# Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude

## TL;DR

This paper presents the first numerical evaluations of the Lorentzian EPRL spin foam amplitude, confirming asymptotic behaviors and oscillations predicted by theoretical models through a novel decomposition method.

## Contribution

The authors introduce a new numerical method based on Clebsch-Gordan coefficient decomposition for evaluating the Lorentzian EPRL vertex amplitude in loop quantum gravity.

## Key findings

- Numerical results support the saddle point approximation asymptotics.
- Power-law decay and oscillations consistent with the Regge action are observed.
- Qualitative matches achieved with simplified models using only the first decomposition terms.

## Abstract

The Lorentzian EPRL spin foam amplitude for loop quantum gravity is a multi-dimensional non-compact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power-law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power-law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used.