Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 067, 32 pages      arXiv:1401.5819      https://doi.org/10.3842/SIGMA.2014.067
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data

Daniele Oriti a and Matti Raasakka b
a) Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam, Germany
b) LIPN, Institut Galilée, CNRS UMR 7030, Université Paris 13, Sorbonne Paris Cité, 99 av. Clement, 93430 Villetaneuse, France

Received February 04, 2014, in final form June 14, 2014; Published online June 26, 2014

Abstract
We apply the non-commutative Fourier transform for Lie groups to formulate the non-commutative metric representation of the Ponzano-Regge spin foam model for 3d quantum gravity. The non-commutative representation allows to express the amplitudes of the model as a first order phase space path integral, whose properties we consider. In particular, we study the asymptotic behavior of the path integral in the semi-classical limit. First, we compare the stationary phase equations in the classical limit for three different non-commutative structures corresponding to the symmetric, Duflo and Freidel-Livine-Majid quantization maps. We find that in order to unambiguously recover discrete geometric constraints for non-commutative metric boundary data through the stationary phase method, the deformation structure of the phase space must be accounted for in the variational calculus. When this is understood, our results demonstrate that the non-commutative metric representation facilitates a convenient semi-classical analysis of the Ponzano-Regge model, which yields as the dominant contribution to the amplitude the cosine of the Regge action in agreement with previous studies. We also consider the asymptotics of the ${\rm SU}(2)$ $6j$-symbol using the non-commutative phase space path integral for the Ponzano-Regge model, and explain the connection of our results to the previous asymptotic results in terms of coherent states.

Key words: Ponzano-Regge model; non-commutative representation; asymptotic analysis.

pdf (607 kb)   tex (121 kb)

