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

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 $D_{M{M}^{\prime }}^J(g)$. 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.

36 pages, 3 figures

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• LQG vertex with finite Immirzi parameter
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