Thank you, Jorge, for the introduction. 1
OK, this is a periodic table of the standard model and gravity -- it's a schematic diagram of all the elementary fields we know of and the algebraic relationships between them.
The gluons are su(3) valued 1-forms interacting with the red green and blue quarks. The electroweak W (which I've colored yellow) is an su(2) interacting with the Higgs and left-chiral fermion doublets, which I've colored yellowish. The electroweak B is a u(1) interacting with the fermions and Higgs according to their weak hypercharges. The gravitational spin connection, omega, is an so(3,1) interacting with the left and right chiral fermions, and with the gravitational frame, e. The frame, and Higgs, phi, interact with the left and right-chiral parts of the fermions, giving them Dirac masses. And this structure is repeated over the three generations of fermions.
This large collection of fields and interactions is pretty complicated. It sure doesn't look like the structure of just one mathematical object, which is what we want in a theory of everything. But today I will describe a uniquely beautiful mathematical object with precisely this structure, and further details are available in this conservatively titled paper.
We will proceed by starting with the most straightforward unification we could hope for.
(next page please) 2
We join all of these fields as parts of one superconnection, over a four dimensional base manifold. This general idea should be familiar from grand unified theories, which combine the gauge fields into a single, larger connection. We're proceeding in the same spirit, but going further by using two unusual tricks. First, we're including gravity -- the connection AND the frame -- as parts of this connection. This reproduces general relativity through the MacDowell-Mansouri approach to gravity, discovered in the late seventies, which I first learned about in Smolin, Freidel, and Starodubtsev's quantum gravity papers. The second trick is that we're also including all the fermions in this superconnection, as Lie algebra valued Grassmann numbers. Now, at first look, this second trick shouldn't work. When we calculate the dynamics of this connection by taking its curvature, the interactions between fields will come from their Lie bracket. But we know gravity and the gauge fields interact with the fermions in fundamental representations. The fermions, such as this Dirac spinor column of spin up and spin down left and right chiral fields, live in a fundamental representation space, and these certainly don't appear to be Lie algebra elements. So how can this possibly work? Well, it turns out that for all five exceptional Lie groups, there are Lie brackets that act like the fundamental action. The structure of these algebras is such that some Lie algebra elements ARE fundamental representation space elements. This fact makes it possible to include the fermions in the connection as Lie algebra valued fields.
Using these two tricks, the unification is very straightforward. But in order to understand it we'll need to review some representation theory.
(next page) 3
The root system of a Lie algebra is a way of describing its structure, independent of the choice of generators. We can start with whatever representation we want, and pick R generators that mutually commute. The R dimensional space, built using these generators, is called a Cartan subalgebra. We get a linear operator on the rest of the Lie algebra by putting the Cartan subalgebra elements in one side of the Lie bracket. This gives a set of eigenvectors, called root vectors, spanning the Lie algebra -- each with a unique eigenvalue, called a root. The root coordinates determine points, the root system, in R dimensional space. Now, the cool thing that happens here is that the Lie bracket between two root vectors gives a third only if their corresponding roots add to give the third. In this way, the pattern of the root system in R dimensions (which can be quite pretty) corresponds to the structure of the Lie algebra -- and it's independent of our choice of generators, which just corresponds to a rotation of the root system.
We can also use our Cartan subalgebra to calculate eigenvectors and eigenvalues, called weight vectors and weights, corresponding to the Lie algebra action on any representation space. The pattern of weights describes the representation. For example, the roots are just the weights in the adjoint representation.
Now, all of this connects to physics because these weight vectors, algebraicly, are the particle states, and their weights are their quantum numbers. This gets a lot clearer with an example.
(next page please) 4
The gluons live in su(3), shown here with the eight Gell-Mann matrices as generators. The Cartan subalgebra, on the diagonal of the matrix, is two dimensional -- producing a root space with coordinates along the g3 and g8 axes. Here's a typical root vector, the green anti-blue gluon, and the eigenequation gives its root coordinates, minus one-half and square root of three over two. These gluons act on the quarks in the fundamenal three representation space, with a red quark represented by this canonical unit vector, with weight coordinates of one half and one over two root three. The anti-quarks are in the dual representation, with the opposite weights.
