Abstract
The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This article presents a new recursion formula for the square of the 6j-symbol. In the asymptotic regime, the new recursion is shown to characterize the closure of the relevant tetrahedron. Since the 6j-symbol is the basic building block of the Ponzano–Regge model for pure three-dimensional quantum gravity, we also discuss how to generalize the method to derive more general recursion relations on the full amplitudes.
Article PDF
Similar content being viewed by others
References
Aquilanti V., Bitencourt A.C.P., Ferreira C.d.S., Marzuoli A., Ragni M.: Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity. Phys. Scripta 78, 058103 (2008) arXiv:0901.1074 [quant-ph]
Wen X.G.: Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405 (1995)
Levin M.A., Wen X.G.: String net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005) arXiv:cond-mat/0404617
Dennis E., Kitaev A., Landahl A., Preskill J.: Topological quantum memory. J. Math. Phys 43, 4452 (2002) arXiv:quant-ph/0110143
Perez, A.: Introduction to loop quantum gravity and spin foams. In: Lectures presented at the II international conference of fundamental interactions, Pedra Azul, Brazil (2004). arXiv:gr-qc/0409061
Perez A.: The spin-foam-representation of loop quantum gravity. In: Oriti, D (ed) Approaches to quantum gravity: toward a new understanding of space, time and matter, Cambridge University Press, UK (2009) arXiv:gr-qc/0601095
Livine, E.R.: A short and subjective introduction to the spinfoam framework for quantum gravity. In: Habilitation thesis 2010, Ecole Normale Supérieure de Lyon, France (2011). arXiv:1101.5061
Baez J.C.: An Introduction to spin foam models of quantum gravity and BF theory. Lect. Notes Phys 543, 25 (2000) arXiv:gr-qc/9905087
Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Spectroscopic and group theoretical methods in physics (Bloch ed.), North-Holland (1968)
Freidel L., Louapre D.: Ponzano–Regge model revisited I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav 21, 5685 (2004) arXiv:hep-th/0401076
Turaev V.G., Viro O.Y.: State sum invariants of 3 manifolds and quantum 6j symbols. Topology 31, 865–902 (1992)
Carfora M., Marzuoli A., Rasetti M.: Quantum tetrahedra. J. Phys. Chem. A 113, 15367 (2009) arXiv:1001.4402 [math-ph]
Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevanjee, N., Littlejohn, R.G., Yu, L.: Semiclassical mechanics of the Wigner 6j-symbol (2010). arXiv:1009.2811 [math-ph]
Varshalovich D.A., Moskalev A.N., Khersonsky V.K.: Quantum Theory of Angular Momentum: Irreducible Tensors, spherical harmonics, vector coupling coefficients, 3nj Symbols, pp. 514. World Scientific, Singapore (1988)
Roberts J.: Classical 6j-symbols and the tetrahedron. Geom. Topol 3, 21–26 (1999) arXiv:math-ph/9812013
Schulten K., Gordon R.G.: Exact recursive evaluation of 3j and 6j coefficients for quantum mechanical coupling of angular momenta. J. Math. Phys 16, 1961–1970 (1975)
Schulten K., Gordon R.G.: Semiclassical approximations to 3j and 6j coefficients for quantum mechanical coupling of angular momenta. J. Math. Phys 16, 1971–1988 (1975)
Freidel L., Louapre D.: Asymptotics of 6j and 10j symbols. Class. Quant. Grav 20, 1267–1294 (2003) arXiv:hep-th/0209134
Gurau R.: The Ponzano–Regge asymptotic of the 6j symbol: an elementary proof. Annales Henri Poincaré 9, 1413 (2008) arXiv:0808.3533 [math-ph]
Dupuis M., Livine E.R.: The 6j-symbol: recursion, correlations and asymptotics. Class. Quant. Grav 27, 135003 (2010) arXiv:0910.2425 [gr-qc]
