Canonical simplicial gravity

TL;DR

This paper develops a canonical formalism for discrete systems like simplicial gravity, enabling consistent evolution and connection between covariant and canonical approaches, with potential applications in quantum gravity and numerical methods.

Contribution

It introduces a general canonical formalism for discrete systems that handles varying phase space dimensions and constraints, applied specifically to simplicial gravity and Regge calculus.

Findings

Canonical formalism reproduces covariant dynamics from the action.

Discrete evolution via Pachner moves is realized as canonical transformations.

Provides a consistent framework connecting covariant and canonical simplicial gravity.

Abstract

A general canonical formalism for discrete systems is developed which can handle varying phase space dimensions and constraints. The central ingredient is Hamilton's principal function which generates canonical time evolution and ensures that the canonical formalism reproduces the dynamics of the covariant formulation following directly from the action. We apply this formalism to simplicial gravity and (Euclidean) Regge calculus, in particular. A discrete forward/backward evolution is realized by gluing/removing single simplices step by step to/from a bulk triangulation and amounts to Pachner moves in the triangulated hypersurfaces. As a result, the hypersurfaces evolve in a discrete `multi-fingered' time through the full Regge solution. Pachner moves are an elementary and ergodic class of homeomorphisms and generically change the number of variables, but can be implemented as canonical transformations on naturally extended phase spaces. Some moves introduce a priori free data which, however, may become fixed a posteriori by constraints arising in subsequent moves. The end result is a general and fully consistent formulation of canonical Regge calculus, thereby removing a longstanding obstacle in connecting covariant simplicial gravity models to canonical frameworks. The present scheme is, therefore, interesting in view of many approaches to quantum gravity, but may also prove useful for numerical implementations.