Geometry & Topology, Vol. 3 (1999) Paper no. 2, pages 21--66.

Classical 6j-symbols and the tetrahedron

Justin Roberts

Abstract. A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.

Keywords. 6j-symbol, asymptotics, tetrahedron, Ponzano-Regge formula, geometric quantization, scissors congruence

AMS subject classification. Primary: 22E99. Secondary: 81R05, 51M20.

DOI: 10.2140/gt.1999.3.21

E-print: arXiv:math-ph/9812013

Submitted to GT on 9 January 1999. Paper accepted 9 March 1999. Paper published 22 March 1999.

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Justin Roberts
Department of Mathematics and Statistics
Edinburgh University, EH3 9JZ, Scotland

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