Classical 6j-symbols and the tetrahedron

TL;DR

This paper proves the asymptotic relation between classical 6j-symbols and tetrahedron volume using geometric quantization, revealing new geometric insights and tetrahedral configurations.

Contribution

It provides a rigorous proof of the Ponzano-Regge asymptotic formula linking 6j-symbols to tetrahedron volume through geometric quantization.

Findings

Proof of the Ponzano-Regge asymptotic formula

Connection between 6j-symbols and Euclidean tetrahedron volume

Discovery of twelve scissors-congruent tetrahedra from a single Euclidean tetrahedron

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.