Hessian and graviton propagator of the proper vertex

TL;DR

This paper computes the graviton propagator in the proper spin-foam model, confirming its consistency with linearized gravity and analyzing the semi-classical limit through Hessian matrix calculations.

Contribution

It introduces a method to compute the two-point function in the proper spin-foam vertex, extending previous work on the EPRL model and analyzing the semi-classical measure factor.

Findings

The two-point function matches the EPRL result in the continuum limit.

The semi-classical measure factor is consistent with the EPRL vertex.

The Hessian matrix is crucial for both propagator and measure calculations.

Abstract

The proper spin-foam vertex amplitude is obtained from the EPRL vertex by projecting out all but a single gravitational sector, in order to achieve correct semi-classical behavior. In this paper we calculate the gravitational two-point function predicted by the proper spin-foam vertex to lowest order in the vertex expansion. We find the same answer as in the EPRL case in the `continuum spectrum' limit, so that the theory is consistent with the predictions of linearized gravity in the regime of small curvature. The method for calculating the two-point function is similar to that used in prior works: we cast it in terms of an action integral and to use stationary phase methods. Thus, the calculation of the Hessian matrix plays a key role. Once the Hessian is calculated, it is used not only to calculate the two-point function, but also to calculate the coefficient appearing in the semi-classical limit of the proper vertex amplitude itself. This coefficient is the effective discrete "measure factor" encoded in the spin-foam model. Through a non-trivial cancellation of different factors, we find that this coefficient is the same as the coefficient in front of the term in the asymptotics of the EPRL vertex corresponding to the selected gravitational sector.