Asymptotics of the spin foam amplitude on simplicial manifold: Euclidean theory

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 {jf, gve, nef} 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 the critical configuration gives a Regge action that contains a sign factor sgn (V4(v)) of the oriented 4-simplex volume. Therefore the Regge action reproduced here can be viewed as a discretized Palatini action with the 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 with different type of geometries.