Geometry & Topology

Classical 6j-symbols and the tetrahedron

Justin Roberts

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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.

Article information

Source
Geom. Topol., Volume 3, Number 1 (1999), 21-66.

Dates
Received: 9 January 1999
Accepted: 9 March 1999
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883138

Digital Object Identifier
doi:10.2140/gt.1999.3.21

Mathematical Reviews number (MathSciNet)
MR1673272

Zentralblatt MATH identifier
0918.22014

Subjects
Primary: 22E99: None of the above, but in this section
Secondary: 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70] 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

Keywords
$6j$–symbol asymptotics tetrahedron Ponzano–Regge formula geometric quantization scissors congruence

Citation

Roberts, Justin. Classical 6j-symbols and the tetrahedron. Geom. Topol. 3 (1999), no. 1, 21--66. doi:10.2140/gt.1999.3.21. https://projecteuclid.org/euclid.gt/1513883138


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