Spin Foam Models of Riemannian Quantum Gravity

TL;DR

This work benchmarks three Barrett–Crane spin foam formulations for 4D Riemannian quantum gravity by computing amplitudes with the $10j$ symbol and examining the partition function $Z(M)=\sum_F Z(F)$ on fixed triangulations. It finds that the DFKR version diverges rapidly, the Perez–Rovelli version converges extremely quickly with pronounced spin-zero dominance, and a new intermediate model sits near the convergence frontier without extreme spin-zero suppression. The authors analyze how convergence properties influence the extraction of physics via observables, using both analytical bounds and Metropolis sampling to study quantities like the average triangle area $O(F)$ and the distribution of spins. These results illuminate how choices of face/edge amplitudes shape the viability of spin foam models for approaching classical gravity and guide future mathematical and group-field-theory directions.

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.