Numerical evaluation of spin foam amplitudes beyond simplices

TL;DR

This paper numerically computes 4D Euclidean spin foam vertex amplitudes with hypercubic combinatorics, compares them to semi-classical approximations, and explores the transition from quantum to semi-classical regimes.

Contribution

First numerical calculation of hypercubic spin foam vertex amplitudes with boundary data, including algorithms and analysis of semi-classical approximation validity.

Findings

Qualitative agreement between full and semi-classical amplitudes at small spins

Oscillation frequency differences and phase shifts observed compared to 4-simplex case

Indications that semi-classical approximation may hold even with some small spins

Abstract

We present the first numerical calculation of the 4D Euclidean spin foam vertex amplitude for vertices with hypercubic combinatorics. Concretely, we compute the amplitude for coherent boundary data peaked on cuboid and frustum shapes. We present the numerical algorithms to explicitly compute the vertex amplitude and compare the results in different cases to the semi-classical approximation of the amplitude. Overall we find good qualitative agreement of the amplitudes and evidence of convergence of the asymptotic formula to the full amplitude already at fairly small spins, yet also differences in the frequency of oscillations and a phase shift absent in the 4-simplex case. However, due to rapidly growing numerical costs, we cannot reach sufficiently high spins to prove agreement of both amplitudes. Lastly, this setup allows us to explore non-uniform vertex amplitudes, where some representations are small while others are large; we find indications that scenarios might exist in which the semi-classical amplitude is a valid approximation even if some spins remain small. This suggests that the transition of the quantum to the semi-classical regime (for a single vertex amplitude) is intricate.