There are very few moments in theoretical physics that qualify as... thrilling -- moments that send shivers of excitement down the spine and make the brain tingle. It's such an abstract pursuit, you wouldn't think the effects would be so visceral. The thoughts take years to accumulate, and are often disjoint and haphazardly organized. On a very rare occasion, a new insight brings a cascade of ideas together at once -- a chain reaction in the mind. It's very cool. Of course, the idea could still be wrong. And usually one needs to set about the hard work of trying to prove it wrong, before airing it in public. But, in this case, with a recent idea I think is significant, that work will likely take me a long time -- and I want to share the main idea now. So here it is:
If we take seriously the idea that fermions may be gauge theory ghosts, there is one gauge theory in particular that stands out: that of a principal E8-bundle. The exceptional group of rank 8 is the largest of the exceptional Lie groups, and perhaps the richest in structure. Pirating an appendix from Superstring Theory, the 248 dimensional Lie algebra of E8 is described as:
e8 = so(16) + S(16)
the special orthogonal group (with 120 elements) acting on the space of 128 dimensional chiral spinors. This is remarkable as it is, since it says there's a Lie algebra in which the Lie bracket of two elements gives one element acting on another as a Clifford algebra element, B, of so(16) acting from the left on a spinor, Psi, of so(16):
[ B, Psi ] = B Psi
There is also a lesser known, equivalent description of e8 that I read about in John Baez's This Week's Finds:
e8 = so(8) + so(8) + (V(8) x V(8)) + (S(8) x S(8)) + (S(8) x S(8))
In this description, the 28 elements, H, of so(8) act from the left on three 64 element blocks, Psi1,Psi2,Psi3:
[ H, Psi123 ] = H Psi123
and the other 28 elements, G, of so(8) act on these from the right. Now, if we build a Yang-Mills theory with E8, and take the three blocks to be ghosts, the BRST extended connection:
A = H + G + Psi1 + Psi2 + Psi3
and its curvature,
F = d A + AA
= (dH+HH) + (dG+GG) + (dPsi1+HPsi1+Psi1G) + (dPsi2+HPsi2+Psi2G) + (dPsi3+HPsi3+Psi3G)
fits the standard model -- complete with three generations of fermions and gravity! This is mostly laid out in my last paper. The gravitational connection, frame, Higgs multiplet, U(1), and SU(2) fit in H, while SU(3) and another piece of U(1) fit in G. And three generations of leptons and quarks fit in the Psi's, related by triality. This is a beautiful thing -- exactly what one would hope for in a TOE!
If it's true, it would explain a lot of complicated structures in the standard model in terms of a simple E8 Yang-Mills field: exactly what and why spinors are, why the particles get the charges they do, why there are three generations, and possibly why the masses are what they are. And there's very little wiggle room. It will have to be a real form of complex E8, since we need a non-compact gauge group for gravity. But there will be only a handful of ways to consider the E8 symmetry breaking to the standard model. After all, it's just a Yang-Mills theory, with no other fancy stuff flying around. It will either clearly work, or it clearly won't.
There is a lot of work to do. I haven't gotten exactly the right particle assignments down yet. And I don't know if someone's tried this before, since the literature is somewhat obfuscated by the use of E8 in heterotic string theory, which is quite different. (I doubt it's been done before though, since it relies on my crazy idea of replacing some gauge fields with fermionic ghosts.) I expect to be working on this for quite a while -- studying the structure of E8, which is quite beautiful, and many other aspects -- trying to see if the fermions will fit properly and the KM matrix pops out of it. It's not a completed theory, which is why I didn't write it up as a paper. But I think it's interesting and exciting enough to put here, for the enjoyment and puzzlement of others.
