Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory

Abstract

The present paper studies the large-j asymptotics of the Lorentzian 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 type of regions R Nondeg , R Deg-A , R Deg-B , where the three regions are simplicial subcomplexes with boundaries. The critical configuration implies different types of geometries in different types of regions, i.e. (1) the critical configuration restricted into R Nondeg implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted into R Deg-A is degenerate of type-A in our definition of degeneracy, but implies a nondegenerate discrete Euclidean geometry in R Deg-A , (3) the critical configuration restricted into R Deg-B is degenerate of type-B, and implies a vector geometry in R Deg-B .With the critical configuration, we further make a subdivision of the regions R Nondeg and R Deg-A into subcomplexes (with boundary) 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 the each sub-complex, the spinfoam amplitude at the critical configuration gives the Regge action in Lorentzian or Euclidean signature respectively in R Nondeg or R Deg-A . The Regge action reproduced here contains a sign prefactor sgn(V 4 (v)) related to the oriented 4-volume of the 4-simplices. Therefore the Regge action reproduced here can be viewed a discretized Palatini action with 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 to different type of geometries.

Semiclassical analysis of spinfoam model with a small Barbero-Immirzi parameter

“…It turns out (in the analysis of the next section) that the "potential" V f g ve , z v f gives exactly the same set of critical points as the one given by the full spinfoam action S, classified in [8,9] 2 . At the critical spinfoam configurations corresponding to nondegenerate simplical geometries, the potential V f g ve , z v f vanishes at the "time-oriented" configurations (defined in [9]), and equals π otherwise, while the critical value of ∑ f j f K f g ve , z v f gives the Regge action evaluated at the corresponding nondegenerate geometry, where the critical value of K f g ve , z v f is the deficit angle at f .…”

“…The semiclassical analysis is carried out by taking into account the sum over spins in the regime where all the spins are uniformly large. Such an analysis is a natural continuation of the previous studies of large spin asymptotics [6][7][8][9], which don't take into account the sum over spins.…”

“…Because the critical equations from the potential V f is identical to the critical equations from the spinfoam action S, the geometrical interpretations of the critical configurations follows in the same way as it was developed in [8,9]. The results is summarized in the following (see [8,9] for details):…”

“…When the critical configuration satisfies the nondegeneracy condition Eq. (2.12), the spinfoam loop holonomy G f (v) along the boundary of the dual face f can be computed at the critical configuration [8] (g ev = g −1 ve ):…”

“…(2.12), the analysis in [8] shows that the critical value of K f [g ve , z v f ] vanishes identically and the critical value…”