A Canonical Approach to the Einstein-Hilbert Action in Two Spacetime Dimensions

The canonical structure of the Einstein-Hilbert Lagrange density $L=\sqrt{-g}R$ is examined in two spacetime dimensions, using the metric density $h^{\mu \nu }\equiv \sqrt{-g}{g}^{\mu \nu }$ and symmetric affine connection ${\mathrm{\Gamma }}_{\sigma \beta }^{\lambda }$ as dynamical variables. The Hamiltonian reduces to a linear combination of three first class constraints with a local SO(2,1) algebra. The first class constraints are used to find a generator of gauge transformations that has a closed off-shell algebra and which leaves the Lagrangian and $\det (h^{\mu \nu })$ invariant. These transformations are distinct from diffeomorphism invariance, and are gauge transformations characterized by a symmetric matrix ${\zeta }_{\mu \nu }$.