P}(z) = \sve{\sxi^{HR}_P}(y)$ in a reductive Lie group geometry. Another way of looking at an Ehresmann homogeneous space geometry is as a [[Cartan H-bundle]] with $\sf{C} = \sf{I}$.\n\nSince the Ehresmann-Maurer-Cartan VVF is the identity projection, its [[FuN curvature]] vanishes, $\sff{\sve{\scal F}} = -\sha \slb \sf{\sve{\scal I}}, \sf{\sve{\scal I}} \srb_L = 0$.

from [[Ehresmann connection]]\n\nequivalent 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]], $\sf{A}(x)=\sf{A^B}T_B=\sf{dx^a}A_a{}^BT_B$, a [[Lieform]] over the base space, describes principal bundle geometry. By choosing a [[reference section|Ehresmann gauge transformation]], $\ssi_0:M \sto E$, this connection may be related to an [[Ehresmann connection]], $\sf{\sve{\scal A}}$, over the total space.\n\nThere 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 $\spi(x,y)=x$. The coordinates are chosen so that $y^p=0$, and hence $g=1$, on the ''canonical reference section'', $\ssi_0(x)=(x,y_0(x))=(x,0) \ssim (x,1)$, which provides the ''canonical local trivialization'', $(x,g) \sin E$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The $y$ [[coordinate basis vectors]] are in the vertical subspace, $\sve{\spa_p} \sin \sve{\sDe_V}$, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\sve{\spa_a} \snotin \sve{\sDe_H}$. 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\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B}(x) \sve{\sxi^L_B}(y) + \sf{\sve{\scal I}}\n$$\nin which $\sve{\sxi^L_B}$ are the [[left action vector fields|Lie group geometry]] for the Lie group geometry, defined by\n$$\n\sve{\sxi^L_B} \sf{\spa} g(y) = T_B g\n$$\nand\n$$\n\sf{\sve{\scal I}} = \sf{\sxi_L^B}(y) \sve{\sxi^L_B}(y) = \sf{\sxi_R^B}(y) \sve{\sxi^R_B}(y) = \sf{dy^p} \sve{\spa_p}\n$$\nis the [[Ehresmann-Maurer-Cartan form|Maurer-Cartan form]] (the [[identity projection|vector projection]] along the fibers). The Ehresmann connection is a projection, $\sf{\sve{\scal A}} \sf{\sve{\scal A}} = \sf{\sve{\scal A}}$, is [[right invariant]], $R_h^*\sf{\sve{\scal A}} = \sf{\sve{\scal A}}$, 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]], $\sf{\scal I}(y) = \sf{\sxi_R^B}(y)T_B = g^- \sf{d} g$, over the total space to get the ''Ehresmann connection form'',\n\sbegin{eqnarray}\n\sf{\scal A}(x,y) &=& \sf{\sve{\scal A}} \sf{\scal I} = \sf{A^B}(x) \sve{\sxi^L_B}(y) \sf{\sxi_R^C}(y) T_C + \sf{\sve{\scal I}} \sf{\scal I} \s\s\n&=& \slp \sf{A^B} L^C{}_B(y) + \sf{\sxi_R^C} \srp T_C \s\s\n&=& g^-(y) \sf{A}(x) g(y) + g^-(y) \sf{d^y} g(y) \n\send{eqnarray}\nusing the defining equation for the [[left-right rotator]],\n$$\nL^C{}_B(y) = \sve{\sxi^L_B}(y) \sf{\sxi_R^C}(y) = \slp T^C, g^-(y) T_B g(y) \srp \n$$\nThis form pulls back along the canonical reference section ($y=0$) to give the principal bundle connection,\n$$\n\ssi_0^*\sf{\scal A} = \sf{A}(x)\n$$\nand satisfies $R_h^* \sf{\scal A} = h^- \sf{\scal A} h$ under the right action.\n\nThe ''[[FuN curvature]] of the Ehresmann principal bundle connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,y) &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L = \slp 1 - \sf{\sve{\scal A}} \srp \slp \sf{\spa} \sf{\sve{\scal A}} \srp = \slp 1 - \sf{\sve{\scal A}} \srp \slp \slp \sf{\spa^x} + \sf{\spa^y} \srp \sf{\sve{\scal A}} \srp \s\s\n&=& \slp 1 - \sf{A^B}(x) \sve{\sxi^L_B}(y) - \sf{dy^p} \sve{\spa_p} \srp \n\slp \slp \sf{d^x} \sf{A^C} \srp \sve{\sxi^L_C}(y) - \sf{A^C} \sf{\spa^y} \sve{\sxi^L_C} \srp \s\s\n&=& \slp \sf{d^x} \sf{A^C} \srp \sve{\sxi^L_C} - \sf{A^B} \sf{A^C} \sve{\sxi^L_B} \sf{\spa^y} \sve{\sxi^L_C} \s\s\n&=& \slp \sf{d^x} \sf{A^D} + \sha \sf{A^B} \sf{A^C} C_{BC}{}^D \srp \sve{\sxi^L_D}(y)\n\send{eqnarray}\nusing the [[Lie bracket for left action vector fields|Lie group geometry]],\n$$\n\sve{\sxi^L_{\slb B \srd}} \slp \sf{\spa} \sve{\sxi^L_{\sld C \srb}} \srp = \sha \slb \sve{\sxi^L_B} , \sve{\sxi^L_C} \srb_L = - \sha C_{BC}{}^D \sve{\sxi^L_D}\n$$\nThis curvature is vector valued in the vertical subspace, and is right invariant, $R_h^* \sff{\sve{\scal F}} = \sff{\sve{\scal F}}$. The ''FuN curvature form of the Ehresmann connection'' is a $Lie(G)$ valued 2-form over $E$,\n\sbegin{eqnarray}\n\sff{\scal F} &=& \sff{\sve{\scal F}} \sf{\scal I} = \slp \sf{d^x} \sf{A^D} + \sha \sf{A^B} \sf{A^C} C_{BC}{}^D \srp g^-(y) T_D g(y) \s\s\n&=& g^-(y) \slp \sf{d^x} \sf{A} + \sha \slb \sf{A}, \sf{A} \srb \srp g(y) \s\s\n&=& g^-(y) \slp \sf{d^x} \sf{A} + \sf{A} \sf{A} \srp g(y)\n\send{eqnarray}\nThis form pulls back along the canonical reference section to give the [[principal bundle]] curvature,\n$$\n\ssi_0^*\sff{\scal F} = \sf{d} \sf{A} + \sf{A} \sf{A} = \sff{F}(x)\n$$\nand satisfies $R_h^* \sff{\scal F} = h^- \sff{\scal F} h$ under the right action.

