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Semiclassical Analysis of the Wigner $9J$-Symbol with Small and Large Angular Momenta

By Liang Yu and Robert G. Littlejohn

Abstract

We derive a new asymptotic formula for the Wigner $9j$-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant non-canonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher $3nj$-symbols. We display without proof some new asymptotic formulas for the $12j$-symbol and the $15j$-symbol in the appendices. This work contributes a new asymptotic formula of the Wigner $9j$-symbol to the quantum theory of angular momentum, and serves as an example of a new general method for deriving asymptotic formulas for $3nj$-symbols.Comment: 18 pages, 16 figures. To appear in Phys. Rev.

Topics: Mathematical Physics
Year: 2011
DOI identifier: 10.1103/PhysRevA.83.052114
OAI identifier: oai:arXiv.org:1104.1499
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