Plotting these weights for the gluons and quarks makes a nice pattern.
(next page) 5
These are the six gluon root vectors and roots of su(3), which are plotted as blue circles, and the three quark and three anti-quark weights of three colors, which are plotted as triangles and inverted triangles.
The interactions correspond to Lie brackets, which correspond to vector addition of the weights in this pattern. If we take the red-anti-green gluon there on the far right, and add it (as a vector) to the green quark, we get a red quark. This is the result of a typical quark-gluon interaction.
Now, this is all standard representation theory. But there's something unusual here. This weight system is not just a weight system -- it's also a root system. It's the root system for the smallest simple exceptional Lie group, G2. We can take the green quark and the red-anti-green gluon to be the simple roots, and build this Dynkin diagram for this root system, corresponding to G2.
So this diagram shows that we don't have to think of quarks as being in the fundamental three representation space, we can treat them as Lie algebra elements of G2.
(next page) 6
The G2 Lie algebra breaks up into an su(3) subalgebra, which are the gluons, and the three and dual three, identified as quarks and anti-quarks. The Lie bracket between gluons and quarks, as root vectors of g2, gives all the interactions -- the Feynman vertices for these particles.
This is a remarkable development -- it implies that use of a particular representation for these elementary particles is not prescribed by nature, as we previously thought. Instead, the quarks and gluons live in the G2 Lie group, a nice symmetric manifold with a geometry described by the relations between its diffeomorphisms. And this doesn't only work for the quarks and gluons in G2 -- it's going to work like this for all other particles in the standard model, which may be identified as Lie algebra elements of the other exceptional Lie groups.
But first I want to show you how we can describe Lie algebra root systems as projections of larger root systems, so we can spot subalgebras.
(next page) 7
The root system of SO(7) can be described as all edge and face midpoints of a cube in three dimensions, here in coordinates x, y, and z. The eight weights of an so(7) spinor are at the vertices of a cube half this size. By rotating to new coordinates, g3 g8 and B2, and projecting along the B2 axis, all of these weights project to the roots of G2 -- a subalgebra of so(7) -- and two project to the origin. This is a graphical representation of the well know embedding of su(n) in so(2n), and is a typical example of how to see a group as a subgroup of a larger one.
These overlapping G2 roots exist at different levels of B2, which we can see by spinning the cube I just described.
(please page forward FOUR pages)
After rotating this cube, we can see there's a grading of this root system along the B2 direction. The B2 coordinate for these weights is related to the weak hypercharge for quarks and leptons.
(next page please) 12
If we go back to our periodic table, we can see the B2 coordinates are the weak hypercharges for the left-chiral leptons and quarks, along the top two rows of the table. In order to get the correct hypercharges for all the fermions, we need another U(1) field, part of B1, that acts to add and subtract one to the weak hypercharges of right chiral fermions, along the bottom two rows of the table. By adding these two B's to get our electroweak U(1) field, we get the correct hypercharges. This is actually an old idea -- it's part of the Pati-Salam grand unified theory.
The Pati-Salam model is a left-right symmetric theory. It adds a right SU(2) partner, B1, to the left SU(2) electroweak W. The third Pauli matrix in this su(2)R is the B13 field that combines with B2. The other two fields in B1 must get large masses somehow, breaking this left-right symmetry, because we haven't seen them.
So, the Pati-Salam model implies the hypercharge comes from two independent sets of quantum numbers, B13 and B2. Once we've made this split, we can see from this pattern of interactions that the electroweak, Higgs, and gravitational fields on the left hand side of this diagram are parts of some larger algebra. The biggest clue is that if the Higgs is going to give Dirac masses to the fermions, it's going to need to interact with their left and right chiral parts. Combining the Higgs with the gravitational frame will do this for us. Let me talk about gravity a bit so we can see how this works.