The [[Ehresmann covariant derivative]] of a [[vector valued form]] field, $\snf{\sve{\scal K}}$, over the total space of a [[principal bundle]] using an [[Ehresmann principal bundle connection]] is\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n$$\nusing the [[FuN derivative]]. The VVF will usually be right invariant and expressible as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi^L_B}(y)\n$$\ncorresponding to the [[Lieform]],\n$$\n\snf{\scal K} = \snf{\sve{\scal K}} \sf{\scal I} = \snf{K^B}(x) g^-(y) T_B g(y) \n$$\nobtained with the [[Maurer-Cartan form]], $\sf{\scal I}(y) = \sf{\sxi_R^B} T_B$, and the [[left-right rotator]]. The [[pullback]] of this form along a section, $\ssi_1=(x,g_1(x))$, gives the Lieform over the base,\n$$\n\snf{K_1}(x) = \ssi_1^* \snf{\scal K} = \snf{K^B}(x) g^-_1(x) T_B g_1(x) = g^-_1(x) \snf{K}(x) g_1(x)\n$$\nin which the form pulled back along the reference section is $\snf{K}(x) = \snf{K^B}(x) T_B$. The Ehresmann covariant derivative of the VVF,\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = \slp \sf{d} \snf{K^C} \srp \sve{\sxi^L_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi^L_B}, \sve{\sxi^L_C} \srb_L\n= \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp \sve{\sxi^L_D}\n$$\ngives a definition for the ''Ehresmann covariant derivative of a Lieform'',\n$$\n\sf{\scal D} \snf{\scal K} = \slp \sf{\scal D} \snf{\sve{\scal K}} \srp \sf{\scal I} = \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp g^-(y) T_D g(y)\n$$\nwhich pulls back along any chosen section to give the [[covariant derivative|principal bundle]] of $\snf{K}$ on the base, \n\sbegin{eqnarray}\n\slp \sf{D_1} \snf{K_1} \srp(x) &=& \ssi_1^* \slp \sf{\scal D} \snf{\scal K} \srp \n= \slp \sf{d} \snf{K^D} + \sf{A^B} \snf{K^C} C_{BC}{}^D \srp g^-_1(x) T_D g_1(x) \s\s\n&=& g^-_1(x) \slp \sf{d} \snf{K} + \slb \sf{A}, \snf{K} \srb \srp g_1(x)\n= g^-_1(x) \slp \sf{\sna} \snf{K} \srp g_1(x)\n\send{eqnarray}\n(This should be equivalent to the usual definition of the Ehresmann covariant derivative of Lieform via\n$$\n\sve{v_1} \sve{v_2} \sdots \sve{v_{k+1}} \sf{\scal D} \snf{\scal K} = \sve{v^H_1} \sve{v^H_2} \sdots \sve{v^H_{k+1}} \sf{d} \snf{\scal K}(x,y)\n$$\nin which the vectors on the right are horizontal projections of the ones on the left, $\sve{v^H} = \sve{v}(1-\sf{\sve{\scal A}})$. //That definition needs to be checked.//)\n\nSimilarly, if a VVF is left invariant and may be expressed as\n$$\n\snf{\sve{\scal K}} = \snf{K^B}(x) \sve{\sxi^R_B}(y)\n$$\ncorresponding to the Lieform,\n$$\n\snf{\scal K} = \snf{\sve{\scal K}} \sf{\scal I} = \snf{K^B}(x) T_B \n$$\nThe [[pullback]] of this form along any section, $\ssi_1=(x,g_1(x))$, gives the same Lieform over the base,\n$$\n\snf{K_1}(x) = \ssi_1^* \snf{\scal K} = \snf{K^B}(x) T_B = \snf{K}(x)\n$$\nThe Ehresmann covariant derivative of this VVF,\n$$\n\sf{\scal D} \snf{\sve{\scal K}} = \slp \sf{d} \snf{K^C} \srp \sve{\sxi^R_C} - \sf{A^B} \snf{K^C} \slb \sve{\sxi^L_B}, \sve{\sxi^R_C} \srb_L\n= \slp \sf{d} \snf{K^C} \srp \sve{\sxi^R_C}\n$$\ngives the Lieform,\n$$\n\sf{\scal D} \snf{\scal K} = \slp \sf{\scal D} \snf{\sve{\scal K}} \srp \sf{\scal I} = \slp \sf{d} \snf{K^C} \srp T_C = \sf{d} \snf{\scal K}\n$$\nwhich pulls back along any chosen section to give the [[exterior derivative]] of $\snf{K}$ on the base, \n$$\n\slp \sf{D_1} \snf{K_1} \srp = \ssi_1^* \slp \sf{\scal D} \snf{\scal K} \srp\n= \sf{d} \snf{K}\n$$\n

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, $\ssi_0:M \sto E$, and principal bundle connection, $\sf{A}$, have been used to build the principal bundle Ehresmann connection, $\sf{\sve{\scal A}}$, a different section, $\ssi_1$, can be introduced and used to pull back a different principal bundle connection, $\sf{A'}$ -- this is a [[gauge transformation]]. Using coordinates adapted to the reference section, the Ehresmann connection is\n$$\n\sf{\sve{\scal A}}(x,y) = \sf{A^B} \sve{\sxi^L_B}(y) + \sf{\sve{\scal I}}\n$$\nand the Ehresmann connection form is\n$$\n\sf{\scal A} = \sf{\sve{\scal A}} \sf{\scal I} = g^-(y) \sf{A}(x) g(y) + g^- \sf{d^y} g(y)\n$$\nusing the [[Maurer-Cartan form]], $\sf{\scal I}(y) = \sf{\sxi_R^B} T_B$, and [[left-right rotator]]. The new section, $\ssi_1 = \sph \scirc \ssi_0$, may be obtained by flowing the reference section by an equivariant vertical [[diffeomorphism]], $\sph(x,y) = (x,y_\sph(x,y))$, satisfying $\sph(ph)=\sph(p)h$ and giving $\ssi_1(x) = (x,y_1(x)) = (x,y_\sph(x,0))$. This is equivalent to transforming the original section by right (//?//) action by an element of $G$ to $\ssi_1(x) = \ssi_0(x) \s, g(y_1(x)) = \ssi_0 \s, g_1(x)$. The [[vector projection onto a section]], $\ssi_1$, is\n$$\n\sf{\sve{P_1}} = \sf{dx^a} \sve{\spa_a} + \sf{dx^a} \sfr{\spa y_1^p}{\spa x^a} \sve{\spa_p}\n$$\nand is used to project the Ehresmann connection to\n$$\n\sf{\sve{{\scal A}_1}} = \sf{\sve{P_1}} \sf{\sve{\scal A}} = \sf{A^B}(x) \sve{\sxi^L_B}(y_1) + \sf{dx^a} \sfr{\spa y_1^p}{\spa x^a} \sve{\spa_p}\n$$\nand the Ehresmann connection form to\n$$\n\sf{{\scal A}_1} = \sf{\sve{P_1}} \sf{\scal A} = \sf{\sve{{\scal A}_1}} \sf{\scal I} = \sf{\sve{P_1}} \sf{\sve{\scal A}} \sf{\scal I} = g^-(y_1(x)) \sf{A}(x) g(y_1(x)) + g^-(y_1(x)) \sf{d^x} g(y_1(x)) \n$$\non the section. This, the gauge transformed connection, gives the [[pullback]] of the Ehresmann connection form along the section,\n$$\n\sf{A'}(x) = \ssi_1^* \sf{\scal A} = \ssi_1^* \sf{{\scal A}_1} = g^-_1(x) \sf{A} g_1(x) + g^-_1(x) \sf{d^x} g_1(x) \n$$\nidentified as the [[principal bundle gauge transformation|principal bundle]] with $g_1(x)=g^-(x)$.\n\nAn alternative way of effecting a gauge transformation is to flow the Ehresmann connection form by the diffeomorphism, $\sf{\scal A'} = \sph^*\sf{\scal A}$, then pull it back along the original section, $\ssi_0$. This ''active gauge transformation'' gives the same result,\n$$\n\ssi_0^* \sf{\scal A'} = \ssi_0^* \slp \sph^* \sf{\scal A} \srp\n= \slp \sph \scirc \ssi_0 \srp^* \sf{\scal A}\n= \ssi_1^* \sf{\scal A} = \sf{A'}\n$$\n\nRef:\n*[[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 $v = v^\sal b_\sal \sin V = F$) and a base manifold, $M$. A [[vector bundle connection]], $\sf{A}{}_\sal{}^\sbe(x) = \sf{dx^i} A_{i \sal}{}^\sbe(x)$, 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]]'', $\ssi_0:M \sto E$, this connection may be related to an [[Ehresmann connection]], $\sf{\sve{\scal A}}$, over the total space.\n\nThere 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, $v^\sal$, 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 $\spi(x,v) = x$. The coordinates are chosen so that $v^\sal = 0$ on the reference section, a ''canonical local trivialization'', $\ssi_0(x) = (x,v_0(x)) = (x,0)$. With these coordinates each fiber corresponds to a coordinate surface of constant $x$. The [[coordinate basis vectors]] for the $v^\sal$ coordinates, $\sve{\spa_\sal} \sin \sve{\sDe_V}$, are in the vertical subspace, but the $x$ coordinate basis vectors are not necessarily in the horizontal subspace, $\sve{\spa_a} \snotin \sve{\sDe_H}$. 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\n$$\n\sf{\sve{\scal A}}(x,v) = \slp \sf{dx^i} A_{i \sal}{}^\sbe(x) v^\sal + \sf{dv^\sbe} \srp \sve{\spa_\sbe}\n$$\n\nIn analogy with the [[Maurer-Cartan form]], we build a vector ($V$) valued 1-form,\n$$\n\sf{\scal I} = \sf{dv^\sbe} b_\sbe\n$$\nand use it to define the ''Ehresmann vector bundle connection form'',\n$$\n\sf{\scal A} = \sf{\sve{\scal A}} \sf{\scal I} = \slp \sf{A}{}_\sal{}^\sbe v^\sal + \sf{dv^\sbe} \srp b_\sbe\n$$\nThis allows us to define the [[vector bundle covariant derivative|vector bundle connection]] of any section, $\ssi_1 : M \smapsto E$, $\ssi_1(x) = (x, v_1(x))$, as the [[pullback]] of $\sf{\scal A}$ along the section to M,\n$$\n\ssi_1^* \sf{\scal A} = \sf{A}{}_\sal{}^\sbe v_1^\sal(x) b_\sbe + \sf{dx^a} \sfr{\spa v_1^\sbe}{\spa x^a} b_\sbe = \sf{\sna} v_1(x)\n$$ \n\nThe ''[[FuN curvature]] of the Ehresmann vector bundle connection'' is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}}(x,y) &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L = \slp 1 - \sf{\sve{\scal A}} \srp \slp \sf{\spa} \sf{\sve{\scal A}} \srp \s\s\n&=& \slp 1 - \sf{A}{}_\sal{}^\sbe v^\sal \sve{\spa_\sbe} - \sf{dv^\sbe} \sve{\spa_\sbe} \srp \n\slp \slp \sf{d^x} \sf{A}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde} - \sf{A}{}_\sga{}^\sde \sf{dv^\sga} \sve{\spa_\sde} \srp \s\s\n&=& \slp \sf{d^x} \sf{A}{}_\sal{}^\sde - \sf{A}{}_\sal{}^\sbe \sf{A}{}_\sbe{}^\sde \srp v^\sal \sve{\spa_\sde} \s\s\n&=& \sff{F}{}_\sal{}^\sde v^\sal \sve{\spa_\sde}\n\send{eqnarray}\nin which the [[vector bundle curvature]], $\sff{F}{}_\sal{}^\sde$, appears.\n\nThe [[Ehresmann covariant derivative]] of any [[vector valued form]] over the total space (such as the curvature above) that can be written as\n$$\n\snf{\sve{\scal K}} = \snf{K}{}_\sga{}^\sde(x) v^\sga \sve{\spa_\sde}\n$$\nis defined using the [[FuN derivative]] as\n\sbegin{eqnarray}\n\sf{\scal D} \snf{\sve{\scal K}} &=& - {\scal L}_{\sf{\sve{\scal A}}} \snf{\sve{\scal K}}\n= - \sf{\sve{\scal A}} \slp \sf{\spa} \snf{\sve{\scal K}} \srp + \slp -1 \srp^k \snf{\sve{\scal K}} \slp \sf{\spa} \sf{\sve{\scal A}} \srp + \sf{\spa} \slp \snf{\sve{\scal K}} \sf{\sve{\scal A}} \srp \s\s\n&=& - \slp \sf{A}{}_\sal{}^\sbe(x) v^\sal + \sf{dv^\sbe} \srp \sve{\spa_\sbe} \slp \slp \sf{d^x} \snf{K}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde} + \slp -1 \srp^k \snf{K}{}_\sga{}^\sde \sf{dv^\sga} \sve{\spa_\sde} \srp \s\s\n& & + \slp -1 \srp^k \snf{K}{}_\sga{}^\sde v^\sga \sve{\spa_\sde} \slp \slp \sf{d^x} \sf{A}{}_\sla{}^\sbe \srp v^\sal \sve{\spa_\sbe} - \sf{A}{}_\sal{}^\sbe \sf{dv^\sal} \sve{\spa_\sbe} \srp\n+ \sf{\spa} \slp \snf{K}{}_\sga{}^\sde v^\sga \sve{\spa_\sde} \srp \s\s\n&=& \slp \sf{d^x} \snf{K}{}_\sga{}^\sde - \sf{A}{}_\sga{}^\sbe \snf{K}{}_\sbe{}^\sde + \sf{A}{}_\sbe{}^\sde \snf{K}{}_\sga{}^\sbe \srp v^\sga \sve{\spa_\sde} \s\s\n&=& \slp \sf{\sna} \snf{K}{}_\sga{}^\sde \srp v^\sga \sve{\spa_\sde}\n\send{eqnarray}\n\nAn [[Ehresmann gauge transformation]] corresponding to a change in local trivialization, $b_\sbe \smapsto b'_\sbe = g_\sbe{}^\sal(x) b_\sal$, gives changes in $\sf{A}{}_\sal{}^\sbe$ 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]], $\sf{R}$, [[Clifford curvature scalar]], $R$, [[frame]], $\sf{e}$, ''cosmological constant'', $\sLa$, and ''Clifford energy-momentum tensor'', $\sf{T}$, for matter are dynamically related by ''Einstein's equation'',\n$$\n\sf{R} - \sha R \sf{e} = \set_{00} ( \sLa \sf{e} - 8 \spi G \sf{T} ) \n$$\nin which $\set_{00}$ specifies the [[Minkowski metric]] sign convention. In a vacuum, $\sf{T} = 0$, Einstein's equation contracted with the coframe, $\sve{e}$, gives\n$$\n\sve{e} \scdot \slp \sf{R} - \sha R \sf{e} \srp = R - \sha R n = \set_{00} \sLa n\n$$\nrequiring the curvature scalar to be constant, $R = - \sfr{2n}{n-2} \set_{00} \sLa = - 4 \set_{00} \sLa$ (with $n=4$), and giving the ''vacuum Einsten's equation'',\n$$\n\sf{R} = - \sfr{2}{n-2} \set_{00} \sLa \sf{e} = - \set_{00} \sLa \sf{e}\n$$\nAny spacetime that satisfies $\sf{R} = \sal \sf{e}$ for some constant, $\sal$, is an ''Einstein space''.\n\nEinstein's equation is derived by extremizing the ''Einstein-Hilbert action'',\n$$\nS = \sint \snf{e} \slp \sfr{1}{16 \spi G} \slp R + 2 \set_{00} \sLa \srp + L_M \srp\n$$\nwith respect to $\sve{e}$, in natural [[units]].

