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Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

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Abstract

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.

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References

  1. Baez, J.: An introduction to spin foam models of Quantum Gravity and BF Theory. In: Geometry and Quantum Physicis, Gausterer, H., Grosse, H., Pittner, L. (eds.) Lect. Notes Phys. 543, Berlin-Heidelberg-New York: Springer Verlag, 2000 pp. 25–94

  2. Baez, J.: Spin foam perturbation theory. In: Diagrammatic Morphisms and Applications (San Francisco, CA, 2000) Contemp. Math. 318, Providence, RI: Amer. Math. Soc., 2003, pp. 9–21

  3. Barrett, J.W., Naish-Guzman, I.: The Ponzano-Regge Model. http://arXiv.org/abs/0803.3319v1 (gr-qc), 2008

  4. Barrett J.W., Garcia-Islas J.M., Faria Martins J.: Observables in the Turaev-Viro and Crane-Yetter models. J. Math. Phys. 48(9), 093508 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  5. Carter, J.S., Flath, D.E., Saito, M.: The Classical and Quantum 6j-Symbols Mathematical Notes 43, Princeton, NJ: Princeton University Press, 1995

  6. Cooper D., Thurston W.P.: Triangulating 3 manifolds using 5 vertex link types. Topology 27(1), 23–25 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Faria Martins J., Miković A.: Invariants of spin networks embedded in three-dimensional manifolds. Commun. Math. Phys. 279, 381–399 (2008)

    Article  MATH  ADS  Google Scholar 

  8. Freidel L., Krasnov K.: Spin foam models and the classical action principle. Adv. Theor. Math. Phys. 2, 1183–1247 (1999)

    MathSciNet  Google Scholar 

  9. Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, 20. Providence, RI: Amer. Mathe. Soc., 1999

  10. Hackett J., Speziale S.: Grasping rules and semiclassical limit of the geometry in the Ponzano-Regge model. Class. Quant. Grav. 24, 1525–1545 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Kauffman, L.H., Lins, S.L.: Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. Annals of Mathematics Studies, 134. Princeton, NJ: Princeton University Press, 1994

  12. Lickorish W.B.R.: The skein method for three-manifold invariants. J. Knot Theory Ram 2(2), 171–194 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Miković A.: Quantum gravity as a deformed topological quantum field theory. J. Phys. Conf. Ser. 33, 266–270 (2006)

    Article  Google Scholar 

  14. Miković, A.: Quantum gravity as a broken symmetry phase of a BF theory. SIGMA 2, 086, (2006) (5 pages)

  15. Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Spectroscopic and Group Theoretical Methods in Physics ed Bloch, F. et al, Amsterdam: North-Holland, 1968

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Roberts J.: Skein theory and Turaev-Viro invariants. Topology 34(4), 771–787 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Reprint. Springer Study Edition. Berlin-New York: Springer-Verlag, 1982

  19. Turaev, V.G.: Quantum Invariants of knots and 3-Manifolds. de Gruyter Studies in Mathematics, 18. Berlin: Walter de Gruyter & Co., 1994

  20. Turaev V.G., Viro O.Ya.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    João Faria Martins

  2. Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia, Av do Campo Grande, 376, 1749-024, Lisboa, Portugal

    Aleksandar Miković

Authors
  1. João Faria Martins
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  2. Aleksandar Miković
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Correspondence to João Faria Martins.

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Communicated by A. Connes

Member of the Mathematical Physics Group, University of Lisbon.

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Martins, J.F., Miković, A. Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity. Commun. Math. Phys. 288, 745–772 (2009). https://doi.org/10.1007/s00220-009-0776-6

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  • DOI: https://doi.org/10.1007/s00220-009-0776-6

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