From covariant to canonical formulations of discrete gravity

TL;DR

This paper develops a canonical formalism for discretized gravity that reproduces covariant dynamics, explores gauge symmetries in linearized Regge calculus, and discusses the breaking of symmetries at higher orders affecting quantum gravity approaches.

Contribution

It introduces a canonical formulation for discrete gravity that captures covariant dynamics and symmetries, and analyzes the impact of non-linearities on constraints and gauge parameters.

Findings

Derived exact constraints for linearized Regge calculus on flat backgrounds.

Identified gauge symmetry breaking at higher orders and resulting pseudo constraints.

Provided a framework connecting path integral and canonical quantizations of 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 non--linear 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 path integral and canonical quantizations of gravity, in particular if one attempts a perturbative expansion.