Classical 6j-symbols and the tetrahedron
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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 6jsymbol, 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.
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In the limit of large j, the 6j-symbol times a phase is asymptotic to a sum of two exponentials involving the Regge action and volume, as in Equation (66).
“Ponzano and Regge noticed, and Roberts later proved rigorously [234], that in the limit of large j,”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [234]
Theorem 1 (Introduction) rigorously gives a large-label asymptotic formula with a volume-dependent amplitude and a cosine of the dihedral-angle phase; the cosine can be written as a sum of two exponentials. But the theorem gives that oscillatory formula only for an admissibly labelled Euclidean tetrahedron. It instead states exponential decay for the Minkowskian case, a condition the claim omits. (LLM, conf=0.87)
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Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134)
Kauffman, L. H., & Lins, S. (1994) · doi:10.1515/9781400882533 · Cited by 245 · see citing works
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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)
- W Fulton, J Harris, Representation theory, Springer GTM 129 (1991)
- A A Kirillov, Geometric quantization, from: "Dynamical systems IV", Ency- clopaedia 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 Uni- versity Press (1997)
- D McDuff, D Salamon, Introduction to symplectic topology, Oxford Univer- sity 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, Ergeb- nisse der Mathematik 34, Springer (1994)
- 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 func- tions, vol. 1, Kluwer (1991)
- E Wigner, Group theory, Academic Press (1959)
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
Asymptotics of 6 <i>j</i> and 10 <i>j</i> symbols
Freidel, L., & Louapre, D. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/7/303 · Cited by 113 · see citing works
Asymptotics of 10 <i>j</i> symbols
Baez, J. C., Christensen, J. D., & Egan, G. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/24/315 · Cited by 80 · see citing works
Path integral measure and triangulation independence in discrete gravity
Dittrich, B., & Steinhaus, S. (2012) · Physical Review D · doi:10.1103/physrevd.85.044032 · Cited by 66 · see citing works
The 6j-symbol: recursion, correlations and asymptotics
Dupuis, M., & Livine, E. R. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/13/135003 · Cited by 19 · see citing works
A New Recursion Relation for the 6j-Symbol
Bonzom, V., & Livine, E. R. (2012) · Annales Henri Poincaré · doi:10.1007/s00023-011-0143-y · Cited by 8 · see citing works
- Authors
- Roberts, J.
- Affiliations
- University of Edinburgh (GB)
- Corresponding author
- Roberts, Justin
- Venue
- Geometry & Topology
- Publisher
- Mathematical Sciences Publishers
- Published
- 1999-03-22
- Volume
- 3
- Issue
- 1
- Pages
- 21-66
- ISSN
- 1364-0380 (ISSN-L)
- 1465-3060
- OpenAlex
- W2115190755
- Cited by
- 92
- References
- 1
- Citations by year
- 2026: 2 · 2025: 2 · 2024: 3 · 2023: 3 · 2022: 4 · 2021: 1 · 2020: 2 · 2019: 2 · 2018: 9 · 2017: 3 · 2016: 2 · 2015: 3 · 2014: 5 · 2013: 3 · 2012: 6
- FWCI
- 4.45
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Geometry & Topology (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Keywords
- 6j symbols, tetrahedron, SU(2) representations, asymptotic formula, geometric quantization, scissors congruence, irreducible representations
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