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

Abstract

In this paper, we employ a 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 DMM′J(g). We use this result to derive asymptotic formulae for the character χJ(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 χJ(g) is in fact exact. The result hints at a "Duistermaat-Heckman like" localization property for discrete sums. © 2011 Springer Basel AG.