The Lorentzian proper vertex amplitude: Asymptotics

TL;DR

This paper analyzes the semi-classical asymptotics of the Lorentzian proper vertex amplitude in quantum gravity spin-foam models, demonstrating the elimination of extra terms present in previous models through advanced stationary phase methods.

Contribution

It introduces new analytical tools to study the asymptotics of a modified spin-foam amplitude, showing the removal of extraneous terms in the semi-classical limit.

Findings

Asymptotics match a single Feynman term for non-degenerate Lorentzian 4-simplices

New stationary phase methods handle discontinuous action integrals

Extra projector term's influence vanishes in the asymptotic limit

Abstract

In previous work, the Lorentzian proper vertex amplitude for a spin-foam model of quantum gravity was derived. In the present work, the asymptotics of this amplitude are studied in the semi-classical limit. The starting point of the analysis is an expression for the amplitude as an action integral with action differing from that in the EPRL case by an extra `projector' term which scales linearly with spins only in the asymptotic limit, and is discontinuous on a submanifold of the integration domain. New tools are introduced to generalize stationary phase methods to this case. For the case of boundary data which can be glued to a non-degenerate Lorentzian 4-simplex, the asymptotic limit of the amplitude is shown to equal the single Feynman term, showing that the extra term in the asymptotics of the EPRL amplitude has been eliminated.