#### 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.

Notes on file formats
Justin Roberts

Department of Mathematics and Statistics

Edinburgh University, EH3 9JZ, Scotland

Email: justin@maths.ed.ac.uk

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