Bonzom V., Livine E.R., Smerlak M., Speziale S.: Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model. Nucl. Phys. B 804, 507 (2008) arXiv:0802.3983 [gr-qc]
Dupuis M., Livine E.R: Pushing further the asymptotics of the 6j-symbol. Phys. Rev. D 80, 024035 (2009) arXiv:0905.4188
Dowdall R.J., Gomes H., Hellmann F.: Asymptotic analysis of the Ponzano– Regge model for handlebodies. J. Phys. A 43, 115203 (2010) arXiv:0909.2027 [gr-qc]
Barrett J.W., Fairbairn W.J., Hellmann F.: Quantum gravity asymptotics from the SU(2) 15j symbol. Int. J. Mod. Phys. A 25, 2897–2916 (2010) arXiv:0912.4907 [gr-qc]
Barrett, J.W., Dowdall, R.J., Fairbairn,W.J., Gomes, H., Hellmann, F., Pereira, R.: Asymptotics of 4d spin foam models (2010). arXiv:1003.1886 [gr-qc]
Littlejohn R.G., Yu L.: Semiclassical Analysis of the Wigner 9J-Symbol with Small and Large Angular Momenta. Phys. Rev. A 83, 052114 (2011) arXiv:1104.1499 [math-ph]
Yu, L.: Semiclassical analysis of the Wigner 12J-symbol with one small angular momentum: part I (2011). arXiv:1104.3275 [math-ph]
Yu, L.: Asymptotic limits of the Wigner 15J-symbol with small quantum numbers (2011). arXiv:1104.3641 [math-ph]
Yu, L.: Asymptotic limits of the Wigner 12J-symbol in terms of the Ponzano–Regge phases (2011). arXiv:1108.1881 [math-ph]
Bonzom, V. and Fleury, P.: Asymptotics of Wigner 3nj-symbols with small and large angular momenta: an elementary method. arXiv:1108.1569 [quant-ph]
Anderson R.W., Aquilanti V., Marzuoli A.: 3nj Morphogenesis and semiclassical disentangling. J. Phys. Chem. A 113, 15106 (2009) arXiv:1001.4386 [quant-ph]
Bonzom V., Livine E.R., Speziale S.: Recurrence relations for spin foam vertices. Class. Quant. Grav 27, 125002 (2010) arXiv:0911.2204 [gr-qc]
Bonzom, V., Freidel, L.: The Hamiltonian constraint in 3d Riemannian loop quantum gravity. arXiv:1101.3524 [gr-qc]
Bonzom V.: Spin foam models and the Wheeler–DeWitt equation for the quantum 4-simplex. Phys. Rev. D 84, 024009 (2011) arXiv:1101.1615 [gr-qc]
Barrett J.W., Naish-Guzman I.: The Ponzano–Regge model. Class. Quant. Grav 26, 155014 (2009) arXiv:0803.3319
Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. arXiv:1008.1476 [math-ph]
Barrett J.W.: First order Regge calculus. Class. Quant. Grav 11, 2723 (1994) arXiv:hep-th/9404124
Barrett J.W., Crane L.: Relativistic spin networks and quantum gravity. J. Math. Phys 39, 3296–3302 (1998) arXiv:gr-qc/9709028
Barrett J.W., Williams R.M.: The asymptotics of an amplitude for the 4-simplex. Adv. Theor. Math. Phys 3, 209–215 (1999) arXiv:gr-qc/9809032
Baez J.C., Christensen J.D., Egan G.: Asymptotics of 10j symbols. Class. Quant. Grav 19, 6489 (2002) arXiv:gr-qc/0208010
Barrett J.W., Steele C.M.: Asymptotics of relativistic spin networks. Class. Quant. Grav 20, 1341–1362 (2003) arXiv:gr-qc/0209023
Christensen J.D., Khavkine I., Livine E.R., Speziale S.: Sub-leading asymptotic behaviour of area correlations in the Barrett–Crane model. Class. Quant. Grav 27, 035012 (2010) arXiv:0908.4476
Freidel L., Livine E.R.: Ponzano–Regge model revisited III: Feynman diagrams and effective field theory. Class. Quant. Grav 23, 2021–2062 (2006) arXiv:hepth/0502106
Baratin A., Freidel L.: Hidden quantum gravity in 3d Feynman diagrams. Class. Quant. Grav 24, 1993–2026 (2007) arXiv:gr-qc/0604016
Bonzom, V., Smerlak M.: BF theory: discrete model and loop quantization (in press)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Carlo Rovelli.
Rights and permissions
About this article
Cite this article
Bonzom, V., Livine, E.R. A New Recursion Relation for the 6j-Symbol. Ann. Henri Poincaré 13, 1083–1099 (2012). https://doi.org/10.1007/s00023-011-0143-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0143-y