Bubble divergences and gauge symmetries in spin foams

TL;DR

This paper investigates divergences and gauge symmetries in spin foam models for quantum gravity, analyzing the Barrett-Crane model, estimating divergence degrees, and exploring broken gauge symmetries related to flat solutions.

Contribution

It introduces a simple method to estimate divergence degrees in spin foam bubbles and identifies gauge symmetries at special solutions, linking divergences to symmetry breaking.

Findings

Barrett-Crane model shows no divergences with delta function weights

A method to estimate divergence degrees of bubbles is proposed

Broken gauge symmetries are identified at special solutions, similar to flat spacetime in discrete gravity

Abstract

Spin foams are candidate state-sum models for transition amplitudes in quantum gravity. An active research subject is to identify the possible divergences of spin foam models, or alternatively to show that models are finite. We will discuss in detail the (non--occurrence of) divergences in the Barrett-Crane model, formulated as an integral of delta function weights only. We will furthermore present a simple method to estimate the divergence degree of the so-called bubbles for general spin foam models. Divergences in spin foams are expected to be related to the existence of gauge symmetries (diffeomorphisms). Thus we have to conclude that such gauge symmetries are not (fully) present in the model we consider. But we will identify a class of gauge symmetries which occur at special solutions of the equations imposed by the delta function weights. This situation is surprisingly similar to the case of broken diffeomorphism symmetries in discrete gravity, which are present around flat solutions. We introduce a method to derive (Ward-identity-like) equations for the vertex amplitude of the model in the case of broken gauge symmetries.