Asymptotics of Wigner 3nj-symbols with Small and Large Angular Momenta: an Elementary Method

TL;DR

This paper introduces a simpler method for deriving asymptotic formulas for Wigner 3nj-symbols with mixed small and large angular momenta, generalizing previous results and providing new asymptotics.

Contribution

It presents an elementary derivation using the Ponzano-Regge formula, simplifying existing formulas and extending asymptotic analysis to more complex Wigner symbols with multiple small angular momenta.

Findings

Simpler formula for asymptotics of Wigner 9j, 12j, 15j-symbols

Generalization to cases with multiple small angular momenta

New asymptotic formulas for complex Wigner 3nj-symbols

Abstract

Yu and Littlejohn recently studied in arXiv:1104.1499 some asymptotics of Wigner symbols with some small and large angular momenta. They found that in this regime the essential information is captured by the geometry of a tetrahedron, and gave new formulae for 9j, 12j and 15j-symbols. We present here an alternative derivation which leads to a simpler formula, based on the use of the Ponzano-Regge formula for the relevant tetrahedron. The approach is generalized to Wigner 3nj-symbols with some large and small angular momenta, where more than one tetrahedron is needed, leading to new asymptotics for Wigner 3nj-symbols. As an illustration, we present 15j-symbols with one, two and four small angular momenta, and give an alternative formula to Yu's recent 15j-symbol with three small spins.