Spin Foam Models of Riemannian Quantum Gravity

TL;DR

This paper compares three versions of the Barrett-Crane spin foam model for 4D Riemannian quantum gravity, analyzing their convergence properties and implications for physical predictions.

Contribution

It provides a numerical comparison of different Barrett-Crane models, introducing a new version with potentially convergent partition function.

Findings

Original model diverges rapidly for many triangulations.

Modified model converges with spin-zero dominance.

New version shows potential for convergence without spin-zero dominance.

Abstract

Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spin-zero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model.