Finiteness of a spinfoam model for euclidean quantum general relativity

TL;DR

The paper proves finiteness for a Euclidean spin-foam model of quantum gravity defined as a group-field theory over $SO(4)^4$, extending the Barrett–Crane construction to sum over arbitrary spinfoams. By bounding the Barrett–Crane vertex amplitude with kernels $K_N(y_i,y_j)=\frac{\sin((N+1)\Theta_{ij})}{\sin\Theta_{ij}}$ and using representation dimensions $\Delta_N$, the authors derive universal bounds that render any pentavalent 2-complex amplitude $A(J)$ finite. They show $|A(J)| \le \prod_f \sum_{N_f} (\Delta_{N_f})^{-1}$ up to controlled face-type considerations, yielding a decay factor $(\zeta(2)-1)^{F_J}$ with the number of faces $F_J$, thus proving finiteness and suggesting favorable convergence for the full sum over 2-complexes. The results indicate that the BF-to-GR constraints induce automatic regularization, with amplitudes diminishing as the combinatorial complexity grows, and point to exponential suppression in the number of faces as a useful feature for the theory’s sum-over-histories approach.

Abstract

We prove that a certain spinfoam model for euclidean quantum general relativity, recently defined, is finite: all its all Feynman diagrams converge. The model is a variant of the Barrett-Crane model, and is defined in terms of a field theory over SO(4) X SO(4) X SO(4) X SO(4).

Figures (1)

• Figure 1: The "singular" face