Spin foam models of Riemannian quantum gravity

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.

Spin foam quantization and anomalies

“…In particular, without modification of the singular edge amplitude (associated with edges bounded by less than four faces), the finite normalizations for both the Riemannian and Lorentzian Barrett-Crane model proposed in [20,34] are to be regarded as formulations where the diffeomorphism gauge symmetry has been broken by an anomalous path integral measure. † Other anomalous formulations are: the new normalization of the Barrett-Crane model proposed in [18] to improve the convergence properties of the previous model and Model A in [35]. This seems to severely limit the physical relevance of such proposals.…”

Asymptotics of 10 j symbols

“…The next simplest case is the '4j symbol': the spin network with two vertices joined by four edges, labelled by positive integers a, b, c and d. Without loss of generality let us assume a ≤ b ≤ c ≤ d. As noted in a previous paper in this series [6], the Riemannian 4j symbol counts the dimension of a space of SU(2) intertwiners. Using this it follows that:…”

Renormalization of discrete models without background

“…While different versions of the Euclidean Barrett-Crane model have been proposed [16,17,18,19,20] we consider a version which appears to be the most natural one in our framework. For ease of terminology we refer to it in the following as "the" Barrett-Crane model.…”

“…However, a numerical study of different versions of the Barrett-Crane model was conducted by Baez et al [20]. This includes in particular the Perez-Rovelli version [17].…”