Path integral measure and triangulation independence in discrete gravity

TL;DR

<3-5 sentence high-level summary>The paper investigates how to define a path integral measure for gravity on a discrete Regge calculus background that preserves (as much as possible) diffeomorphism symmetry, by demanding triangulation independence under Pachner moves. Focusing on linearized Regge calculus, it derives the Hessian structure for 3D and 4D Pachner moves, identifies gauge (null) modes, and constructs a local measure μ(l) that yields triangulation independence in 3D, matching the asymptotics of the Ponzano–Regge model. In 4D, invariance under 4-2 and 5-1 moves can be achieved up to a nonlocal factor D, while the 3-3 move generally breaks invariance; nevertheless the Hessian retains a factorized form that clarifies how conformal-mode issues arise and how gauge degrees of freedom can be handled. These results illuminate how to relate discrete gravity path integrals to spin foam amplitudes and point toward routes (e.g., perfect discretizations or BF-based formulations) to achieve broader triangulation independence in higher dimensions.

Abstract

A path integral measure for gravity should also preserve the fundamental symmetry of general relativity, which is diffeomorphism symmetry. In previous work, we argued that a successful implementation of this symmetry into discrete quantum gravity models would imply discretization independence. We therefore consider the requirement of triangulation independence for the measure in (linearized) Regge calculus, which is a discrete model for quantum gravity, appearing in the semi--classical limit of spin foam models. To this end we develop a technique to evaluate the linearized Regge action associated to Pachner moves in 3D and 4D and show that it has a simple, factorized structure. We succeed in finding a local measure for 3D (linearized) Regge calculus that leads to triangulation independence. This measure factor coincides with the asymptotics of the Ponzano Regge Model, a 3D spin foam model for gravity. We furthermore discuss to which extent one can find a triangulation independent measure for 4D Regge calculus and how such a measure would be related to a quantum model for 4D flat space. To this end, we also determine the dependence of classical Regge calculus on the choice of triangulation in 3D and 4D.

Figures (7)

• Figure 1: $3 - 2$ move. The two tetrahedra can be split into three by connecting the two vertices separated by the shared triangle. The dashed line in the three tetrahedra configuration is the dynamical edge.
• Figure 2: $4 - 1$ move. The tetrahedron is split into four by placing one additional vertex inside the tetrahedron and connecting it to the remaining vertices in the boundary giving four internal edges (dashed).
• Figure 3: $4 - 2$ move. By connecting the vertices $(0)$ and $(1)$ the two 4-simplices are split into four with one bulk edge, here drawn dashed.
• Figure 4: $5 - 1$ move. The 4-simplex is split into five 4-simplices by placing one vertex inside the 4-simplex and connecting it to the boundary vertices, hence obtaining five bulk edges (dashed lines).
• Figure 5: $3 - 3$ move. Three 4--simplices sharing the triangle $(0 1 2)$ and not containing $(3 4 5)$ are rebuilt into three 4--simplices sharing the triangle $(3 4 5)$ and not including triangle $(0 1 2)$. The shared triangles are drawn dashed in this figure. Note that all edges are boundary edges and are contained in both configurations, so the configurations are determined by the shared triangle.