A Canonical Analysis of the Einstein-Hilbert Action in First Order Form

Abstract

Using the Dirac constraint formalism, we examine the canonical structure of the Einstein-Hilbert action $S_d = \frac{1}{16\pi G} \int d^dx \sqrt{-g} R$, treating the metric $g_{\alpha\beta}$ and the symmetric affine connection $\Gamma_{\mu\nu}^\lambda$ as independent variables. For $d > 2$ tertiary constraints naturally arise; if these are all first class, there are $d(d-3)$ independent variables in phase space, the same number that a symmetric tensor gauge field $\phi_{\mu\nu}$ possesses. If $d = 2$, the Hamiltonian becomes a linear combination of first class constraints obeying an SO(2,1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when $d = 2$; it is characterized by a symmetric matrix $\xi_{\mu\nu}$. We also show that the canonical analysis is different if $h^{\alpha\beta} = \sqrt{-g} g^{\alpha\beta}$ is used in place of $g^{\alpha\beta}$ as a dynamical variable when $d = 2$, as in $d$ dimensions, $\det h^{\alpha\beta} = - (\sqrt{-g})^{d-2}$. A comparison with the formalism used in the ADM analysis of the Einstein-Hilbert action in first order form is made by applying this approach in the two dimensional case with $h^{\alpha\beta}$ and $\Gamma_{\mu\nu}^\lambda$ taken to be independent variables.