The December issue of Scientific American is out, and it has an article by Garrett Lisi and Jim Weatherall about geometry and unification entitled A Geometric Theory of Everything. Much of the article is about the geometry of Lie groups, fiber-bundles and connections that underpins the Standard Model as well as general relativity, and it promotes the idea of searching for a unified theory that would involve embedding the SU(3)xSU(2)xU(1) of the Standard Model and the Spin(3,1) Lorentz group in a larger Lie group.
The similarities between (pseudo)-Riemannian geometry in the “vierbein” formalism where there is a local Spin(3,1) symmetry, and the Standard Model with its local symmetries makes the idea of trying to somehow unify these into a single mathematical structure quite appealing. There’s a long history of such attempts and an extensive literature, sometimes under the name of “graviGUT”s. For a recent example, see here for some recent lectures by Roberto Percacci. The Scientific American article discusses two related unification schemes of this sort, one by Nesti and Percacci that uses SO(3,11), another by Garrett that uses E8. Garrett’s first article about this is here, the latest version here.
While I’m very sympathetic to the idea of trying to put these known local symmetry groups together, in a set-up close to our known formalism for quantizing theories with gauge symmetry, it still seems to me that major obstructions to this have always been and are still there, and I’m skeptical that the ideas about unification mentioned in the Scientific American article are close to success. I find it more likely that some major new ideas about the relationship between internal and space-time symmetry are still needed. But we’ll see, maybe the LHC will find new particles, new dimensions, or explain electroweak symmetry breaking, leading to a clear path forward.
For a really skeptical and hostile take on why these “graviGUT” ideas can’t work, see blog postings here and here by Jacques Distler, and an article here he wrote with Skip Garibaldi. For a recent workshop featuring Lisi, as well as many of the most active mathematicians working on representations of exceptional groups, see here. Some of the talks feature my favorite new mathematical construction, Dirac Cohomology.
One somewhat unusual aspect of Garrett’s work on all this, and of the Scientific American article, is that his discussion of Lie groups puts their maximal torus front and center, as well as the fascinating diagrams you get labeling the weights of various representations under the action of these maximal tori. He has a wonderful fun toy to play with that displays these things, which he calls the Elementary Particle Explorer. I hear that t-shirts will soon be available…
Update: T-shirts are available here.