Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, $3j$ Symbols, and Character Localization

TL;DR

This paper develops a novel method combining the Euler Maclaurin formula and saddle point approximation to derive exact asymptotic formulas for SU(2) Wigner matrix elements, characters, and 3j symbols, revealing a localization property.

Contribution

It introduces a new technique for asymptotic analysis of SU(2) recoupling coefficients that yields exact formulas and demonstrates a localization phenomenon.

Findings

Asymptotic formula for SU(2) characters is exact.

New method combining Euler Maclaurin and saddle point approximations.

Evidence of a localization property similar to Duistermaat-Heckman.

Abstract

In this paper we employ a novel technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index $J$) of generic Wigner matrix elements $D^{J}_{MM'}(g)$. We use this result to derive asymptotic formulae for the character $\chi^J(g)$ of an SU(2) group element and for Wigner's $3j$ symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for $\chi^J(g)$ is in fact exact. This result provides a non trivial example of a Duistermaat-Heckman like localization property for discrete sums.