Asymptotic of Lorentzian Polyhedra Propagator

TL;DR

This paper analyzes the asymptotic behavior of a key Lorentzian operator in spin-foam models, showing its leading order is proportional to the identity and demonstrating the convergence of related amplitudes.

Contribution

It provides the large $j$ asymptotic analysis of the Lorentzian Polyhedra Propagator and develops technical tools for Lorentzian Spin-Foam calculations.

Findings

Leading order of the operator is proportional to the identity.

The sum of amplitudes with increasing vertices converges.

Tools for Lorentzian Spin-Foam calculations are introduced.

Abstract

A certain operator $\T=\int_{\SL}\dd g\, Y^{\dagger}gY$ can be found in various Lorentzian EPRL calculations. The properties of this operator has been studied here in large $j$ limit. The leading order of $\T$ is proportional to the identity operator. Knowing the operator $\T$ one can renormalize spin-foam's edge self-energy by computing the amplitude of sum of a series of edges with increasing number of vertices and bubbles. This amplitude is calculated and is shown to be convergent. Moreover some technical tools useful in Lorentzian Spin-Foam calculation has been developed.