Skip to main content
SpringerLink
Log in

State Sum Models and Observables

  • Chapter
  • First Online:
Quantum Triangulations

Part of the book series: Lecture Notes in Physics ((LNP,volume 942))

  • 998 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 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. 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. 3.

    The \(\tfrac {1}{2}\)–shift is crucial in the analysis developed in [52]: for high quantum numbers the length [j(j + 1)]1/2 of an angular momentum vector is close to \(j+ \tfrac {1}{2}\), see the semiclassical analysis given in Sect. 6.1.2 below.

  4. 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. 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

  1. Ambjørn, J., Carfora, M., Marzuoli, A.: The Geometry of Dynamical Triangulations. Lecture Notes in Physics Monographs, vol. 50. Springer, Berlin (1997)

    Google Scholar 

  2. Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum Geometry. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  3. Anderson, R.W., Aquilanti, V., Marzuoli, A.: 3nj morphogenesis and asymptotic disentangling. J. Phys. Chem. A 113, 15106–15117 (2009)

    Google Scholar 

  4. 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)

    Google Scholar 

  5. 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)

    Google Scholar 

  6. Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Poppe, S., Yu, L.: Asymptotics of the Wigner 6j symbol in a 4j model. Preprint (2010)

    Google Scholar 

  7. Arcioni, G., Carfora, M., Dappiaggi, C., Marzuoli, A.: The WZW model on random Regge triangulations. J. Geom. Phys. 52, 137–173 (2004)

    Google Scholar 

  8. Arcioni, G., Carfora, M., Marzuoli, A., O’ Loughin, M.: Implementing holographic projections in Ponzano–Regge gravity. Nucl. Phys. B 619, 690–708 (2001)

    Google Scholar 

  9. 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/

  10. Atiyah, M.F.: The Geometry and Physics of Knots. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  11. Barrett, J.W., Crane, L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39, 3296–3302 (1998)

    Google Scholar 

  12. Beliakova, A., Durhuus, B.: Topological quantum field theory and invariants of graphs for quantum groups. Commun. Math. Phys. 167, 395–429 (1995)

    Google Scholar 

  13. Biedenharn, L.C., Lohe, M.A.: Quantum Group Symmetry and Q-tensor Algebra. World Scientific, Singapore (1995)

    Google Scholar 

  14. 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)

    Google Scholar 

  15. 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)

    Google Scholar 

  16. Carbone, G.: Turaev–Viro invariant and 3nj symbols. J. Math. Phys. 41, 3068–3084 (2000)

    Google Scholar 

  17. Carbone, G., Carfora, M., Marzuoli, A.: Wigner symbols and combinatorial invariants of three–manifolds with boundary. Commun. Math. Phys. 212, 571–590 (2000)

    Google Scholar 

  18. Carbone, G., Carfora, M., Marzuoli, A.: Hierarchies of invariant spin models. Nucl. Phys. B 595, 654–688 (2001)

    Google Scholar 

  19. Carfora, M., Marzuoli, A., Rasetti, M.: Quantum tetrahedra. J. Phys. Chem. A 113, 15376–15383 (2009)

    Google Scholar 

  20. Carlip, S.: Quantum Gravity in 2+1 Dimensions. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  21. Carter, J.S., Flath, D.E., Saito, M.: The Classical and Quantum 6j-Symbol. Princeton University Press, Princeton (1995)

    Google Scholar 

  22. 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)

    Google Scholar 

  23. Crane, L., Kauffman, L.H., Yetter, D.N.: State sum invariants of 4 manifolds. arXiv:hep–th/9409167

    Google Scholar 

  24. 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)

    Google Scholar 

  25. Durhuus, B., Jakobsen, H.P., Nest, R.: Topological quantum field theories from generalized 6j-symbols. Rev. Math. Phys. 5, 1–67 (1993)

    Google Scholar 

  26. Freed, D.S.: Remarks on Chern–Simons theory. Bull. Am. Math. Soc. 46, 221–254 (2009)

    Google Scholar 

  27. Freidel, L., Krasnov, K., Livine, E.R.: Holomorphic factorization for a quantum tetrahedron. Commun. Math. Phys. 297, 45–93 (2010)

    Google Scholar 

  28. 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)

    Google Scholar 

  29. Gomez C., Ruiz–Altaba M., Sierra, G.: Quantum Group in Two–Dimensional Physics. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  30. Guadagnini, E.: The link invariants of the Chern–Simons field theory. W. de Gruyter, Berlin/Boston (1993)

    Google Scholar 

  31. Haggard, H.M., Littlejohn, R.G.: Asymptotics of the Wigner 9j symbol. Class. Quant. Grav. 27, 135010 (2010)

    Google Scholar 

  32. Ionicioiu, R., Williams, R.M.: Lens spaces and handlebodies in 3D quantum gravity. Class. Quant. Grav. 15, 3469–3477 (1998)

    Google Scholar 

  33. Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993)

    Google Scholar 

  34. Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985)