http://arxiv.org/abs/gr-qc/0606062\n*looks to be a good reveiw of ECT\n\nIn ECT...\nThe 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.\n\nIn teleparallel theories of gravity the spin connection is purely torsional ($\sf{\snu}=0$, $\sf{d} \sf{e}=0$, $\sf{\ska} \sneq 0$) and the [[spacetime]] is, in that sense, flat, with the gravitational field a force represented solely by torsion.\n\nRef:\n[[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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<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td><div class="math">\n\sbegin{array}{c}\nW =\n\slb \sbegin{array}{cc}\n{\ssmall \sfrac{i}{2}} W^3 & W^+ \s\s\nW^- & {\ssmall - \s! \sfrac{i}{2}} W^3\n\send{array} \srb \svp{|_{\sBig(}}\n\squad\nB_1 =\n\slb \sbegin{array}{cc}\n{\ssmall \sfrac{i}{2}} B_1^3 & B_1^+ \s\s\nB_1^- & {\ssmall - \s! \sfrac{i}{2}} B_1^3\n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\sbig[\n\slb \sbegin{array}{cc}\nW & \s\s\n& B_1\n\send{array} \srb\n,\n\slb \sbegin{array}{cc}\n & \sph_B \s\s\n\sph_W &\n\send{array} \srb\n\sbig] \svp{|_{\sBig(}}\n\s\s\n\sqquad \sqquad \sqquad \sqquad \squad\n\sph_{W/B} =\n\slb \sbegin{array}{cc}\n- \sph_{0/1} & \sph_+ \s\s\n\sph_- & \sph_{1/0}\n\send{array} \srb \svp{|_{\sBig(}}\n\s\s\n\slb \sbegin{array}{cc}\nW & \s\s\n& B_1\n\send{array} \srb\n\squad\n\slb \sbegin{array}{c}\n\snu_{eL} \s\s e_L \s\s \snu_{eR} \s\s e_R\n\send{array} \srb\n\squad\n\slb \sbegin{array}{c}\nu_L \s\s d_L \s\s u_R \s\s d_R\n\send{array} \srb \s\s[.5em]\n\sbig( \sfr{\ssqrt{3}}{\ssqrt{5}} B_1^3 - \sfr{\ssqrt{2}}{\ssqrt{5}} B_2 \sbig)\n= (\sfr{\ssqrt{3}}{\ssqrt{5}}) \sha Y \n \s;\sto\s; g_1=\sfr{\ssqrt{3}}{\ssqrt{5}}\n\send{array}\n</div></td>\n\n<td>&nbsp;&nbsp;&nbsp;</td>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th ALIGN=CENTER COLSPAN="2"><SPAN class="math">SO(4)</SPAN></th>\n<th></th>\n<th ALIGN=CENTER><SPAN class="math">W^3</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">B_1^3</SPAN></th>\n<th></th>\n<th ALIGN=CENTER><SPAN class="math">\sfr{\ssqrt{2}}{\ssqrt{3}} B_2</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">\sha Y</SPAN></th>\n<th ALIGN=CENTER><SPAN class="math">Q</SPAN></th>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W^+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W^-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFFFF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">B_1^+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFFFF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">B_1^-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">- 1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#B2B200} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_+</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smdia{#F2F200} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_-</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smsqu{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smdia{#4D4D4D}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph_1</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#B2B200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_{eL}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_{eR}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sha</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-1</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#BF6000}}{\smtri{#668000}}{\smtri{#8F00B2}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#F77C00}}{\smtri{#99BF00}}{\smtri{#AD00F7}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d_L</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#990000}}{\smtri{#009900}}{\smtri{#0000B2}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sfr{2}{3}</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\strip{\smtri{#D90000}}{\smtri{#00BF00}}{\smtri{#0000F7}}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d_R</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">0</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{6}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n</table>\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>

