Finiteness of Lorentzian 10j symbols and partition functions

TL;DR

This paper proves the finiteness of the Lorentzian 10j symbol, a key component in quantum gravity models, and extends the result to related integrals, ensuring the finiteness of certain partition functions.

Contribution

It provides a simple, general proof of the finiteness of Lorentzian 10j symbols and related integrals, impacting quantum gravity models.

Findings

Lorentzian 10j symbol is finite

Lorentzian and Riemannian causal 10j symbols are finite

Partition functions for certain triangulations are finite

Abstract

We give a short and simple proof that the Lorentzian 10j symbol, which forms a key part of the Barrett-Crane model of Lorentzian quantum gravity, is finite. The argument is very general, and applies to other integrals. For example, we show that the Lorentzian and Riemannian causal 10j symbols are finite, despite their singularities. Moreover, we show that integrals that arise in Cherrington's work are finite. Cherrington has shown that this implies that the Lorentzian partition function for a single triangulation is finite, even for degenerate triangulations. Finally, we also show how to use these methods to prove finiteness of integrals based on other graphs and other homogeneous domains.