    Google Scholar 

  35. Karowski, M., Schrader, R.: A combinatorial approach to topological quantum field theories and invariants of graphs. Commun. Math. Phys. 167, 355–402 (1993)

    Google Scholar 

  36. Kauffman, L.: Knots and Physics. World Scientific, Singapore (2001)

    Google Scholar 

  37. Kauffman, L., Lins, S.: Temperley–Lieb recoupling theory and invariants of 3-Manifolds. Princeton University Press, Princeton (1994)

    Google Scholar 

  38. 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)

    Google Scholar 

  39. Kirby, R., Melvin, P.: The 3-manifold invariant of Witten and Reshetikhin–Turaev for \(sl(2, \mathbb {C})\). Invent. Math. 105, 437–545 (1991)

    Google Scholar 

  40. 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)

    Google Scholar 

  41. Mizoguchi, S., Tada, T.: 3-dimensional gravity from the Turaev–Viro invariant. Phys. Rev. Lett. 68, 1795–1798 (1992)

    Google Scholar 

  42. Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin/New York (1991)

    Google Scholar 

  43. 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)

    Google Scholar 

  44. Nomura, M.: Relations for Clebsch–Gordan and Racah coefficients in su q (2) and Yang–Baxter equations. J. Math. Phys. 30, 2397–2405 (1989)

    Google Scholar 

  45. Ohtsuki T. (ed.): Problems on invariants of knots and 3–manifolds, RIMS geometry and topology monographs, vol. 4 (eprint arXiv: math.GT/0406190)

    Google Scholar 

  46. Ooguri, H.: Topological lattice models in four dimensions. Mod. Phys. Lett. A 7, 2799–2810 (1992)

    Google Scholar 

  47. Ooguri, H.: Schwinger–Dyson equation in three-dimensional simplicial quantum gravity. Prog. Theor. Phys. 89, 1–22 (1993)

    Google Scholar 

  48. Pachner, U.: Ein Henkel Theorem für geschlossene semilineare Mannigfaltigkeiten [A handle decomposition theorem for closed semilinear manifolds]. Result. Math. 12, 386–394 (1987)

    Google Scholar 

  49. Pachner, U.: Shelling of simplicial balls and P.L. manifolds with boundary. Discr. Math. 81, 37–47 (1990)

    Google Scholar 

  50. Pachner, U.: Homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12, 129–145 (1991)

    Google Scholar 

  51. 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)

    Google Scholar 

  52. 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)

    Google Scholar 

  53. 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)

    Google Scholar 

  54. Regge, T.: Symmetry properties of Racah’s coefficients. Nuovo Cimento 11, 116–117 (1958)

    Google Scholar 

  55. Regge, T.: General relativity without coordinates. Nuovo Cimento 19, 558–571 (1961)

    Google Scholar 

  56. Regge, T., Williams, R.M.: Discrete structures in gravity. J. Math. Phys. 41, 3964–3984 (2000)

    Google Scholar 

  57. Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)

    Google Scholar 

  58. Roberts, J.D.: Skein theory and Turaev–Viro invariants. Topology 34, 771–787 (1995)

    Google Scholar 

  59. Roberts, J.D.: Classical 6j-symbols and the tetrahedron. Geom. Topol. 3, 21–66 (1999)

    Google Scholar 

  60. Rolfsen, D.: Knots and Links. Publish or Perish, Inc., Berkeley (1976)

    Google Scholar 

  61. Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise–Linear Topology. Springer, Berlin (1972)

    Google Scholar 

  62. Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  63. 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)

    Google Scholar 

  64. 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)

    Google Scholar 

  65. 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)

    Google Scholar 

  66. ’t Hooft, G.: The scattering matrix approach for the quantum black hole, an overview. Int. J. Mod. Phys. A11, 4623–4688 (1996)

    Google Scholar 

  67. Turaev, V.G.: Quantum invariants of links and 3-valent graphs in 3-manifolds. Publ. Math. IHES 77, 121–171 (1993)

    Google Scholar 

  68. Turaev, V.G.: Quantum Invariants of Knots and 3-manifolds. W. de Gruyter, Berlin (1994)

    Google Scholar 

  69. Turaev, V.G., Viro, O.Y.: State sum invariants and quantum 6j symbols. Topology 31, 865–902 (1992)

    Google Scholar 

  70. Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore/Philadelphia (1988)

    Google Scholar 

  71. Walker, K.: On witten’s 3-manifolds invariant. Preprint (1991). (An extended version dated 2001 is available on the web)

    Google Scholar 

  72. Williams, R.M., Tuckey, P.A.: Regge calculus: a bibliography and brief review. Class. Quant. Grav. 9, 1409–1422 (1992)

    Google Scholar 

  73. Witten, E.: (2+1)-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 49–78 (1988/89)

    Google Scholar 

  74. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)

    Google Scholar 

  75. 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)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università degli Studi di Pavia, Pavia, Italy

    Mauro Carfora

  2. Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy

    Annalisa Marzuoli

Authors
  1. Mauro Carfora
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Annalisa Marzuoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

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

Publish with us

Policies and ethics

Search

Navigation