(next page) 13
The so(3,1) valued spin connection, with generators written here as Clifford algebra bivectors, can be broken up into left and right chiral parts, built with Pauli matrices -- more familiar as the self-dual and anti-self-dual parts of the connection. The Cartan subalgebra is the diagonal of this four by four matrix, and it gives up and down root vectors for the left and right chiral spin connection fields, with corresponding root coordinates as shown in the table. The gravitational frame is a Clifford algebra vector, and breaks up into these weight vectors, with the weights shown in the table.
When we write out the Dirac operator in curved spacetime, the spin connection acts on the fermions in the fundamental four representation space -- they're Dirac spinors, built from stacking two Vile spinors, with spin-up and spin-down parts as the weight vectors, and the spin quantum numbers shown in the table.
When the frame interacts with the fermions it mixes the left and right-chiral parts, which we can see by multiplying the frame matrix times the Dirac spinor, or by adding their weights in the table. For example, we could add the weights of the spatial-up frame field to those of a left-chiral-down fermion to get a right-chiral-up fermion, the result of this interaction.
The structure of the electroweak interactions is very similar.
(next page) 14
We can identify the left-chiral acting su(2), the W, and right-chiral acting su(2), the B1, of the Pati-Salam model as parts of an SO(4) electroweak connection. If we put the four Higgs fields in the vector of this SO(4) we get the correct weights for the standard model Higgs. And the Pati-Salam model gives us the correct weights for the fermions.
When we rotate the B2 coordinates for the fermions in with the B13 coordinate (the equation there at the bottom), we get the correct standard model hypercharges, with the electroweak, U(1), coupling constant equal to the square root of three fifths. This gives the same prediction for the Weinberg angle as almost all grand unified theories. The electric charge quantum numbers come from adding W3 with half Y, and are shown in the table.
Since we have gravity described by the SO(3,1) spin connection acting on the frame as a vector, and we have the electroweak connection as an SO(4) acting on the four Higgs in a vector, it's very natural to combine all these as parts of a unified, SO(7,1) graviweak connection.
(next page please) 15
This is the combined, SO(7,1) graviweak connection, with the frame and Higgs combined as a simple bivector. This is a very unusual feature, the frame and Higgs are parts of one composite field in this theory, with this frame-Higgs field occupying sixteen Lie algebra elements.
We can pick a chiral Clifford matrix representation for Cl(7,1) and use that (there on the bottom) to calculate the roots for the SO(7,1) fields and the weights for the fermions, here as elements of the positive-chiral eight dimensional spinor. But that's the hard way to do it. The easy way is to just combine the two pairs of coordinates we already found to get the roots in four dimensions, as shown in the table.
Now, these 24 roots of the D4 root system that we end up with have a nice set of symmetries called triality. There are many planes through this root system such that if we rotate in one of these planes by two pi over three, it gives us the same set of roots. But this doesn't work if we include this eight-spinor at the bottom of the table -- the same triality rotation of these weights gives us eight new weights, and another application of the same rotation gives eight more. But if we include these two sets of triality partners in a larger group, we do have a root system with triality -- it's the root system of the rank four exceptional Lie group, F4.
(next page) 16
This particular triality rotation permutes three of the F4 coordinates. Applying it three times to any root gets us back where we started. Applying this triality rotation to the positive-chiral eight-spinor gives the triality-equivalent eight vector, and applying this triality rotation to that gives the negative-chiral eight-spinor. Since these new fields have the same quantum numbers under this triality symmetry, we label them as the three generations of fermions -- with smaller triangles for their their symbols in the tables and plots. This assignment of particles to triality partners is tentative, and I expect the three physical generations of fermions to have some more complicated relationship to these triality partners.
But for now, we can plot the 48 roots of the F4 root system, rotated and projected down to two dimensions.
(next page) 17
This shows the graviweak gauge fields and three generations of leptons, with lines connecting the triality partners. Since projection is a linear operation, we can still compute interactions by adding the roots visually. For example, the yellow W+, on the far right, interacts with any of the left-chiral electrons (the yellow triangles on the left) to give left-chiral electron-neutrinos (the darker yellow triangles on the right).
We're limited in what we can do in this diagram, because there are no anti-fermions here, and no quarks or gluons. To get the whole picture, we have to combine F4 and G2.
