Geometry & Topology

Classical 6j-symbols and the tetrahedron

Justin Roberts

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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

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  • R Carter, G Segal, I MacDonald, Lectures on Lie groups and Lie algebras, LMS Student Texts 32, Cambridge University Press (1995)
  • P Cartier, Décomposition des polyèdres: le point sure le troisième problème de Hilbert, Astérisque 133–134 (1986) 261–288
  • W Fulton, J Harris, Representation theory, Springer GTM 129 (1991)
  • V Guillemin, S Sternberg, Geometric quantization and multiplicities of group representations, Inventiones Math. 67 (1982) 515–538
  • N Hitchin, Metrics on moduli spaces, Contemp. Math. 58 part 1 (1986) 157–178
  • L Kauffman, S Lins, Temperley–Lieb recoupling theory and invariants of 3–manifolds, Annals of Maths Studies 134, Princeton University Press (1994)
  • A A Kirillov, Geometric quantization, from: “Dynamical systems IV”, Encyclopaedia of Mathematical Sciences 4 (V I Arnol'd and S P Novikov, editors) Springer (1990)
  • D A Klain, G-C Rota, Introduction to geometric probability, Cambridge University Press (1997)
  • D McDuff, D Salamon, Introduction to symplectic topology, Oxford University Press (1995)
  • J Milnor, Euler characteristic and finitely additive Steiner measures, from: “Collected papers vol. 1”, Publish or Perish (1993)
  • J Milnor, The Schläfli differential equality, from: “Collected papers vol. 1”, Publish or Perish (1993)
  • D Mumford, J Fogerty, F C Kirwan, Geometric invariant theory, Ergebnisse der Mathematik 34, Springer (1994)
  • W D Neumann, Hilbert's 3rd problem and invariants of 3–manifolds, from: “The Epstein birthday schrift”, (Igor Rivin, Colin Rourke, Caroline Series, editors) Geometry and Topology Mongraphs, 1 (1998) 383–411
  • G Ponzano and T Regge, Semi-classical limit of Racah coefficients, from: “Spectroscopic and group theoretical methods in physics” (F Bloch, editor) North-Holland (1968)
  • D A Varshalovich, A N Moskalev, V K Khersonskii, Quantum theory of angular momentum: irreducible tensors, spherical harmonics, vector coupling coefficients, $3nj$–symbols, World Scientific (1988)
  • N Y Vilenkin, A U Klimyk, Representation of Lie groups, and special functions, vol. 1, Kluwer (1991)
  • E Wigner, Group theory, Academic Press (1959)