Abstract
From a historical viewpoint the Ponzano–Regge asymptotic formula for the 6j symbol of the group SU(2) (Ponzano and Regge, Semiclassical limit of Racah coefficients. In: Bloch et al (eds) Spectroscopic and group theoretical methods in physics. North–Holland, Amsterdam, pp 1–58, 1968), together with Penrose’s original idea of combinatorial spacetime out of coupling of angular momenta –or spin networks – Penrose (Angular momentum: an approach to combinatorial space–time. In: Bastin (ed) Quantum theory and beyond. Cambridge University Press, 151–180, 1971), is the precursor of the discretized approaches to 3–dimensional Euclidean quantum gravity collectively referred to as ‘state sum models’ after the 1992 paper by Turaev and Viro (State sum invariants and quantum 6j symbols. Topology 31:865–902, 1992). The prominent role here is played by the colored tetrahedron encoding the tetrahedral symmetry of the 6j symbol – reminiscent of the Platonic solid shown in the reproduction of Fig. 6.1 – and recognized in the semiclassical limit as a geometric 3–simplex whose edge lengths are irreps labels from the representation ring of either SU(2) or its universal enveloping algebra \({\mathcal {U}}_q(sl(2))\) with deformation parameter q = root of unity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Actually the above expression should contain the Racah W–coefficient W(j 1 j 2 j 3 j; j 12 j 23) which differs from the 6j by the factor \((-)^{j_1 + j_2 + j_3 + j}\).
- 2.
According to Condon–Shortely conventions adopted here, the 6j is a real orthogonal matrix, and the same holds true for Clebsch–Gordan and Wigner coefficients.
- 3.
- 4.
By consistency we mean that the discretized counterpart of the functional measure in the Euclidean path integral would be proportional to \(\prod (2j+1)dj\), according to the identification between ‘colors’ and edge lengths. Triangular (tetrahedral) inequalities, quite difficult to be implemented within a purely simplicial PL background, are automatically fulfilled since the 6j symbol vanishes whenever a constraint of this kind is violated, cfr. the introductory part of this section and further remarks on regularized functional measures in Sect. 6.2.2.
- 5.
Actually the geometric content of the q-6j symbol comes out in such a perturbative (not quite ‘semiclassical’) limit, and interestingly its emerging geometry is spherical at q a root of unity and hyperbolic in case of q real positive [65].
References
Ambjørn, J., Carfora, M., Marzuoli, A.: The Geometry of Dynamical Triangulations. Lecture Notes in Physics Monographs, vol. 50. Springer, Berlin (1997)
Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum Geometry. Cambridge University Press, Cambridge (1997)
Anderson, R.W., Aquilanti, V., Marzuoli, A.: 3nj morphogenesis and asymptotic disentangling. J. Phys. Chem. A 113, 15106–15117 (2009)
Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Yu, L.: Semiclassical analysis of Wigner 3j-symbol. J. Phys. A Math. Theor. 40, 5637–5674 (2007)
Aquilanti, V., Bitencourt, A.P.C., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications. Theor. Chem. Acc. 123, 237–247 (2009)
Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Poppe, S., Yu, L.: Asymptotics of the Wigner 6j symbol in a 4j model. Preprint (2010)
Arcioni, G., Carfora, M., Dappiaggi, C., Marzuoli, A.: The WZW model on random Regge triangulations. J. Geom. Phys. 52, 137–173 (2004)
Arcioni, G., Carfora, M., Marzuoli, A., O’ Loughin, M.: Implementing holographic projections in Ponzano–Regge gravity. Nucl. Phys. B 619, 690–708 (2001)
Askey, R.: Ortogonal Polynomials and Special Functions, Society for Industrial and Applied Mathematics. Philadelphia (1975); R. Koekoek, R.F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its Q-Analogue. Technische Universiteit Delft, Delft (1998). http://aw.twi.tudelft.nl/~koekoek/askey/
Atiyah, M.F.: The Geometry and Physics of Knots. Cambridge University Press, Cambridge (1990)
