Second-order amplitudes in loop quantum gravity

TL;DR

This paper investigates second-order amplitudes in loop quantum gravity, revealing peculiar large-distance behaviors and potential issues with the Barrett-Crane vertex related to flatness versus Ricci-flatness.

Contribution

It computes second-order contributions to the graviton propagator using the old Barrett-Crane vertex and analyzes their geometric and large-distance properties.

Findings

Second-order amplitudes show peculiar large-distance behavior.

Old Barrett-Crane vertex may lead to flatness instead of Ricci-flatness.

Potential issues with the vertex's ability to reproduce expected geometries.

Abstract

We explore some second-order amplitudes in loop quantum gravity. In particular, we compute some second-order contributions to diagonal components of the graviton propagator in the large distance limit, using the old version of the Barrett-Crane vertex amplitude. We illustrate the geometry associated to these terms. We find some peculiar phenomena in the large distance behavior of these amplitudes, related with the geometry of the generalized triangulations dual to the Feynman graphs of the corresponding group field theory. In particular, we point out a possible further difficulty with the old Barrett-Crane vertex: it appears to lead to flatness instead of Ricci-flatness, at least in some situations. The observation raises the question whether this difficulty remains with the new version of the vertex.