Lorentzian quantum gravity via Pachner moves: one-loop evaluation

TL;DR

This paper investigates Lorentzian quantum gravity using Regge calculus and Pachner moves, demonstrating how to handle the indefinite action and construct a measure in 3D, advancing towards a non-perturbative Lorentzian path integral.

Contribution

It introduces a discretization-invariant measure for 3D Lorentzian Regge calculus and analyzes the role of Pachner moves in addressing the indefinite action problem.

Findings

Pachner moves isolate the indefinite nature of the Lorentzian gravitational action.

A discretization-invariant measure is constructed for 3D Lorentzian Regge calculus.

Obstructions are identified in defining such a measure in 4D.

Abstract

Lorentzian quantum gravity is believed to cure the pathologies encountered in Euclidean quantum gravity, such as the conformal factor problem. We show that this is the case for the Lorentzian Regge path integral expanded around a flat background. We illustrate how a subset of local changes of the triangulation, so-called Pachner moves, allow to isolate the indefinite nature of the gravitational action at the discrete level. The latter can be accounted for by oppositely chosen deformed contours of integration. Moreover, we construct a discretization-invariant local path integral measure for 3D Lorentzian Regge calculus and point out obstructions in defining such a measure in 4D. We see the work presented here as a first step towards establishing the existence of the non-perturbative Lorentzian path integral for Regge calculus and related frameworks such as spin foams. An extensive appendix provides an overview of Lorentzian Regge calculus, using the recently introduced concept of the complexified Regge action, and derives useful geometric formulae and identities needed in the main text.