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\n<html>\n<center>\n<table class="ptable">\n<tr>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Forces (bosons)</SPAN></th>\n<th><SPAN class="math">\s;\s;\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Matter (fermions)</SPAN></th>\n<th><SPAN class="math">\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Second generation</SPAN></th>\n<th><SPAN class="math">\s;\s;</SPAN></th>\n<th ALIGN=CENTER COLSPAN="3"><SPAN>Third generation</SPAN></th>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#000000} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{(}_{\sbig(}} \sga \sp{{\sBig(}^{(}_{\sbig(}} </SPAN></td>\n<td ALIGN=CENTER><SPAN>electromagnetism</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#F2F200} \sbtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">e</SPAN></td>\n<td ALIGN=CENTER><SPAN>electron</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smtri{#F2F200} \smtri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN>muon</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sstri{#F2F200} \sstri{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN>tau</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#FFFF00} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">W,Z</SPAN></td>\n<td ALIGN=CENTER><SPAN>weak</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbutr{#F2F200} \sbutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{e}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{\sBig(}}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smutr{#F2F200} \smutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\smu}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\ssutr{#F2F200} \ssutr{#D9D9D9}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\stau}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">g</SPAN></td>\n<td ALIGN=CENTER><SPAN>strong</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbtri{#B2B200} \sbtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_e</SPAN></td>\n<td ALIGN=CENTER><SPAN>electron <br> neutrino</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smtri{#B2B200} \smtri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN>muon <br> neutrino</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sstri{#B2B200} \sstri{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\snu_\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN>tau <br> neutrino</SPAN></td>\n</tr>\n<tr class="butt">\n<td ALIGN=CENTER><SPAN class="math">\smcir{#59FF59} </SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}^{\sbig(}_{\sbig(}} \som \sp{{\sBig(}^{\sbig(}_{\sbig(}}</SPAN></td>\n<td ALIGN=CENTER><SPAN>gravity</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\sbutr{#B2B200} \sbutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_e</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sp{{\sBig(}_(}</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\smutr{#B2B200} \smutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_\smu</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\ssutr{#B2B200} \ssutr{#999999}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{\snu}{}_\stau</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smdia{#F2F200} \s, \smdia{#BF6000} \s\s[-.5em]\n\smsqu{#B2B200} \s, \smsqu{#F77C00}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sph</SPAN></td>\n<td ALIGN=CENTER><SPAN>Higgs</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbtri{#BF6000} \sbtri{#990000} \s\s[-.8em]\n\sbtri{#668000} \sbtri{#009900} \s\s[-.8em]\n\sbtri{#8F00B2} \sbtri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">u</SPAN></td>\n<td ALIGN=CENTER><SPAN>up <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smtri{#BF6000} \smtri{#990000} \s\s[-.8em]\n\smtri{#668000} \smtri{#009900} \s\s[-.8em]\n\smtri{#8F00B2} \smtri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">c</SPAN></td>\n<td ALIGN=CENTER><SPAN>charm <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sstri{#BF6000} \sstri{#990000} \s\s[-.8em]\n\sstri{#668000} \sstri{#009900} \s\s[-.8em]\n\sstri{#8F00B2} \sstri{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">t</SPAN></td>\n<td ALIGN=CENTER><SPAN>top <br> quark</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbutr{#BF6000} \sbutr{#990000} \s\s[-.8em]\n\sbutr{#668000} \sbutr{#009900} \s\s[-.8em]\n\sbutr{#8F00B2} \sbutr{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{u}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smutr{#BF6000} \smutr{#990000} \s\s[-.8em]\n\smutr{#668000} \smutr{#009900} \s\s[-.8em]\n\smutr{#8F00B2} \smutr{#0000B2}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{c}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\ssutr{#BF6000} \ssutr{#990000} \s\s[-.8em]\n\ssutr{#668000} \ssutr{#009900} \s\s[-.8em]\n\ssutr{#8F00B2} \ssutr{#0000B2}\n\send{array}</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{t}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbtri{#F77C00} \sbtri{#D90000} \s\s[-.8em]\n\sbtri{#99BF00} \sbtri{#00BF00} \s\s[-.8em]\n\sbtri{#AD00F7} \sbtri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">d</SPAN></td>\n<td ALIGN=CENTER><SPAN>down <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smtri{#F77C00} \smtri{#D90000} \s\s[-.8em]\n\smtri{#99BF00} \smtri{#00BF00} \s\s[-.8em]\n\smtri{#AD00F7} \smtri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">s</SPAN></td>\n<td ALIGN=CENTER><SPAN>strange <br> quark</SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sstri{#F77C00} \sstri{#D90000} \s\s[-.8em]\n\sstri{#99BF00} \sstri{#00BF00} \s\s[-.8em]\n\sstri{#AD00F7} \sstri{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">b</SPAN></td>\n<td ALIGN=CENTER><SPAN>bottom <br> quark</SPAN></td>\n</tr>\n<tr>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN class="math"></SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\sbutr{#F77C00} \sbutr{#D90000} \s\s[-.8em]\n\sbutr{#99BF00} \sbutr{#00BF00} \s\s[-.8em]\n\sbutr{#AD00F7} \sbutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{d}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\smutr{#F77C00} \smutr{#D90000} \s\s[-.8em]\n\smutr{#99BF00} \smutr{#00BF00} \s\s[-.8em]\n\smutr{#AD00F7} \smutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{s}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n<td></td>\n<td ALIGN=CENTER><SPAN class="math">\n\sbegin{array}{c}\n\ssutr{#F77C00} \ssutr{#D90000} \s\s[-.8em]\n\ssutr{#99BF00} \ssutr{#00BF00} \s\s[-.8em]\n\ssutr{#AD00F7} \ssutr{#0000F7}\n\send{array}\n</SPAN></td>\n<td ALIGN=CENTER><SPAN class="math">\sbar{b}</SPAN></td>\n<td ALIGN=CENTER><SPAN></SPAN></td>\n</tr>\n</table>\n</center>\n</html>

Images can be included by their filename or full URL. It's good practice to include a title to be shown as a tooltip, and when the image isn't available. An image can also link to another note or or a URL\n[img[Romanesque broccoli|images/fractalveg.jpg][http://www.flickr.com/photos/jermy/10134618/]]\n{{{\n[img[Romanesque broccoli|images/fractalveg.jpg]\n [http://www.flickr.com/photos/jermy/10134618/]]\n[img[title|filename]]\n[img[filename]]\n[img[title|filename][link]]\n[img[filename][link]]\n}}}\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]][>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]You can also float images to the left or right: the forest is left aligned with {{{[<img[}}}, and the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):You can use CSS to clear the floats@@\n{{{\n[<img[Forest|images/forest.jpg][http://www.flickr.com/photos/jermy/8749660/]]\n[>img[Field|images/field.jpg][http://www.flickr.com/photos/jermy/8749285/]]\nYou can also float images to the left or right:\n the forest is left aligned with {{{[<img[}}},\nand the field is right aligned with {{{[>img[}}}.\n@@clear(left):clear(right):display(block):\nYou can use CSS to clear the floats@@\n}}}

Decent intros using MaxEnt:\nhttp://arxiv.org/abs/cond-mat/0507388\nhttp://arxiv.org/abs/physics/9805024\n\nsome Q.A.Wang papers\n*http://arxiv.org/abs/cond-mat/0312329\n**I don't like his use of ergodicity in defining the long time average equal to the Bayesian expectation value.\n**nice: uses fixed average action and MaxEnt to get partition function with action per path instead of energy per state\n***I might like using Rovelli's Hamiltonian constraint dynamics better.\n*http://arxiv.org/abs/cond-mat/0407515\n**seems inferior to previous paper\n\nGood Hawking paper on it:\nhttp://arxiv.org/abs/gr-qc/9501014\n*boundary term in depth\n*Hamiltonian formulation\n*relationship between partition functions for static spacetimes (timelike Killing vector).\n\nrelated discussion on time and Tomita flow in Rovelli book

