Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory

This paper studies the large-j asymptotics of the Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitude on a 4D simplicial complex with an arbitrary number of simplices. The asymptotics of the spinfoam amplitude is determined by the critical configurations. Here we show that, given a critical configuration in general, there exists a partition of the simplicial complex into three types of regions R-Nondeg, RDeg-A and RDeg-B, where the three regions are simplicial sub-complexes with boundaries. The critical configuration implies different types of geometries in different types of regions, i.e. (1) the critical configuration restricted to R-Nondeg implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted to RDeg-A is degenerate of type-A in our definition of degeneracy, but it implies a nondegenerate discrete Euclidean geometry in RDeg-A, (3) the critical configuration restricted to RDeg-B is degenerate of type-B, and it implies a vector geometry in RDeg-B. With the critical configuration, we further make a subdivision of the regions RNondeg and RDeg-A into sub-complexes (with boundaries) according to their Lorentzian/Euclidean oriented 4-volume V-4(v) of the 4-simplices, such that sgn(V-4(v)) is a constant sign on each sub-complex. Then in each sub-complex R-Nondeg orR(Deg-A), the spinfoam amplitude at the critical configuration gives the Regge action in a Lorentzian signature or an Euclidean signature respectively. The Regge action reproduced here contains a sign prefactor sgn(V4(v)) related to the oriented 4-volume of the 4-simplices. Therefore the R-egge action reproduced here can be viewed as a discretized Palatini action with an on-shell connection. Finally, the asymptotic formula of the spinfoam 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 types of geometries.