Canonical simplicial gravity

TL;DR

This work introduces a general canonical framework for discrete gravity with evolving phase-space dimensions, built on Hamilton's principal function $\tilde{S}$ as a generating function for time evolution on constraint surfaces. It applies the framework to Regge calculus by encoding space-time evolution through Pachner moves (gluing/removing single simplices), implementing local, multi-fingered time while preserving additivity and covariant dynamics via momentum matching and extended phase spaces. The construction yields a fully consistent canonical Regge calculus in both 3D (flat vacua, gauge freedoms tied to vertex displacements) and 4D (curved solutions with nontrivial constraints and potential pseudo-constraints), with a detailed account of how each Pachner move affects edge lengths, momenta, and constraints. The formalism offers a route to quantization and numerical implementations, and it provides a bridge between covariant spin-foam/CDT approaches and canonical loop-quantum-gravity frameworks, while highlighting how discretization breaks diffeomorphism symmetry and how gauge structure emerges in the flat limit. Overall, the paper provides a principled, versatile method to evolve discrete geometries canonically and to analyze the role of constraints in discrete gravity models.

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.

Figures (14)

• Figure 1: Schematic illustration of a fat slicing of a triangulation. The triangulation can be built up step by step by single simplices where we count such elementary steps by $k \in {\mathbbm Z}$. The elementary steps can be grouped up into fat slices which we now count by $n \in {\mathbbm Z}$.
• Figure 2: Hypersurface $\Sigma_k$ separating 'past' and 'future' region at step $k$.
• Figure 3: 3D example: gluing a single tetrahedron onto a single triangle in the 2D boundary hypersurface of a 3D bulk triangulation. From the perspective of the 2D hypersurface this gluing move appears as a subdivision of the triangle $t$. That is, the move appears as a 1--3 Pachner move in the hypersurface. This is a specific example of the situation described earlier where the numbers of variables associated to the two hypersurfaces before and after the new move differ due to the three new edges.
• Figure 4: Removal moves for the 'past triangulation' are equivalent to gluing moves for the 'future triangulation' and vice versa.
• Figure 5: a) The 1--3 gluing Pachner move, b) The 1--3 removal Pachner move. The dashed edges are the three new edges.

Theorems & Definitions (3)

• Definition 6.1
• Theorem 6.1
• proof