<<note HideTags>>$$\n\sbegin{array}{llcl}\n\n1918, \s!\s!&\s!\s! {\srm Weyl} \s!\s!&\s!\s! : & \sf{A} \sin \sf{\srm Lie}(G) \sp{{}_{\sbig(}} \s\s\n\n1954, \s!\s!&\s!\s! {\srm Y.M.} \s!\s!&\s!\s! : & \sf{A} = \sf{B} + \sf{W} + \sf{G}\n\s;\s; \sin \s; \sf{\srm Lie}(G) = \sf{su}(1) + \sf{su}(2) + \sf{su}(3) \sp{{}_{\sbig(}} \s\s\n\n1967, \s!\s!&\s!\s! {\srm F.P.} \s!\s!&\s!\s! : & \sudf{A} = \sf{A} + \sud{g} \sp{{}_{\sbig(}}\n\s;\s; \sin \s; \sudf{\srm Lie}(G) \s\s\n\n1977, \s!\s!&\s!\s! {\srm M.M.} \s!\s!&\s!\s! : & \sf{A} = \sf{\som} + \sf{e}\n\s;\s; \sin \s; \sf{\srm Lie}(G) = \sf{so}(1,4) \sp{{}_{\sBig(_(}} \s\s\n\n\n2002, \s!\s!&\s!\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sud{\sps} = \sud{g} \sp{{}_{\sbig(}} \s\s\n\n2005, \s!\s!&\s!\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sudf{A} = {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{B} + \sf{W} + \sf{G} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d} \s\s\n & & & \s;\s;\s;\s,\s, \sin \s; \sf{\srm Lie}(G) = \sf{Cl}(1,7) \sp{{}_{\sbig(}} \s\s\n\n{\srm now}, \s!&\s! {\srm Y.T.} \s!\s!&\s!\s! : & \sudf{A} = {\ssmall \sfrac{1}{2}} \sf{\som} + {\ssmall \sfrac{1}{4}} \sf{e} \sph + \sf{B} + \sf{W} + \sf{G} + \sud{\snu^e} + \sud{e} + \sud{u} + \sud{d} \s\s\n & & & \s;\s;\s;\s;\s;\s;\s;\s; + \sud{\snu^\smu} + \sud{\smu} + \sud{c} + \sud{s}\n+ \sud{\snu^\sta} + \sud{\sta} + \sud{t} + \sud{b} \s\s\n & & & \s;\s;\s;\s,\s, \sin \s; \sf{\srm Lie}(G) = \sf{e8} ? \sp{{}_{\sbig(_(}}\n\send{array}\n$$\n

<<note HideTags>>The $14$ Lie algebra elements of the smallest exceptional Lie group, $G2$:\n$$\n\sbegin{array}{rcccccccl}\ng2 \s!\s!&\s!\s!=\s!\s!&\s!\s! su(3) \s!\s!&\s!\s! + \s!\s!&\s!\s! 3 \s!\s!&\s!\s! + \s!\s!&\s!\s! \sbar{3} \s!\s!&& \s\s\n&&\s!\s! \sf{g} \s!\s!&\s!\s! + \s!\s!&\s!\s! \sud{q} \s!\s!&\s!\s! + \s!\s!&\s!\s! \sud{\sbar{q}} \s!\s!&\s! \sin \s! &\s! \sudf{g2} \n\send{array}\n$$\nStructure of $G2$ implies Lie bracket equivalent to fundamental action,\n$$\n[ g,q ] = \sbig[ g^A T_A,q^B T_B \sbig] = g \s, q =\n\slb\n\smatrix{\n\s! \sfr{i}{2} g^3 \s!+\s! {\sscriptsize \sfrac{i}{2\ssqrt{3}}} g^8 \s!\s! & g^{r\sbar{g}} & g^{r\sbar{b}} \s\s\ng^{\sbar{r}g} & \s!\s! {\sscriptsize -\s!\sfrac{i}{2}} g^3 \s!+\s! {\sscriptsize \sfrac{i}{2\ssqrt{3}}} g^8 \s!\s! & g^{g\sbar{b}} \s\s\ng^{\sbar{r}b} & g^{\sbar{g}b} & {\sscriptsize -\s!\sfrac{i}{\ssqrt{3}}} g^8\n}\n\srb\n\slb \smatrix{\nq^r \s\s q^g \s\s q^b\n} \srb\n$$\ncorresponding to the strong interactions, such as\n<html>\n<center>\n<table class="gtable">\n<tr>\n<td>\n<div class="math">\n\sbig[ g^{r\sbar{g}}, q^g \sbig] = q^r\n</div>\n</td>\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n<td ALIGN=CENTER><img SRC="images/png/quark gluon vertex.png" height=160px></td>\n</tr>\n</table>\n</center>\n</html>

''Bold''\n{{{''Bold''}}}\n==Strikethrough==\n{{{==Strikethrough==}}}\n__Underline__ \n{{{__Underline__}}}\n//Italic// \n{{{//Italic//}}}\n2^^3^^=8 \n{{{2^^3^^=8}}}\na~~ij~~ = -a~~ji~~ \n{{{a~~ij~~ = -a~~ji~~}}}\n@@highlight@@ \n{{{@@highlight@@}}}\n\n//The highlight can also accept CSS syntax to directly style the text://\n@@color:green;green coloured@@\n{{{@@color:green;green coloured@@}}}\n@@background-color:#ff0000;color:#ffffff;red coloured@@\n{{{@@background-color:#ff0000;color:#ffffff;red coloured@@}}}\n@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@\n{{{@@text-shadow:black 3px 3px 8px;font-size:18pt;display:block;margin:1em 1em 1em 1em;border:1px solid black;Access any CSS style@@}}}\n@@display:block;text-align:center;centered text or image@@\n{{{@@display:block;text-align:center;centered text or image@@}}}\n\n//For backwards compatibility, the following highlight syntax is also accepted://\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n{{{\n@@bgcolor(#ff0000):color(#ffffff):red coloured@@\n}}}

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]].

$$\n\sbegin{array}{rcll}\nF4 &:& ( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3 ) \s;\s;\s; \sp{(B_2, g^3, g^8)} &\sleft\s{ \n\sbegin{array}{l}\stext{graviweak interactions} \s\s \stext{three generations}\send{array} \sright.\n\s\s\nG2 &:& \sp{( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3 ) \s;\s;\s;}(B_2, g^3, g^8) &\sleft\s{ \n\sbegin{array}{l}\stext{strong interactions} \s\s \stext{anti-particles}\send{array} \sright.\n\s\s[-.3em]\n\srlap{\shbox{@(hr noshade size="1" style="position:relative; left:-2em;\n width:30em; border:0px; border-top:1px solid black")}}\s\s[-.5em]\nE8 &:& ( \sha \som_L^3, \sha \som_R^3, W^3, B_1^3, w, B_2, g^3, g^8) & \s, \sleft\s{ \s; \stext{everything} \sright.\n\send{array}\n$$\n\nBreakdown of E8 to the standard model and gravity:\n\sbegin{eqnarray}\ne8 &=& f4 + g2 + 26 \s! \stimes \s! 7 \s\s\n&=& so(7,1) + su(3) + (8_{S+}\s!+\s!8_V+\s!8_{S-})\s!\stimes\s!(1\s!+\s!1\s!+\s!3\s!+\s!\sbar{3}) + 3\s!\stimes\s!(3\s!+\s!\sbar{3}) + 2 \s\s[.4em]\nA &=& \sbig( {\sscriptsize \sfrac{1}{2}} \som + {\sscriptsize \sfrac{1}{4}} e \sph + W + B_1 \sbig) + g + 3 \s! \stimes \s! \sPs + x \sPh + B_2 + w\n\send{eqnarray}\nTwo new quantum numbers and some non-standard particles:\n$$\n\s{ \s; w \squad (B_1^3\s!+\s!B_2) \squad B_1^\spm \squad x_{1/2/3} \sPh^{r/g/b} \squad x_{1/2/3} \sPh^{\sbar{r}/\sbar{g}/\sbar{b}} \s; \s} \svp{\sbig(}\n$$\n<<note HideTags>>

