Influence of the Measure on Simplicial Quantum Gravity in Four Dimensions

TL;DR

This paper studies how different measures affect the phase structure of four-dimensional simplicial quantum gravity using Regge calculus, revealing measure-dependent inhomogeneities and curvature behaviors.

Contribution

It introduces a scale fixing method and compares scale invariant and uniform measures, highlighting their impact on quantum gravity phases in four dimensions.

Findings

Low $eta$ phase has negative curvature and homogeneous link lengths.

High $eta$ phase shows inhomogeneous link lengths with spikes.

Curvature depends on the measure used.

Abstract

We investigate the influence of the measure in the path integral for Euclidean quantum gravity in four dimensions within the Regge calculus. The action is bounded without additional terms by fixing the average lattice spacing. We set the length scale by a parameter $\beta$ and consider a scale invariant and a uniform measure. In the low $\beta$ region we observe a phase with negative curvature and a homogeneous distribution of the link lengths independent of the measure. The large $\beta$ region is characterized by inhomogeneous link lengths distributions with spikes and positive curvature depending on the measure.