In the first order formulation of general relativity, the frame field $e_{\mu}^a$ and $\mathrm{SO}(3,1)$ spin connection $\omega_{\mu c}^b$ are independent variables. In the Hamiltonian formulation of this theory, one finds that there are second-class constraints.

According to Dirac, the way to deal with these second-class constraints when quantising is to first define the Dirac bracket, which is essentially a new Poisson bracket that 'respects the constraints', in the sense that the Dirac bracket of any two constraints is another constraint, and then proceed with the quantisation procedure.

After looking a little bit in the literature, I have been unable to find any paper that actually attempts to construct the Dirac bracket for the first-order formulation of general relativity. And indeed it seems people go to lengths to reformulate gravity so that it doesn't have any second class constraints from the get-go (e.g. using the Ashtekar variables). My question is, has the Dirac bracket for first-order gravity been constructed? If so, a reference would be great.

This post imported from StackExchange Physics at 2014-08-12 09:33 (UCT), posted by SE-user Steven