(next page) 18
If we fit F4 and G2 together, they give the root system of the most mathematically and aesthetically beautiful Lie group there is -- the largest simple exceptional Lie group, E8. Physically, F4 has the quantum numbers for the graviweak interactions, and the three generation structure, while G2 has the strong interactions, a hypercharge piece, and the anti-particle structure. Consistently combining the roots of these two Lie groups gives the quantum numbers of all particles in the standard model. In order to match them up, we need to introduce a new quantum number, w, with values determined by the values of the others. Knowing the quantum numbers of the standard model allows us to identify each elementary particle as one of the roots of E8.
If we start with E8, we can break its Lie algebra down to the subalgebras of the standard model. It breaks up into f4 and g2, which operate on their smallest irreducible representations: the 26 is the traceless exceptional Jordan algebra and the smallest irreducible representation space of f4, and the 7 is the smallest irreducible representation of g2. These break up further to give the algebras of the standard model gauge fields, gravity, three generations of fermions, and a handful of new fields. The weights of the standard model and gravitational fields fit perfectly to the 240 roots of E8. And we can plot it.
(next page) 19
This is a projection of the E8 root system from eight dimensions down to two, with each root labeled by an elementary particle symbol. We can see every allowed interaction in the standard model by adding these roots together. That's kind of hard to do in this plot, which shows the vertices of the E8 polytope spread out into eight concentric circles, as first drawn by hand in 1964. To see the standard model structure here, we'll want to rotate our viewing plane in eight dimensions until we can see the F4 and G2 subalgebras.
(please page forward four pages) 23
We are rotating in eight dimensions until we see the F4 subalgebra of E8. Now we'll rotate the viewing plane a little towards the G2 root system, which is orthogonal to the F4 system in E8.
(please page forward two pages) 25
The gluons emerge here from the center, and the quarks and anti quarks spread away from the leptons. Now we continue rotating from the plane of F4 to that of G2.
(page forward five pages) 30
We see the 27 elements of the exceptional Jordan algebra converge for each color and anti-color. The central cluster of leptons and gauge fields are the 72 roots of the E6 subgroup of E8.
(page forward three pages) 33
Here's the G2 subalgebra in E8. Now back to where we started.
(page forward four pages, to page 37) 37
That's a brief tour of the E8 root system -- it's very beautiful, and there are many other wonderful structures hiding in here. One can get lost in it.
And we couldn't of asked for a better fit of the standard model and gravity into a Lie algebra. We can assemble a periodic table from the structure of E8, and compare it to the periodic table we built for the standard model.
(next page please. this page should be the E8 periodic table) 38
This table displays the algebraic structure of E8. The so(7,1) subalgebra is the cluster of elements on the left, and there's an so(8) subalgebra cluster at the bottom. This so(7,1) and so(8) act on three generations of sixty-four fermion fields. It's a perfect match to the standard model. There are only a handful of new new fields, which are labeled here as w and x Phi. These new fields are suspiciously similar to the gravitational spin connection and frame-Higgs fields, but I'm not sure what to make of that yet. The x interacts weakly, and the scalar big Phi has color, interacting with the quarks, and the w is a new U(1) field. We haven't seen these fields in accelerators (so if they exist they must have large masses), but I don't think they're bad, since a set of Higgs fields like this seems promising for producing the CKM matrix. I've also colored these new, weakly interacting massive particles a suggestively DARK color, but I won't say more about that.
So, this is an amazingly good match of every field we know of to one of the most beautiful algebraic structures in mathematics. But let me turn away from the pretty pictures and get to the nitty gritty of the dynamics, which is also very pretty, in its own way.
(next page) 39
The E8 connection breaks up into the standard model pieces of the Lie algebra. The H1 is the so(7,1), graviweak connection, which is made up of the so(3,1) spin connection, the frame-Higgs in the mixed electroweak part, and the left and right su(2) electroweak fields. The H2 is the so(8) connection, containing the su(3) gluons, an electroweak u(1) and a new u(1), and the new x Phi fields. The Grassmann fields in the connection are the three generations of fermions, related by triality. When we compute the curvature of this connection, it will break up into these same algebraic parts.
