Numerical evaluation of spin foam amplitudes beyond simplices

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 semiclassical 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 nonuniform vertex amplitudes, where some representations are small while others are large; we find indications that scenarios might exist in which the semiclassical amplitude is a valid approximation even if some spins remain small. This suggests that the transition of the quantum to the semiclassical regime (for a single vertex amplitude) is intricate.