Barrett, J.W., Crane, L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39, 3296–3302 (1998)
Beliakova, A., Durhuus, B.: Topological quantum field theory and invariants of graphs for quantum groups. Commun. Math. Phys. 167, 395–429 (1995)
Biedenharn, L.C., Lohe, M.A.: Quantum Group Symmetry and Q-tensor Algebra. World Scientific, Singapore (1995)
Biedenharn, L.C., Louck, J.D.: Angular momentum in quantum physics: theory and applications. In: Rota G.-C. (ed.) Encyclopedia of Mathematics and Its Applications, vol. 8. Addison–Wesley Publications Co., Reading (1981)
Biedenharn, L.C., Louck, J.D.: The Racah–Wigner algebra in quantum theory. In: Rota G.-C. (ed.) Encyclopedia of Mathematics and Its Applications, vol. 9. Addison–Wesley Publications Co., Reading (1981)
Carbone, G.: Turaev–Viro invariant and 3nj symbols. J. Math. Phys. 41, 3068–3084 (2000)
Carbone, G., Carfora, M., Marzuoli, A.: Wigner symbols and combinatorial invariants of three–manifolds with boundary. Commun. Math. Phys. 212, 571–590 (2000)
Carbone, G., Carfora, M., Marzuoli, A.: Hierarchies of invariant spin models. Nucl. Phys. B 595, 654–688 (2001)
Carfora, M., Marzuoli, A., Rasetti, M.: Quantum tetrahedra. J. Phys. Chem. A 113, 15376–15383 (2009)
Carlip, S.: Quantum Gravity in 2+1 Dimensions. Cambridge University Press, Cambridge (1998)
Carter, J.S., Flath, D.E., Saito, M.: The Classical and Quantum 6j-Symbol. Princeton University Press, Princeton (1995)
Cattaneo, A.S., Cotta–Ramusino, P., Frölich, J., Martellini, M.: Topological BF theories in 3 and 4 dimensions. J. Math. Phys. 36, 6137–6160 (1995)
Crane, L., Kauffman, L.H., Yetter, D.N.: State sum invariants of 4 manifolds. arXiv:hep–th/9409167
De Pietri, R., Freidel, L., Krasnov, K., Rovelli, C.: Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space. Nucl. Phys. B 574, 785–806 (2000)
Durhuus, B., Jakobsen, H.P., Nest, R.: Topological quantum field theories from generalized 6j-symbols. Rev. Math. Phys. 5, 1–67 (1993)
Freed, D.S.: Remarks on Chern–Simons theory. Bull. Am. Math. Soc. 46, 221–254 (2009)
Freidel, L., Krasnov, K., Livine, E.R.: Holomorphic factorization for a quantum tetrahedron. Commun. Math. Phys. 297, 45–93 (2010)
Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246 (1985)
Gomez C., Ruiz–Altaba M., Sierra, G.: Quantum Group in Two–Dimensional Physics. Cambridge University Press, Cambridge (1996)
Guadagnini, E.: The link invariants of the Chern–Simons field theory. W. de Gruyter, Berlin/Boston (1993)
Haggard, H.M., Littlejohn, R.G.: Asymptotics of the Wigner 9j symbol. Class. Quant. Grav. 27, 135010 (2010)
Ionicioiu, R., Williams, R.M.: Lens spaces and handlebodies in 3D quantum gravity. Class. Quant. Grav. 15, 3469–3477 (1998)
Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)
Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985)
Karowski, M., Schrader, R.: A combinatorial approach to topological quantum field theories and invariants of graphs. Commun. Math. Phys. 167, 355–402 (1993)
Kauffman, L.: Knots and Physics. World Scientific, Singapore (2001)
Kauffman, L., Lins, S.: Temperley–Lieb recoupling theory and invariants of 3-Manifolds. Princeton University Press, Princeton (1994)
Kaul, R.,K., Govindarajan, T. R., P. Ramadevi, P.: Schwarz type topological quantum field theories. In: Encyclopedia of Mathematical Physics. Elsevier (2005) (eprint hep–th/0504100)
Kirby, R., Melvin, P.: The 3-manifold invariant of Witten and Reshetikhin–Turaev for \(sl(2, \mathbb {C})\). Invent. Math. 105, 437–545 (1991)
Kirillov, A.N., Reshetikhin, N.Y.: In: Kac, V.G. (ed.) Infinite Dimensional Lie Algebras and Groups. Advanced Series in Mathematical Physics, vol. 7, pp. 285–339. World Scientific, Singapore (1988)
Mizoguchi, S., Tada, T.: 3-dimensional gravity from the Turaev–Viro invariant. Phys. Rev. Lett. 68, 1795–1798 (1992)
Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin/New York (1991)
