Path integral measure and triangulation independence in discrete gravity

TL;DR

This paper develops a triangulation-independent measure for 3D linearized Regge calculus, linking it to spin foam models, and explores the possibility of similar measures in 4D for quantum gravity.

Contribution

It introduces a local measure for 3D Regge calculus that ensures triangulation independence and relates it to the Ponzano Regge Model, while discussing extensions to 4D.

Findings

A simple, factorized structure of the linearized Regge action under Pachner moves.

A local measure for 3D Regge calculus matching Ponzano Regge asymptotics.

Discussion on the feasibility of triangulation-independent measures in 4D.

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.