For the last couple years I’ve been working on the idea of using what mathematicians call “Dirac Cohomology” to replace the standard BRST formalism for handling gauge symmetries. So far this is just in a toy model: gauge theory in 0+1 dimensions, with a finite dimensional Hilbert space. Over the last few months I’ve finally got this to the point where I think I understand completely how this should work, at least for this toy model. I talked about this last week in St. Petersburg, and have a preliminary version of a paper on the subject, which is available here. Next weekend I’m leaving for a trip to Shanghai and Hong Kong (the plan is to be in Shanghai for the July 22 total solar eclipse, which will be visible there). After I get back at the beginning of August I’ll work on the paper a bit more, hoping to have a final version done by the beginning of September, when the academic year starts.
The paper uses quite a lot of mathematical technology, so I fear most people will find it hard to read. This fall I hope to get back to finishing the Notes on BRST I was writing up, the idea behind those was to give a more expository account of this subject. That project got bogged down when I realized there was something I was still confused about, and after getting unconfused it seemed like a good idea to get the basic ideas down on paper, since the expository project might take a while to complete.
First of all, what this is and what it isn’t. It’s a toy quantum mechanical model, with gauge symmetry treated using some new ideas from representation theory which are related to BRST, but different. It’s not a QFT, and not a treatment of gauge symmetry in the physical case of four space-time dimensions. I’ve been thinking about how to extend this to higher dimensions, but this requires some new ideas. Next on the agenda is to try and get something that works in 1+1 dimensions, where one can exploit a lot that is known about affine lie algebras and coset models. There also appear to be interesting possible connections to geometric Langlands in that case.
Given a quantum system with G-symmetry, the BRST method allows one to gauge a subgroup H, picking out the H-invariant subspace of the original Hilbert space using Lie algebra cohomology methods. The proposal here is to do something different, picking out a subgroup H of symmetries one wants to keep, and gauging the rest. In the special case where Lie G/Lie H is the sum of a Lie subalgebra and its conjugate, the method proposed here reduces to the standard BRST method, but it is more general.
An algebraic version of the Dirac operator plays a role here somewhat like that of the BRST operator in the standard formalism. One difference is that the square of this operator is not zero. However, it is in the center of the algebra of operators acting on the Hilbert space, so its action on operators squares to zero. This sort of thing has been studied a bit before in the physics literature, in the context of supersymmetric quantum mechanics models, but I do believe that the interpretation here as a method for handling gauge symmetry is new.
One thing I want to add to the paper is some comments about the relation to the physical Dirac operator. The point of view on the Dirac operator explained here that comes out of representation theory seems to me perhaps the most intriguing part of this story. Remarkably, this Dirac operator is in some sense a quantization of the Chern-Simons form. The full story of how to use this in higher dimensions remains obscure to me, but there is some hope it will bring together the physical Dirac operator, something like BRST, and something like supersymmetry in a new way.
