Partial absence of cosine problem in 3d Lorentzian spin foams

TL;DR

This paper investigates the semi-classical limit of a (2+1) Lorentzian spin foam model, revealing that the cosine problem is partially absent depending on the spacetime signature of the triangles.

Contribution

It demonstrates that the cosine problem in spin foam models is mitigated when triangles are uniformly spacelike or timelike, leading to a single exponential instead of a cosine.

Findings

Two solutions for uniform spacelike or timelike triangles

Single solution (Regge exponential) in mixed signature cases

Nuanced behavior of stationary phase solutions in Lorentzian quantum gravity

Abstract

We study the semi-classical limit of the recently proposed coherent spin foam model for (2+1) Lorentzian quantum gravity. Specifically, we analyze the gluing equations derived from the stationary phase approximation of the vertex amplitude. Typically these exhibit two solutions yielding a cosine of the Regge action. However, by inspection of the algebraic equations as well as their geometrical realization, we show in this note that the behavior is more nuanced: when all triangles are either spacelike or timelike, two solutions exist. In any other case, only a single solution is obtained, thus yielding a single Regge exponential.