Volume 3, issue 1 (1999)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Classical 6j-symbols and the tetrahedron

Justin Roberts

Geometry & Topology 3 (1999) 21–66

arXiv: math-ph/9812013


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.

$6j$–symbol, asymptotics, tetrahedron, Ponzano–Regge formula, geometric quantization, scissors congruence
Mathematical Subject Classification
Primary: 22E99
Secondary: 81R05, 51M20
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Received: 9 January 1999
Accepted: 9 March 1999
Published: 22 March 1999
Proposed: Robion Kirby
Seconded: Vaughan Jones, Walter Neumann
Justin Roberts
Department of Mathematics and Statistics
Edinburgh University