From covariant to canonical formulations of discrete gravity

TL;DR

This work develops a canonical framework for discretized gravity based on tent moves, reproducing covariant Regge dynamics and exposing how gauge symmetries manifest in the linearized, flat-background regime through Abelian first-class constraints. It shows that while linearized Regge calculus preserves exact gauge invariances with a clear geometric interpretation, higher-order dynamics introduce symmetry breaking and yield background-gauge dependent pseudo constraints. The analysis provides explicit constraint forms for four- and higher-valent vertices, proves an Abelian constraint algebra, and demonstrates the construction of observables that isolate gravitons from gauge modes. These results illuminate the link between path integral and canonical quantizations in discrete gravity and offer a pathway toward anomaly-free quantization or discovery of perfect actions that retain continuum symmetries. The findings have implications for connecting spin-foam models with canonical loop quantum gravity and for understanding perturbative expansions around discrete backgrounds.

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.

Figures (5)

• Figure 1: The tent move in 3d applied to a vertex $v_n$ in a 2d Cauchy hypersurface.
• Figure 2: The local tent move evolution of a vertex $v_n$ in 3d.
• Figure 3: Construction of the 3d three-valent analogue of the 4d four-valent tent move where both tetrahedra are oriented in the same (future) direction.
• Figure 4: Illustration of the symmetry--reduced 3d $\text{star}(v)$ of a five--valent vertex $v$, consisting of the six tetrahedra $\tau(v124), \ \tau(v134), \ \tau(v234), \ \tau(v125), \ \tau(v135)$ and $\tau(v235)$.
• Figure 5: Schematic illustration of the alternating tent moves at neighbouring vertices $v$ and $v'$, starting from the Cauchy hypersurface $\Sigma_{n-1}$.