<<note HideTags>>\n@@display:block;text-align:center;\n<html><center><embed src="images/png/f4.png" width="510" height="510"></embed></center></html>\n@@

confirmed attendees:\n*[[Scott Aaronson|http://www.scottaaronson.com/]], QMI, quantum computing\n*[[Fred Adams|http://www.physics.lsa.umich.edu/department/directory/bio.asp?ID=1]]${}^*$, constant change, astrophysics\n*[[Anthony Aguirre|http://scipp.ucsc.edu/~aguirre/]], cosmology\n*[[Stephon Alexander|http://www.phys.psu.edu/people/display/index.html?person_id=4901]], astrophysics\n**Just put out a new paper: Isogravity\n**friends with James Bjorken (and everyone else, apparently)\n*[[Markus Aspelmeyer|http://homepage.univie.ac.at/Markus.Aspelmeyer/]]${}^*$, QM foundations\n*[[Paul A Benioff|http://www.phy.anl.gov/theory/staff/pab.html]], QM foundations, older guy\n*[[Caslav Brukner|http://homepage.univie.ac.at/Caslav.Brukner/index.htm]], QM foundations\n*[[Dmitry Budker|http://www.fqxi.org/aw-budker2.html]]${}^*$, constant change (experimental)\n*[[Gregory Chaitin|http://www.umcs.maine.edu/~chaitin/]], math, complexity, and philosophy of science\n*[[Hyung Choi|http://www.zoominfo.com/people/Choi_Hyung_78134925.aspx]], metanexus, QM foundations, science and religion (uh oh)\n*[[Louis Crane|http://www.fqxi.org/aw-crane2.html]]${}^*$, QGR, QM histories\n*[[Paul Davies|http://cosmos.asu.edu/]], QMI, astrophysics, popular author\n*[[John Donoghue|http://www.fqxi.org/aw-donoghue2.html]]${}^*$, emergent symmetry\n*[[Richard Easther|http://www.fqxi.org/aw-easther.html]]${}^*$, superstring cosmology\n*[[David Ritz Finkelstein|http://www.physics.gatech.edu/people/faculty/dfinkelstein.html#personal]], older particle physicist\n**Lie algebra expert. Proponent of stable Lie algebras.\n*[[Rodolfo Gambini|http://www.fqxi.org/aw-pullin2.html]]${}^*$, QM GR\n*[[Jaume Garriga|http://www.ffn.ub.es/gcg/personal/jaume.html]], cosmology, branes\n*[[Steven Gratton|http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+ea+gratton%2C+steven]], cosmology, inflation\n*[[Alan Guth|http://web.mit.edu/physics/facultyandstaff/faculty/alan_guth.html]], err, invented inflation\n*[[Lucien Hardy|http://www.perimeterinstitute.ca/index.phpindex.php?option=com_content&task=view&id=30&Itemid=7&view_directory=1&pi=1078]], Causaloids\n*[[Adrian Kent|http://www.damtp.cam.ac.uk/user/apak/]], QM foundations\n*[[Lawrence Krauss|http://www.phys.cwru.edu/~krauss/]], astrophysics and cosmology, popular author, dislikes KK and strings, lectures a LOT\n*[[Matthew Leifer|http://www.fqxi.org/aw-leifer2.html]], QM foundations\n*[[Eugene Lim|http://pantheon.yale.edu/~eal48/papers.html]]${}^*$, cosmology\n*A. [[Garrett Lisi]]${}^*$\n*[[Abraham Loeb|http://www.fqxi.org/aw-loeb2.html]]${}^*$, SETI, astronomy\n*[[Fotini Markopoulou|http://www.fqxi.org/aw-markopoulou2.html]]${}^*$, quantum graphity\n*[[Laura Mersini|http://en.wikipedia.org/wiki/Laura_Mersini]], cosmology\n*[[Farzad Nekoogar|http://www.fqxi.org/aw-nekoogar.html]]${}^*$, popularizer of theoretical physics -- [[multiversal journeys|http://www.multiversaljourneys.com/]]\n*[[Ken Olum|http://www.fqxi.org/aw-olum2.html]]${}^*$, GR, wants to rule out wormholes and other GR exotics\n*[[Maulik Parikh|http://www.fqxi.org/aw-khoury2.html]]${}^*$, GR boundaries, mach's principle, hep-th and strings\n*[[Philip Pearle|http://physerver.hamilton.edu/people/]], QM foundations, older guy\n*[[Ekkehard Peik|http://www.fqxi.org/aw-peik2.html]]${}^*$, constant change (experimental)\n*[[Simon Saunders|http://www.fqxi.org/aw-saunders2.html]]${}^*$, QM foundations\n*Lee Smolin (not going)\n*[[Robert Spekkens|http://www.fqxi.org/aw-spekkens2.html]]${}^*$, QM foundations\n*[[Max Tegmark|http://web.mit.edu/physics/facultyandstaff/faculty/max_tegmark.html]], astrophysics, cosmology, trouble maker...\n*[[Mark Trodden|http://physics.syr.edu/~trodden/]], cosmology, particle physics -- QFT\n*[[Roderich Tumulka|http://www.fqxi.org/aw-tumulka2.html]]${}^*$, Bohmian QM\n*[[Jos Uffink|http://www.phys.uu.nl/igg/jos/]], QM foundations\n*[[Vitaly Vanchurin|http://cosmos.phy.tufts.edu/~vitaly/]], cosmic strings \n*[[Xiao-Gang Wen|http://www.fqxi.org/aw-wen2.html]]${}^*$, gravity and light emergent from substrate :P\n*[[Serge Winitzki|http://www.theorie.physik.uni-muenchen.de/~serge/]], quantum cosmology\n**Likes wiki, and likes ToE.\n**Inflation expert -- says $R \sph^2$ term would be great, among others.\n***Strong constraints on these coefficients.\n*[[Toby Wiseman|http://schwinger.harvard.edu/~wiseman/]], string theory\n*[[Wojciech Zurek|http://public.lanl.gov/whz/]], cosmology and astrophysics, chaos, QM foundations\n\n${}^*$ grant winners (19)\n\nPress and Foundation people\n*[[Graham P Collins|http://www.sff.net/people/GPC/]], Scientific American Magazine\n*[[Valerie Jamieson|http://www.scienceinpublic.com/scienceweek/speakers.htm#Valerie%20Jamieson%20background]], New Scientist Magazine\n**particle physics background\n*[[Wade Davis|http://en.wikipedia.org/wiki/Wade_Davis]], National Geographic Explorer-in-Residence, ethnobiologist\n*[[Charles Harper|http://www.templeton.org/about_us/who_we_are/leadership_team/charles_harper/]], Senior Vice-President, John Templeton Foundation\n**He's paying, try not to insult him.\n*[[Amanda High|http://www.nptrust.org/about_npt/key_staff.asp#high]], Vice President, National Philanthropic Trust\n**What's she doing at the FQXi conference\n*[[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\n**Just got ousted from PI position, even though he founded it. Used to be main PI talent scout.\n*[[Christopher Liedel|http://executiveeducation.wharton.upenn.edu/fellows/feb_info/roster_detail.cfm?id=KRSM00000024468]], Executive Vice President & Chief Financial Officer, National Geographic Society\n*Robert Kuhn, Kuhn foundation -- makes science documentaries for PBS

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Choosing the anti-Grassmann 3-form to be $\sfff{\sod{B}} = \snf{e} \sod{\sPs} \sve{e} \s,$ gives the massive Dirac action in curved spacetime:\n\sbegin{eqnarray}\nS_f &=& \sint \sbig< \sfff{\sod{B}} \sudff{F} \sbig>\n= \sint \sbig< \sfff{\sod{B}} \sf{D} \sud{\sPs} \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sPs} \sve{e} \sbig( \sf{d} \sud{\sPs} + \sf{H}{}_1 \sud{\sPs} - \sud{\sPs} \sf{H}{}_2 \sbig) \sbig> \s\s\n&=& \sint \sbig< \snf{e} \sod{\sPs} \sve{e} \sbig( ( \sf{d} + {\sscriptsize \sfrac{1}{2}} \sf{\som} + {\sscriptsize \sfrac{1}{4}} \sf{e}\sph + \sf{W} + \sf{B}{}_1 ) \sud{\sPs}\n- \sud{\sPs} ( \sf{w} + \sf{B}{}_2 + \sf{x} \sPh + \sf{g} ) \sbig) \sbig> \s\s\n&=& \sint \snf{d^4 x} \s, |e| \s, \sbig< \sod{\sPs} \sga^\smu (e_\smu)^i \sbig( \spa_i \sud{\sPs} + {\sscriptsize \sfrac{1}{4}} \som_i^{\sp{i} \smu \snu} \sga_{\smu \snu} \sud{\sPs} + W_i \sud{\sPs} + B_{1i} \sud{\sPs} \s\s\n&& \shphantom{\sint \snf{d^4 x} \s, |e| \s, \sbig< \sod{\sPs} \sga^\smu (e_\smu)^i \sbig(} + \sud{\sPs} w_i + \sud{\sPs} B_{2i} + \sud{\sPs} x_i \sPh + \sud{\sPs} g_i \sbig) + \sod{\sPs} \s, \sph \s, \sud{\sPs} \sbig>\n\send{eqnarray}\nThe $\sod{\sPs} \s, \sph \s, \sud{\sPs}$ is the standard Higgs mass term.$\svp{\sHuge(}$\nThe $\sod{\sPs} \sga^\smu \sud{\sPs} x_\smu \sPh$ term... I don't understand yet -- promising for CKM.\n<<note HideTags>>

