Searching for classical geometries in spin foam amplitudes: a numerical method

TL;DR

This paper introduces a numerical approach to identify classical geometries within spin foam amplitudes, focusing on the semiclassical limit and stationary phase points, with applications to 3D and potential 4D models.

Contribution

It presents a novel numerical method for analyzing the semiclassical limit of spin foam models, including techniques to locate classical geometries and extend to models with local curvature.

Findings

Identified stationary phase points corresponding to classical geometries in the Ponzano-Regge model.

Demonstrated the method on a three-vertex transition amplitude.

Discussed potential generalizations to four-dimensional EPRL models.

Abstract

We develop a numerical method to investigate the semiclassical limit of spin foam amplitudes with many vertices. We test it using the Ponzano-Regge model, a spin foam model for three-dimensional euclidean gravity, and a transition amplitude with three vertices. We study the summation over bulk spins, and we identify the stationary phase points that dominate it and that correspond to classical geometries. We complement with the numerical analysis of a four vertex transition amplitude and with a modification of the model that includes local curvature. We discuss the generalization of our results to the four-dimensional EPRL spin foam model, and we provide suggestions for new computations.