Semi-Classical Limits of Simplicial Quantum Gravity

TL;DR

This paper explores the semi-classical limits of the Ponzano-Regge model for 3D quantum gravity, interpreting asymptotic formulas in terms of Lorentzian and Euclidean geometries, and analyzing stationary points of the state-sum.

Contribution

It extends the geometric interpretation of the Ponzano-Regge model to Lorentzian geometry and provides insights into the semi-classical behavior of simplicial quantum gravity.

Findings

Asymptotic formulas relate 6j-symbols to Lorentzian and Euclidean geometries.

Stationary points correspond to locally flat Lorentzian or Euclidean manifolds.

The model supports solutions for both Lorentzian and Euclidean 3D gravity.

Abstract

We consider the simplicial state-sum model of Ponzano and Regge as a path integral for quantum gravity in three dimensions. We examine the Lorentzian geometry of a single 3-simplex and of a simplicial manifold, and interpret an asymptotic formula for $6j$-symbols in terms of this geometry. This extends Ponzano and Regge's similar interpretation for Euclidean geometry. We give a geometric interpretation of the stationary points of this state-sum, by showing that, at these points, the simplicial manifold may be mapped locally into flat Lorentzian or Euclidean space. This lends weight to the interpretation of the state-sum as a path integral, which has solutions corresponding to both Lorentzian and Euclidean gravity in three dimensions.