The ''FuN curvature'' (//''Frölicher-Nijenhuis curvature''//), $\sff{\sve{\scal F}}=\sf{dx^i} \sf{dx^j} \sha {\scal F}_{ij}{}^k \sve{\spa_k}$, of a [[vector valued form]], $\sf{\sve{\scal A}}=\sf{dx^i} {\scal A}_i{}^j \sve{\spa_j}$, is its [[FuN bracket|FuN derivative]] with itself,\n\sbegin{eqnarray}\n\sff{\sve{\scal F}} &=& - \sha \slb \sf{\sve{\scal A}}, \sf{\sve{\scal A}} \srb_L\n= - \slb \sf{\sve{\scal A}}, \sf{\spa} \srb \sf{\sve{\scal A}} \s\s\n&=& - \sf{\sve{\scal A}} \slp \sf{\spa} \sf{\sve{\scal A}} \srp + \sf{\spa} \slp \sf{\sve{\scal A}} \sf{\sve{\scal A}} \srp \s\s\n&=& - \slp \sf{\sve{\scal A}} \sf{\spa} \srp \sf{\sve{\scal A}} + \slp \sf{\spa} \sf{\sve{\scal A}} \srp \sf{\sve{\scal A}} \n\send{eqnarray}\nIn components, this is\n$$\n{\scal F}_{ij}{}^k = - {\scal A}_i{}^m \spa_m {\scal A}_j{}^k + {\scal A}_j{}^m \spa_m {\scal A}_i{}^k + {\scal A}_m{}^k \spa_i {\scal A}_j{}^m - {\scal A}_m{}^k \spa_j A_i{}^m\n$$\n\nIf, as often happens, a vector valued form is a [[vector projection]], $\sf{\sve{\scal P}} = \sf{\sve{\scal P}} \sf{\sve{\scal P}}$, its FuN curvature is\n\sbegin{eqnarray}\n\sff{\sve{\scal F}} &=& \sf{\spa} \sf{\sve{\scal P}} - \sf{\sve{\scal P}} \slp \sf{\spa} \sf{\sve{\scal P}} \srp \s\s\n&=& \slp 1 - \sf{\sve{\scal P}} \srp \slp \sf{\spa} \sf{\sve{\scal P}} \srp \n\send{eqnarray}\nwhich satisfies $\sf{\sve{\scal P}} \sff{\sve{\scal F}}=0$ -- 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,\n$$\n\sve{u} \sve{v} \sff{\sve{\scal F}} = - \sha {\slb \sve{u_H} , \sve{v_H} \srb_L}_V = - \sha \slb \sve{u} \slp 1- \sf{\sve{\scal P}} \srp , \sve{v} \slp 1- \sf{\sve{\scal P}} \srp \srb_L \sf{\sve{\scal P}}\n$$\n//check that//

