---
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.'
address:
- 'Dipartimento di Chimica, Università di Perugia, Perugia, Italy 06100'
- 'Department of Physics, University of California, Berkeley, California 94720 USA'
author:
- Vincenzo Aquilanti
- 'Hal M. Haggard, Austin Hedeman, Nadir Jeevanjee, Robert G. Littlejohn, and Liang Yu'
title: 'Semiclassical Mechanics of the Wigner $6j$-Symbol'
---

Introduction
============

The Wigner $6j$-symbol (or Racah $W$-coefficient) is a central object in angular momentum theory, with many applications 