Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory

Abstract

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 and , 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 implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted to is degenerate of type-A in our definition of degeneracy, but it implies a nondegenerate discrete Euclidean geometry in , (3) the critical configuration restricted to is degenerate of type-B, and it implies a vector geometry in . With the critical configuration, we further make a subdivision of the regions and into sub-complexes (with boundaries) according to their Lorentzian/Euclidean oriented 4-volume V4(v) of the 4-simplices, such that sgn(V4(v)) is a constant sign on each sub-complex. Then in each sub-complex or , 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 Regge 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.