Semiclassical regime of Regge calculus and spin foams

TL;DR

This paper demonstrates that boundary states in quantum Regge calculus effectively peak interior geometries in the semiclassical regime, aligning with spin foam models and suppressing non-classical contributions.

Contribution

It shows that boundary states can successfully induce semiclassical interior geometries in quantum Regge calculus, extending previous single-simplex results to general triangulations.

Findings

Boundary states peak interior geometries in semiclassical Regge calculus.

Replacing exponential with cosine of the Regge action suppresses non-classical contributions.

Results match conventional Regge calculus in the semiclassical limit.

Abstract

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine -- as expected from the semiclassical limit of spin foam models -- then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.