References

  1. Alexandrov S., Geiller M., Noui K., Spin foams and canonical quantization, SIGMA 8 (2012), 055, 79 pages, arXiv:1112.1961.
  2. Baez J.C., An introduction to spin foam models of $BF$ theory and quantum gravity, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
  3. Bahr B., Dittrich B., (Broken) gauge symmetries and constraints in Regge calculus, Classical Quantum Gravity 26 (2009), 225011, 34 pages, arXiv:0905.1670.
  4. Baratin A., Dittrich B., Oriti D., Tambornino J., Non-commutative flux representation for loop quantum gravity, Classical Quantum Gravity 28 (2011), 175011, 19 pages, arXiv:1004.3450.
  5. Baratin A., Girelli F., Oriti D., Diffeomorphisms in group field theories, Phys. Rev. D 83 (2011), 104051, 22 pages, arXiv:1101.0590.
  6. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
  7. Baratin A., Oriti D., Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model, New J. Phys. 13 (2011), 125011, 28 pages, arXiv:1108.1178.
  8. Baratin A., Oriti D., Group field theory and simplicial gravity path integrals: a model for Holst-Plebański gravity, Phys. Rev. D 85 (2012), 044003, 15 pages, arXiv:1111.5842.
  9. Barrett J.W., Crane L., Relativistic spin networks and quantum gravity, J. Math. Phys. 39 (1998), 3296-3302, gr-qc/9709028,.
  10. Barrett J.W., Dowdall R.J., Fairbairn W.J., Hellmann F., Pereira R., Lorentzian spin foam amplitudes: graphical calculus and asymptotics, Classical Quantum Gravity 27 (2010), 165009, 34 pages, arXiv:0907.2440.
  11. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2011), 155014, 48 pages, arXiv:0803.3319.
  12. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  13. Caselle M., D'Adda A., Magnea L., Regge calculus as a local theory of the Poincaré group, Phys. Lett. B 232 (1989), 457-461.
  14. Chaichian M., Demichev A., Path integrals in physics. Vol. I. Stochastic processes and quantum mechanics, Series in Mathematical and Computational Physics, Institute of Physics Publishing, Bristol, 2001.
  15. Conrady F., Freidel L., Semiclassical limit of 4-dimensional spin foam models, Phys. Rev. D 78 (2008), 104023, 18 pages, arXiv:0809.2280.
  16. Dittrich B., Guedes C., Oriti D., On the space of generalized fluxes for loop quantum gravity, Classical Quantum Gravity 30 (2013), 055008, 24 pages, arXiv:1205.6166.
  17. Dowdall R.J., Gomes H., Hellmann F., Asymptotic analysis of the Ponzano-Regge model for handlebodies, J. Phys. A: Math. Theor. 43 (2010), 115203, 27 pages, arXiv:0909.2027.
  18. Dupuis M., Girelli F., Livine E., Spinors and Voros star-product for group field theory: first contact, Phys. Rev. D 86 (2012), 105034, 18 pages, arXiv:1107.5693.
  19. Dupuis M., Livine E.R., Holomorphic simplicity constraints for 4D spinfoam models, Classical Quantum Gravity 28 (2011), 215022, 32 pages, arXiv:1104.3683.
  20. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  21. Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
  22. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, \mboxhep-th/0505016.
  23. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  24. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
  25. Freidel L., Majid S., Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity, Classical Quantum Gravity 25 (2008), 045006, 37 pages, hep-th/0601004.
  26. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  27. Guedes C., Oriti D., Raasakka M., Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups, J. Math. Phys. 54 (2013), 083508, 31 pages, arXiv:1301.7750.
  28. Han M., On spinfoam models in large spin regime, Classical Quantum Gravity 31 (2013), 015004, 21 pages, arXiv:1304.5627.
  29. Han M., Semiclassical analysis of spinfoam model with a small Barbero-Immirzi parameter, Phys. Rev. D 88 (2013), 044051, 13 pages, arXiv:1304.5628.
  30. Han M., Krajewski T., Path integral representation of Lorentzian spinfoam model, asymptotics and simplicial geometries, Classical Quantum Gravity 31 (2014), 015009, 34 pages, arXiv:1304.5626.
  31. Han M., Zhang M., Asymptotics of the spin foam amplitude on simplicial manifold: Euclidean theory, Classical Quantum Gravity 29 (2012), 165004, 40 pages, arXiv:1109.0500.
  32. Han M., Zhang M., Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory, Classical Quantum Gravity 30 (2013), 165012, 57 pages, arXiv:1109.0499.
  33. Hellmann F., Kamiński W., Geometric asymptotics for spin foam lattice gauge gravity on arbitrary triangulations, arXiv:1210.5276.
  34. Hellmann F., Kamiński W., Holonomy spin foam models: asymptotic geometry of the partition function, J. High Energy Phys. 2013 (2013), no. 10, 165, 63 pages, arXiv:1307.1679.
  35. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  36. Kamiński W., Steinhaus S., Coherent states, $6j$ symbols and properties of the next to leading order asymptotic expansions, J. Math. Phys. 54 (2013), 121703, 58 pages, arXiv:1307.5432.
  37. Kawamoto N., Nielsen H.B., Lattice gauge gravity, Phys. Rev. D 43 (1991), 1150-1156.
  38. Magliaro E., Perini C., Regge gravity from spinfoams, Internat. J. Modern Phys. D 22 (2013), 1350001, 21 pages, arXiv:1105.0216.
  39. Majid S., Schroers B.J., $q$-deformation and semidualization in 3D quantum gravity, J. Phys. A: Math. Gen. 42 (2009), 425402, 40 pages, arXiv:0806.2587.
  40. Mizoguchi S., Tada T., Three-dimensional gravity from the Turaev-Viro invariant, Phys. Rev. Lett. 68 (1992), 1795-1798, hep-th/9110057.
  41. Noui K., Perez A., Three-dimensional loop quantum gravity: physical scalar product and spin-foam models, Classical Quantum Gravity 22 (2005), 1739-1761, gr-qc/0402110.
  42. Noui K., Perez A., Pranzetti D., Canonical quantization of non-commutative holonomies in $2+1$ loop quantum gravity, J. High Energy Phys. 2011 (2011), no. 10, 036, 21 pages, arXiv:1105.0439.
  43. Noui K., Perez A., Pranzetti D., Non-commutative holonomies in $2+1$ LQG and Kauffman's brackets, J. Phys. Conf. Ser. 360 (2012), 012040, 4 pages, arXiv:1112.1825.
  44. Oriti D., The microscopic dynamics of quantum space as a group field theory, in Foundations of Space and Time: Reflections on Quantum Gravity, Editors J. Murugan, A. Weltman, G. Ellis, Cambridge University Press, Cambridge, 2012, 257-320, arXiv:1110.5606.
  45. Oriti D., Raasakka M., Quantum mechanics on ${\rm SO}(3)$ via non-commutative dual variables, Phys. Rev. D 84 (2011), 025003, 18 pages, arXiv:1103.2098.
  46. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  47. Perez A., The new spin foam models and quantum gravity, Papers Phys. 4 (2012), 040004, 37 pages, arXiv:1205.0911.
  48. Perez A., The spin foam approach to quantum gravity, Living Rev. Relativ. 16 (2013), 3, 128 pages, arXiv:1205.2019.
  49. Ponzano G., Regge T., Semiclassical limit of Racah coefficients, in Spectroscopy and Group Theoretical Methods in Physics, Editor F. Block, North Holland, Amsterdam, 1968, 1-58.
  50. Pranzetti D., Turaev-Viro amplitudes from $2+1$ loop quantum gravity, Phys. Rev. D 89 (2014), 084058, 14 pages, arXiv:1402.2384.
  51. Regge T., Williams R.M., Discrete structures in gravity, J. Math. Phys. 41 (2000), 3964-3984, gr-qc/0012035.
  52. Reisenberger M.P., Rovelli C., ''Sum over surfaces'' form of loop quantum gravity, Phys. Rev. D 56 (1997), 3490-3508, gr-qc/9612035.
  53. Reshetikhin N., Turaev V.G., Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
  54. Rovelli C., Basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum-gravity model is the loop representation basis, Phys. Rev. D 48 (1993), 2702-2707, hep-th/9304164.
  55. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  56. Sahlmann H., Thiemann T., Chern-Simons theory, Stokes' theorem, and the Duflo map, J. Geom. Phys. 61 (2011), 1104-1121, arXiv:1101.1690.
  57. Sahlmann H., Thiemann T., Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map, Phys. Rev. Lett. 108 (2012), 111303, 5 pages, arXiv:1109.5793.
  58. Schroers B.J., Combinatorial quantization of Euclidean gravity in three dimensions, in Quantization of Singular Symplectic Quotients, Progr. Math., Vol. 198, Editors N. Landsman, M. Pflaum, M. Schlichenmaier, Birkhäuser, Basel, 2001, 307-327, math.QA/0006228.
  59. Sengupta A.N., The volume measure for flat connections as limit of the Yang-Mills measure, J. Geom. Phys. 47 (2003), 398-426.
  60. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  61. Turaev V.G., Viro O.Y., State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), 865-902.
  62. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.
  63. Witten E., On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153-209.

Previous article  Next article   Contents of Volume 10 (2014)