Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory

TL;DR

This paper analyzes the large-spin asymptotics of Euclidean spin foam amplitudes on 4d simplicial complexes, revealing geometric interpretations and deriving a Regge action with sign factors, thus connecting quantum amplitudes to classical geometry.

Contribution

It provides a detailed classification of critical configurations into geometric and degenerate types, and derives an asymptotic formula linking spin foam amplitudes to Regge and Palatini actions.

Findings

Critical configurations correspond to Euclidean or vector geometries.

The asymptotic amplitude sums over all geometric configurations.

A Regge action with sign factors emerges in the large-j limit.

Abstract

We study the large-j asymptotics of the Euclidean EPRL/FK spin foam amplitude on a 4d simplicial complex with arbitrary number of simplices. We show that for a critical configuration (j_f, g_{ve}, n_{ef}) in general, there exists a partition of the simplicial complex into three regions: Non-degenerate region, Type-A degenerate region and Type-B degenerate region. On both the non-degenerate and Type-A degenerate regions, the critical configuration implies a non-degenerate Euclidean geometry, while on the Type-B degenerate region, the critical configuration implies a vector geometry. Furthermore we can split the Non-degenerate and Type-A regions into sub-complexes according to the sign of Euclidean oriented 4-simplex volume. On each sub-complex, the spin foam amplitude at critical configuration gives a Regge action that contains a sign factor sgn(V_4(v)) of the oriented 4-simplices volume. Therefore the Regge action reproduced here can be viewed as a discretized Palatini action with on-shell connection. The asymptotic formula of the spin foam amplitude is given by a sum of the amplitudes evaluated at all possible critical configurations, which are the products of the amplitudes associated to different type of geometries.