From covariant to canonical formulations of discrete gravity

Abstract

Starting from an action for discretized gravity, we derive a canonical formalism that exactly reproduces the dynamics and (broken) symmetries of the covariant formalism. For linearized Regge calculus on a flat background—which exhibits exact gauge symmetries—we derive local and first-class constraints for arbitrary triangulated Cauchy surfaces. These constraints have a clear geometric interpretation and are a first step towards obtaining anomaly-free constraint algebras for canonical lattice gravity. Taking higher order dynamics into account, the symmetries of the action are broken. This results in consistency conditions on the background gauge parameters arising from the lowest nonlinear equations of motion. In the canonical framework, the constraints to quadratic order turn out to depend on the background gauge parameters and are therefore pseudo constraints. These considerations are important for connecting the path integral and canonical quantizations of gravity, in particular if one attempts a perturbative expansion.