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

TL;DR

This paper analyzes the canonical structure of the Einstein-Hilbert action in two dimensions, revealing a new gauge symmetry distinct from diffeomorphisms, with implications for understanding gravitational theories in lower dimensions.

Contribution

It introduces a novel gauge transformation characterized by a symmetric matrix, derived from the Hamiltonian formulation using metric density and affine connection as variables.

Findings

Hamiltonian reduces to three first class constraints with SO(2,1) algebra.

Identifies a gauge transformation distinct from diffeomorphisms.

Provides a generator with a closed off-shell algebra that preserves the Lagrangian.

Abstract

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 $\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 $\zeta_{\mu \nu}$.