(next page) 40
The calculation of the curvature is straightforward, and it breaks up into many familiar pieces inhabiting various parts of the Lie algebra. The so(3,1) part is the Riemann curvature plus an area bivector term times phi squared -- the square of the Higgs amplitude. In the mixed electroweak part, we get non-dynamical torsion coupled to our Higgs, and the covariant derivative of the Higgs. And we get the curvatures of the electroweak gauge fields. The su(3) part is the curvature for the gluons, and the rest of so(8) is the curvature of these new fields that I don't completely understand yet, but I can write it down.
The coolest thing that happens is in the fermionic part of the curvature. The exceptional structure of E8 implies the bosons act on the fermions as left and right multiplication of matrices in fundamental representation spaces. This gives us exactly the correct covariant Dirac derivative operator on the fermions in curved spacetime.
Now, using this curvature, we can write the action for everything in a very concise form.
(next page) 41
The most natural and efficient way to express the action for the standard model and gravity in this model is as a modified BF action. We introduce a combined two form plus anti-Grassman three form field, B, as a Lagrange multiplier. We then hand pick two modifying terms, quadratic in B, in order to match the action of the standard model. (It would be nice if there were a better mathematical justification for this symmetry breaking, but for now we just do it because it works.) Varying the B field and plugging it back in gives us the correct Dirac action, the Einstein-Hilbert action for gravity, and the action for the other bosons.
The cosmological constant in this theory comes out equal to three quarters the Higgs vev squared. The resulting vacuum solution of Einstein's equation is de Sitter spacetime, and it's worth noting that this vacuum solution for the spin connection, frame, and Higgs, corresponds to a vanishing E8 curvature.
We skipped several steps in this computation, and it's worth looking at each part of this action in more detail.
(next page) 42
Gravity is described here using the MacDowell-Mansouri approach. The gravitational part of the action involves the so(3,1) part of the e8 curvature, which includes the Riemann curvature and area bivector terms. Varying B and plugging it back in gives the action quadratic in the curvature. Multiplying this out gives three terms: a Chern-Simons boundary term, which we toss, the spacetime volume element, and the curvature scalar. Adding these last two terms gives the Palatini action, equivalent to the Einstein-Hilbert action, with a cosmological constant related to the Higgs vev.
(next page) 43
If we choose the anti-Grassmann 3-form, by hand, to include the spacetime volume form, anti-fermions, and coframe, then the fermionic part of the BF action is the massive Dirac action in curved spacetime. This is a really neat result, because all the interactions we want are here, including this correct factor of one fourth in front of the spin connection. And the coframe cancels with the frame to give exactly the standard interaction term between the fermions and the Higgs, so they can get Dirac masses. There is also an interaction with the new fields that I don't understand yet, but it seems promising for getting the CKM matrix.
(OK, next page please, for the summary) 44
The idea proposed here is an extreme unification: everything is described with one E8 principal bundle connection. All known fields of the standard model and gravity match the algebraic structure of E8, and their interactions and dynamics come from the curvature and action of this connection. Since a principal bundle, geometrically, consists of the E8 Lie group manifold twisting and dancing over our four dimensional spacetime, everything here is described by pure geometry. I think it's a very appealing picture of the universe, but, of course, it could be wrong. This picture is so simple that it leaves no room for error; it's either going to be right, and make many successful predictions as it develops, or it's spectacularly wrong. So far the theory looks very good, but it's young and there's still much to understand about it.
(next page please) 45
Currently, the action is assembled by hand to match the standard model. It's efficient, but mathematically unjustified and kind of ugly -- it needs improvement. Also, the way the three physical fermion generations work, and how these w and x Phi fields fit in, is not perfectly understood.
Finally, and most importantly, this theory requires a fundamental quantum description. Since it is a background independent theory of a connection, it is philosophically and technically aligned with the goals and methods of loop quantum gravity. Successful techniques developed to quantize the gravitational connection should extend to describe a quantum theory of this very large connection. A background independent, Quantum E8 Theory depends on the success of the quantum gravity program. If these two theories both work out, and they work together, we will have a fully successful theory of everything.
(thank you, I'm sure there are some questions)
-------------------------------------
46
47
48