Path integral measure and triangulation independence in discrete gravity

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

3D holography: from discretum to continuum

“…As it is (at least formally) invariant under changes of triangulation [42][43][44][45], it also has a triangulation invariant measure built in. In the semiclassical limit it moreover coincides with the one found in [38,39]. The Ponzano-Regge model is the first example of a spin foam model.…”

“…To make it also a topological, and for 3D Regge calculus, triangulation invariant theory in the quantum realm, one needs to identify the correct path integral measure. Such a measure is not known for the original quantum Regge calculus, but has been identified for the linearized theory in [38,39].…”

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…However such a discrete dynamics for four-dimensional gravity, which is implemented (via the 4-simpex gluing) in a local way, breaks four-dimensional diffeomorphism invariance [100][101][102][103]. To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58].…”

On background-independent renormalization in spin foam models

“…Incorrect implementation of diffeomorphism symmetry on the discrete level can lead to the emergence of spurious degrees of freedom, as one would expect from a breaking of classical gauge symmetry in the quantum theory [30,31]. Still, from lower-dimensional models there are hints that at the IR fixed point diffeomorphism symmetry can emerge [32,33], and that one can even construct the discrete theories with full continuum diffeomorphism symmetry, leading to the quantization of discretizations of the correct continuum degrees of freedom [32,34,35], although this might be highly nontrivial in four space-time dimensions [36,37].…”