Asymptotic analysis of spin foam amplitude with timelike triangles

TL;DR

This paper analyzes the large spin asymptotics of a 4D Lorentzian spin foam model with timelike elements, showing that dominant contributions correspond to classical Lorentzian geometries with phases matching the Regge action.

Contribution

It provides a detailed asymptotic analysis of the extended spin foam model including timelike tetrahedra, identifying the dominant Lorentzian geometries and their relation to Regge calculus.

Findings

Critical configurations correspond to Lorentzian simplicial geometries.

Amplitude phases match the Regge action of gravity.

Split signature and degenerate configurations are suppressed in certain cases.

Abstract

The large $j$ asymptotic behavior of $4$-dimensional spin foam amplitude is investigated for the extended spin foam model (Conrady-Hnybida extension) on a simplicial complex. We study the most general situation in which timelike tetrahedra with timelike triangles are taken into account. The large $j$ asymptotic behavior is determined by critical configurations of the amplitude. We identify the critical configurations that correspond to the Lorentzian simplicial geometries with timelike tetrahedra and triangles. Their contributions to the amplitude are phases asymptotically, whose exponents equal to Regge action of gravity. The amplitude may also contains critical configurations corresponding to non-degenerate split signature $4$-simplices and degenerate vector geometries. But for vertex amplitudes containing at least one timelike tetrahedron and one spacelike tetrahedron, critical configurations only give Lorentzian $4$-simplices, while the split signature and degenerate $4$-simplices do not appear.