Neville, D.: A technique for solving recurrence relations approximately and its application to the 3-j and 6-J symbols. J. Math. Phys. 12, 2438–2453 (1971)
Nomura, M.: Relations for Clebsch–Gordan and Racah coefficients in su q (2) and Yang–Baxter equations. J. Math. Phys. 30, 2397–2405 (1989)
Ohtsuki T. (ed.): Problems on invariants of knots and 3–manifolds, RIMS geometry and topology monographs, vol. 4 (eprint arXiv: math.GT/0406190)
Ooguri, H.: Topological lattice models in four dimensions. Mod. Phys. Lett. A 7, 2799–2810 (1992)
Ooguri, H.: Schwinger–Dyson equation in three-dimensional simplicial quantum gravity. Prog. Theor. Phys. 89, 1–22 (1993)
Pachner, U.: Ein Henkel Theorem für geschlossene semilineare Mannigfaltigkeiten [A handle decomposition theorem for closed semilinear manifolds]. Result. Math. 12, 386–394 (1987)
Pachner, U.: Shelling of simplicial balls and P.L. manifolds with boundary. Discr. Math. 81, 37–47 (1990)
Pachner, U.: Homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12, 129–145 (1991)
Penrose, R.: Angular momentum: an approach to combinatorial space–time. In: Bastin, T. (ed.) Quantum Theory and Beyond, pp. 151–180. Cambridge University Press, Cambridge (1971)
Ponzano, G., Regge, T.: Semiclassical limit of racah coefficients. In: Bloch F. et al. (eds.) Spectroscopic and Group Theoretical Methods in Physics, pp. 1–58. North–Holland, Amsterdam (1968)
Ragni, M., Bitencourt, A.P.C., da S. Ferreira, C. Aquilanti, V., Anderson, R.W., Littlejohn, R.G.: Exact computation and asymptotic approximations of 6j symbols: illustration of their semiclassical limits. Int. J. Quant. Chem. 110, 731–742 (2009)
Regge, T.: Symmetry properties of Racah’s coefficients. Nuovo Cimento 11, 116–117 (1958)
Regge, T.: General relativity without coordinates. Nuovo Cimento 19, 558–571 (1961)
Regge, T., Williams, R.M.: Discrete structures in gravity. J. Math. Phys. 41, 3964–3984 (2000)
Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)
Roberts, J.D.: Skein theory and Turaev–Viro invariants. Topology 34, 771–787 (1995)
Roberts, J.D.: Classical 6j-symbols and the tetrahedron. Geom. Topol. 3, 21–66 (1999)
Rolfsen, D.: Knots and Links. Publish or Perish, Inc., Berkeley (1976)
Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise–Linear Topology. Springer, Berlin (1972)
Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)
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)
Taylor, Y.U., Woodward, C.T.: 6j symbols for U q (sl 2) and non–Euclidean tetrahedra. Sel. Math. New Ser. 11, 539–571 (2005)
’t Hooft, G.: The scattering matrix approach for the quantum black hole, an overview. Int. J. Mod. Phys. A11, 4623–4688 (1996)
Turaev, V.G.: Quantum invariants of links and 3-valent graphs in 3-manifolds. Publ. Math. IHES 77, 121–171 (1993)
Turaev, V.G.: Quantum Invariants of Knots and 3-manifolds. W. de Gruyter, Berlin (1994)
Turaev, V.G., Viro, O.Y.: State sum invariants and quantum 6j symbols. Topology 31, 865–902 (1992)
Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore/Philadelphia (1988)
Walker, K.: On witten’s 3-manifolds invariant. Preprint (1991). (An extended version dated 2001 is available on the web)
Williams, R.M., Tuckey, P.A.: Regge calculus: a bibliography and brief review. Class. Quant. Grav. 9, 1409–1422 (1992)
Witten, E.: (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 49–78 (1988/89)
Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)
Yutsis, A.P., Levinson, I.B., Vanagas, V.V.: The Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Scientific Translations Ltd (1962)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Carfora, M., Marzuoli, A. (2017). State Sum Models and Observables. In: Quantum Triangulations. Lecture Notes in Physics, vol 942. Springer, Cham. https://doi.org/10.1007/978-3-319-67937-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-67937-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67936-5
Online ISBN: 978-3-319-67937-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)