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.


2 Answers 2


In the particle-physics-oriented part of the theoretical physics community, it was becoming increasingly clear that the Dirac bracket is at most a complicated piece of formalism that isn't able to solve any real physical problems and make theories well-defined or finite or renormalizable etc.

So the people who are playing with such tools applied to quantized gravity are still close to the loop quantum gravity community. The Dirac bracket quantization for the Ashtekar-Barbero form of gravity has been attempted in papers by Sergei Alexandrov, e.g.


The paper has 77 respectable followups but I think that none of them really uses the results in any meaningful way.


D.G.C. McKeon, The Canonical Structure of the First Order Einstein-Hilbert Action, arXiv:1005.3001, Journal-ref: Int.J.Mod.Phys.A25:3453-3480,2010.


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