Asymptotics of 6j and 10j symbols

TL;DR

This paper analyzes the asymptotic behavior of 6j and 10j symbols, key components in quantum gravity models, using group integral techniques to derive their asymptotics and explore implications for the Barrett-Crane model.

Contribution

It introduces new techniques for studying the asymptotics of 6j and 10j symbols via group integrals and invariant variables, providing detailed asymptotic expansions for Euclidean and Lorentzian cases.

Findings

Computed asymptotics of Euclidean and Lorentzian 6j-symbols.

Derived the non-oscillating asymptotics of the 10j symbol.

Discussed modifications to the Barrett-Crane model to address non-oscillating behavior.

Abstract

It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe the measure involved in terms of invariant variables and develop new technics in order to study their asymptotics. Using these technics we compute the asymptotics of the various Euclidean and Lorentzian 6j-symbols. Finally we compute the asymptotic expansion of the 10j symbol which is shown to be non-oscillating in agreement with a recent result of Baez et al. We discuss the physical origin of this behavior and a way to modify the Barrett-Crane model in order to cure this disease.