The ''Frölicher-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, $\snf{\sve{K}}$, with respect to a vector field, $\sve{v}$, is written terms of [[partial derivative]]s as\n\sbegin{eqnarray}\n{\scal L}_{\sve{v}} \snf{\sve{K}} &=& \slim_{t \sto 0} \sfr{\sph_t^*\snf{\sve{K}} - \snf{\sve{K}}}{t} = \sve{v} \slp \sf{\spa} \snf{\sve{K}} \srp + \sf{\spa} \slp \sve{v} \snf{\sve{K}} \srp - \slp \snf{\sve{K}} \sf{\spa} \srp \sve{v} \s\s\n&=& \slp \sve{v} \sf{\spa} \srp \snf{\sve{K}} + \slp \sf{\spa} \sve{v} \srp \snf{\sve{K}} - \slp \snf{\sve{K}} \sf{\spa} \srp \sve{v}\n\send{eqnarray}\nThis defines the ''Frölicher-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.,\n$$\n{\scal L}_{\snf{\sve{K}}} {\sve{v}} = \slb \snf{\sve{K}} , \sve{v} \srb_L = - \slb \sve{v} , \snf{\sve{K}} \srb_L = - {\scal L}_{\sve{v}} \snf{\sve{K}} \n$$ \nSimilarly, generalizing Cartan's formula for the Lie derivative, the FuN derivative of a differential form is,\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{K}}} \snf{F} &=& \slb \snf{\sve{K}} , \snf{F} \srb_L = \snf{\sve{K}} \slp \sf{d} \snf{F} \srp + \slp -1 \srp^k \sf{d} \slp \snf{\sve{K}} \snf{F} \srp \s\s\n&=& \slp \snf{\sve{K}} \sf{\spa} \srp \snf{F} + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \snf{F}\n\send{eqnarray}\nwhich 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,\n$$\n\slb \snf{\sve{K}} , \snf{F} \srb_L = \slb \snf{\sve{K}} , \sf{d} \srb \snf{F}\n$$\nthereby 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 \n\sbegin{eqnarray}\n\slb \snf{\sve{K}} , \snf{\sve{L}} \srb_L &=& {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \s\s\n&=& \snf{\sve{K}} \slp \sf{\spa} \snf{\sve{L}} \srp - \slp -1 \srp^{kl} \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp + \slp -1 \srp^k \sf{\spa} \slp \snf{\sve{K}} \snf{\sve{L}} \srp - \slp -1 \srp^{kl+l} \sf{\spa} \slp \snf{\sve{L}} \snf{\sve{K}} \srp \s\s\n&=& \slp \snf{\sve{K}} \sf{\spa} \srp \snf{\sve{L}} - \slp -1 \srp^{kl} \slp \snf{\sve{L}} \sf{\spa} \srp \snf{\sve{K}} + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \snf{\sve{L}} - \slp -1 \srp^{kl+l} \slp \sf{\spa}\snf{\sve{L}} \srp \snf{\sve{K}} \s\s\n&=& \slb \snf{\sve{K}} , \sf{\spa} \srb \snf{\sve{L}} - \slp -1 \srp^{kl} \slb \snf{\sve{L}} , \sf{\spa} \srb \snf{\sve{K}}\n\send{eqnarray}\nwhich 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\n$$\n{\scal L}_{\slb \snf{\sve{K}} , \snf{\sve{L}} \srb_L} = \slb {\scal L}_{\snf{\sve{K}}} , {\scal L}_{\snf{\sve{L}}} \srb\n$$\nwhen acting on vectors or forms.\n\nThe FuN derivative has a number of other nice [[properties|FuN identities]].\n\n//(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]], ${\scal L}_{\snf{\sve{K}}}$, 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.\n\nThe FuN Lie bracket may or may not change sign under the exchange of its vector valued $k$-form and vector valued $l$-form arguments,\n$$\n\slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L = - \slp -1 \srp^{kl} \slb \snf{\sve{L}}, \snf{\sve{K}} \srb_L\n$$\nAs a derivation, the FuN derivative operates on products of forms via the graded Liebniz rule,\n$$\n{\scal L}_{\snf{\sve{K}}} \slp \snf{F} \snf{G} \srp \n= \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp \snf{G} + \slp -1 \srp^{kf} \snf{F} \slp {\scal L}_{\snf{\sve{K}}} \snf{G} \srp\n$$\nBut it is not a derivation over products of VVFs and forms. Using some [[vector valued form identities]] we get\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{F} \srp &=& \slp \snf{\sve{K}} \sf{\spa} \srp \slp \snf{\sve{L}} \snf{F} \srp + \slp -1 \srp^k \slp \sf{\spa} \snf{\sve{K}} \srp \slp \snf{\sve{L}} \snf{F} \srp \s\s\n&=& \slp \slp \snf{\sve{K}} \sf{\spa} \srp \snf{\sve{L}} \srp \snf{F} \n+ \slp-1\srp^{k\slp l-1\srp} \slb \snf{\sve{L}} \slp \slp \snf{\sve{K}} \sf{\spa} \srp \snf{F} \srp\n- \slp \snf{\sve{L}} \slp \snf{\sve{K}} \sf{\spa} \srp \srp \snf{F} \srb \s\s\n& &\n+ \slp-1\srp^k \slp \slp \sf{\spa} \snf{\sve{K}} \srp \snf{\sve{L}} \srp \snf{F}\n+ \slp-1\srp^{kl} \snf{\sve{L}} \slp \slp \sf{\spa} \snf{\sve{K}} \srp \snf{F} \srp\n- \slp-1\srp^{kl} \slp \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp \srp \snf{F} \s\s\n&=& \n\slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{F} \n+ \slp-1\srp^{k\slp l-1\srp} \snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp \n - \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{F}\n\send{eqnarray}\nand\n\sbegin{eqnarray}\n{\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{F} &=& \slp \slp \snf{\sve{L}} \snf{\sve{K}} \srp \sf{\spa} \srp \snf{F} - \slp-1\srp^{\slp l+k\srp} \slp \sf{\spa} \slp \snf{\sve{L}} \snf{\sve{K}} \srp \srp \snf{F} \s\s\n&=& \slp \snf{\sve{L}} \slp \snf{\sve{K}} \sf{\spa} \srp \srp \snf{F} - \slp-1\srp^k \slp\n\slp-1\srp^l \slp \sf{\spa} \snf{\sve{L}} \srp \snf{\sve{K}} - \snf{\sve{L}} \slp \sf{\spa} \snf{\sve{K}} \srp + \slp \snf{\sve{L}} \sf{\spa} \srp \snf{\sve{K}} \n\srp \snf{F} \s\s\n&=& \n\snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{F} \srp\n+ \slp-1\srp^{k\slp l-1\srp} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{F}\n- \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{F} \srp \n\send{eqnarray}\nwhich 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:\n$$\n{\scal L}_{\snf{\sve{L}}\snf{\sve{K}}} \snf{\sve{M}}\n= \snf{\sve{L}} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{M}} \srp\n+ \slp-1\srp^{k\slp l-1\srp} \slp {\scal L}_{\snf{\sve{K}}} \snf{\sve{L}} \srp \snf{\sve{M}}\n- \slp-1\srp^{k\slp l-1\srp} {\scal L}_{\snf{\sve{K}}} \slp \snf{\sve{L}} \snf{\sve{M}} \srp\n- \slp-1\srp^{m\slp k+ l-1\srp} \slp {\scal L}_{\snf{\sve{M}}} \snf{\sve{L}} \srp \snf{\sve{K}}\n$$\n\nWhen the two VVF's are written as $\snf{\sve{K}}=\snf{K^A} \sve{X_A}$ and $\snf{\sve{L}}=\snf{L^A} \sve{Y_A}$ their FuN bracket is\n\sbegin{eqnarray}\n\slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L &=& \snf{K^A} \snf{L^B} \slb \sve{X_A}, \sve{Y_B} \srb_L + \snf{K^A} \slp {\scal L}_{\sve{X_A}} \snf{L^B} \srp \sve{Y_B}\n- \slp {\scal L}_{\sve{Y_B}} \snf{K^A} \srp \snf{L^B} \sve{X_A} \s\s\n&+& \slp -1 \srp^k \slp \sf{d} \snf{K^A} \srp \sve{X_A} \snf{L^B} \sve{Y_B}\n+ \slp -1 \srp^k \slp \sve{Y_B} \snf{K^A} \srp \slp \sf{d} \snf{L^B} \srp \sve{X_A}\n\send{eqnarray}\nand, for the FuN bracket of a vector valued 1-form with itself,\n\sbegin{eqnarray}\n\slb \sf{\sve{K}}, \sf{\sve{K}} \srb_L &=& \sf{K^A} \sf{K^B} \slb \sve{X_A}, \sve{X_B} \srb_L + 2 \sf{K^A} \slp {\scal L}_{\sve{X_A}} \sf{K^B} \srp \sve{X_B}\n- 2 \slp \sf{d} \sf{K^A} \srp \sve{X_A} \sf{K^B} \sve{X_B}\n\send{eqnarray}\n\nWhen acting on forms, the FuN derivative commutes with the [[exterior derivative]],\n$$\n0 = \slb {\scal L}_{\snf{\sve{K}}}, \sf{d} \srb = {\scal L}_{\snf{\sve{K}}} \sf{d} + \slp -1 \srp^k \sf{d} {\scal L}_{\snf{\sve{K}}} \n$$\nIn fact, the FuN derivative of a form with respect to the [[identity projection|vector projection]] is the exterior derivative,\n$$\n{\scal L}_{\snf{\sve{I}}} \snf{F} = \sf{d} \snf{F}\n$$\nand of a VVF is zero, ${\scal L}_{\snf{\sve{I}}} \snf{\sve{K}} = 0$.\n\nActing on itself twice, the FuN bracket satisfies the ''graded Jacobi identity'',\n$$\n\slb \snf{\sve{K}}, \slb \snf{\sve{L}} , \snf{\sve{M}} \srb_L \srb_L = \slb \slb \snf{\sve{K}}, \snf{\sve{L}} \srb_L, \snf{\sve{M}} \srb_L - \slp -1 \srp^{kl} \slb \snf{\sve{L}}, \slb \snf{\sve{K}} , \snf{\sve{M}} \srb_L \srb_L\n$$

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>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#00BF00} </SPAN></td>\n<td><SPAN class="math">q^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbtri{#0000F7} </SPAN></td>\n<td><SPAN class="math">q^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#D90000} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\sbutr{#00BF00} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^g_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\ssmash{-\sfr{1}{2\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\sbutr{#0000F7} </SPAN></td>\n<td><SPAN class="math">\sbar{q}{}^b_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">{\sfr{1}{\ssqrt{3}}}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\smutr{#0000F7}}{\smtri{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{II}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\smp 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\spm \sfr{2}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sstri{#0000F7}}{\ssutr{#0000F7}}</SPAN></td>\n<td><SPAN class="math">q_{III}</SPAN></td>\n<td></td>\n<td COLSPAN="3"><SPAN class="math">\spm 1 \s;\s; \spm \s! 1</SPAN></td>\n<td></td>\n<td>"</td>\n<td>"</td>\n<td><SPAN class="math">\smp \sfr{4}{3}</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\srlap{\sbutr{#999999}}{\sbtri{#999999}}</SPAN></td>\n<td><SPAN class="math">l</SPAN></td>\n<td></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\smp \sfr{1}{2}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">\spm 1</SPAN></td>\n</tr>\n</table>\n</td>\n\n<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>\n\n<td>\n<embed src="talks/Perimeter07/anim/g2spin.mov" width="452" height="452" controller="false" autoplay="false" loop="false"></embed>\n<!-- <embed src="talks/Perimeter07/anim/g2spin/p1.png" width="462" height="462"></embed> -->\n</td>\n\n</tr>\n</table>\n</center>\n</html>\n<<note HideTags>>

<html>\n<center>\n<table class="gtable">\n<tr border=none>\n\n<td border=none>\n<table class="ptable">\n<tr>\n<th COLSPAN="2"><SPAN class="math">G2+</SPAN></th>\n<th></th>\n<th><SPAN class="math">x</SPAN></th>\n<th><SPAN class="math">y</SPAN></th>\n<th><SPAN class="math">z</SPAN></th>\n<th></th>\n<th><SPAN class="math">g^3</SPAN></th>\n<th><SPAN class="math">g^8</SPAN></th>\n<th><SPAN class="math">\sfr{\ssqrt{8}}{\ssqrt{3}} B_2</SPAN></th>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{g}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}g}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{r\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{r}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr>\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{\sbar{g}b}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td></td>\n<td><SPAN class="math">\sha</SPAN></td>\n<td><SPAN class="math">-\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n<tr class="butt">\n<td><SPAN class="math">\smcir{#6666FF} </SPAN></td>\n<td><SPAN class="math">g^{g\sbar{b}}</SPAN></td>\n<td></td>\n<td><SPAN class="math">0</SPAN></td>\n<td><SPAN class="math">-1</SPAN></td>\n<td><SPAN class="math">1</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{\ssqrt{3}}{2}</SPAN></td>\n<td><SPAN class="math">0</SPAN></td>\n</tr>\n\n<tr>\n<td><SPAN class="math">\sbtri{#D90000} </SPAN></td>\n<td><SPAN class="math">q^r_I</SPAN></td>\n<td></td>\n<td><SPAN class="math">-\sfr{1}{2}</SPAN></td>\n<td><SPAN class="math">\sfr{1}{2}&l