Several people have been posting, here and more broadly, about Garrett Lisi’s new candidate unified theory of fields and gravity. (Even slashdot seems to have picked it up) This got encouraged by Lee Smolin, of loop quantum gravity fame, getting publicly excited about it, and it makes for great news because Lisi isn’t currently a practicing physicist — he’s currently a surf bum with a PhD. The biggest problem is that this paper is wrong in some rather key ways.
He starts with the realization that you can express the equations of (classical) general relativity as a gauge theory of SO(3, 1), which is both true and very important, and then tries to do the natural operation of unifying this SO(3, 1) gauge group with various other groups to form a nice unified theory. He ends up with something he calls E8, which seems awfully nice from a mathematical perspective, and is very elegant. There are only two problems.
- His group isn’t E8. Since SO(3,1) is noncompact, it should be pretty obvious that it can’t embed in a compact Lie group, and it doesn’t. He ends up with something that looks sort of like a noncompact cousin of E8… except that all of the Dynkin diagram–based classification that he uses for his calculations doesn’t actually work properly for noncompact groups. (The basic theorem that Dynkin diagrams can describe Lie algebras is very dependent on compactness — and to see why, if you work out the Dynkin diagram for the Virasoro algebra, it looks the same as the diagram for SU(2). Despite the Virasoro algebra being infinite-dimensional and SU(2) being only 3-dimensional.)
- But assuming that this is fixable, his group isn’t really E8 but some other group, and everything else with his group theory is OK (I didn’t sit down to check this)… gauge theories of noncompact gauge groups aren’t renormalizable. Not even slightly. This E8 unification is pretty and all from a classical perspective, but if you try to quantize it everything diverges. (After all, if you could do that to a noncompact group, you could do it to SO(3, 1) as well and write a working theory of quantum gravity in ten minutes)
Anyway, that was really technical and is mostly for the reference of any physicists who still read this. The non-technical version is that it makes for a great news story and all, but this is the sort of idea that most high-energy physicists come up with sometime during grad school, think about for a few minutes, and then realize why it doesn’t work.
What’s more amusing is watching Lee Smolin go off and praise it, just because it’s a non-string-theory theory of quantum gravity. 🙂