Semiclassical Mechanics of the Wigner 6j-Symbol

TL;DR

This paper explores the semiclassical mechanics of the Wigner 6j-symbol using WKB theory, focusing on geometric and symplectic aspects, and introduces a new approach based on angular momentum recoupling.

Contribution

It presents a novel method for analyzing the 6j-symbol through recoupling of four angular momenta and generalizes spin network techniques for semiclassical analysis.

Findings

Derived a new semiclassical approach based on recoupling angular momenta.

Analyzed the reduced phase space as the 2-sphere of Kapovich and Millson.

Established principles for semiclassical study of arbitrary spin networks.

Abstract

The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano-Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is devoted to symplectic reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich and Millson), and the reduction of Poisson bracket expressions for semiclassical amplitudes. General principles for the semiclassical study of arbitrary spin networks are laid down; some of these were used in our recent derivation of the asymptotic formula for the Wigner 9j-symbol.