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


SIGMA 8 (2012), 009, 50 pages      arXiv:1110.2157      http://dx.doi.org/10.3842/SIGMA.2012.009
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity

Valentin Bonzom a and Alok Laddha b
a) Perimeter Institute for Theoretical Physics, 31 Caroline St. N, ON N2L 2Y5, Waterloo, Canada
b) Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, PA 16802-6300, USA

Received October 11, 2011, in final form February 24, 2012; Published online March 07, 2012

Abstract
We review some approaches to the Hamiltonian dynamics of (loop) quantum gravity, the main issues being the regularization of the Hamiltonian and the continuum limit. First, Thiemann's definition of the quantum Hamiltonian is presented, and then more recent approaches. They are based on toy models which provide new insights into the difficulties and ambiguities faced in Thiemann's construction. The models we use are parametrized field theories, the topological BF model of which a special case is three-dimensional gravity which describes quantum flat space, and Regge lattice gravity.

Key words: Hamiltonian constraint; loop quantum gravity; parametrized field theories; topological BF theory; discrete gravity.

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References

  1. Alesci E., Noui K., Sardelli F., Spin-foam models and the physical scalar product, Phys. Rev. D 78 (2008), 104009, 16 pages, arXiv:0807.3561.
  2. Alesci E., Rovelli C., Regularization of the Hamiltonian constraint compatible with the spinfoam dynamics, Phys. Rev D 82 (2010), 044007, 17 pages, arXiv:1005.0817.
  3. Alesci E., Thiemann T., Zipfel A., Linking covariant and canonical LQG: new solutions to the Euclidean scalar constraint, arXiv:1109.1290.
  4. Ambjørn J., Durhuus B., Jónsson T., Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Phys. Lett. A 6 (1991), 1133-1146.
  5. Anderson R.W., Aquilanti V., Marzuoli A., 3nj morphogenesis and semiclassical disentangling, J. Phys. Chem. A 113 (2009), 15106-15117, arXiv:1001.4386.
  6. Aquilanti V., Bitencourt A.C.P., da S. Ferreira C., Marzuoli A., Ragni M., Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity, Phys. Scr. 78 (2008), 058103, 7 pages, arXiv:0901.1074.
  7. Aquilanti V., Haggard H.M., Hedeman A., Jeevanjee N., Littlejohn R., Yu L., Semiclassical mechanics of the Wigner 6j-symbol, arXiv:1009.2811.
  8. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  9. Ashtekar A., Pawlowski T., Singh P., Quantum nature of the big bang: improved dynamics, Phys. Rev. D 74 (2006), 084003, 23 pages, gr-qc/0607039.
  10. 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, 25-93, gr-qc/9905087.
  11. Baez J.C., Perez A., Quantization of strings and branes coupled to BF theory, Adv. Theor. Math. Phys. 11 (2007), 451-469, gr-qc/0605087.
  12. Baez J.C., Wise D.K., Crans A.S., Exotic statistics for strings in 4D BF theory, Adv. Theor. Math. Phys. 11 (2007), 707-749, gr-qc/0603085.
  13. Bahr B., Dittrich B., (Broken) gauge symmetries and constraints in Regge calculus, Classical Quantum Gravity 26 (2009), 225011, 34 pages, arXiv:0905.1670.
  14. Bahr B., Dittrich B., Improved and perfect actions in discrete gravity, Phys. Rev. D 80 (2009), 124030, 15 pages, arXiv:0907.4323.
  15. Bahr B., Dittrich B., He S., Coarse graining free theories with gauge symmetries: the linearized case, New J. Phys. 13 (2011), 045009, 34 pages, arXiv:1011.3667.
  16. Bahr B., Dittrich B., Ryan J.P., Spin foam models with finite groups, arXiv:1103.6264.
  17. Bahr B., Dittrich B., Steinhaus S., Perfect discretization of reparametrization invariant path integrals, Phys. Rev. D 83 (2011), 19 pages, arXiv:1101.4775.
  18. Baratin A., Girelli F., Oriti D., Diffeomorphisms in group field theories, Phys. Rev. D 83 (2011), 104051, 22 pages, arXiv:1101.0590.
  19. Barrett J.W., Crane L., An algebraic interpretation of the Wheeler-DeWitt equation, Classical Quantum Gravity 14 (1997), 2113-2121, gr-qc/9609030.
  20. Barrett J.W., Dowdall R.J., Fairbairn W.J., Gomes H., Hellmann F., Pereira R., Asymptotics of 4d spin foam models, Gen. Relativity Gravitation 43 (2011), 2421-2436, arXiv:1003.1886.
  21. Barrett J.W., Fairbairn W.J., Hellmann F., Quantum gravity asymptotics from the SU(2) 15j-symbol, Internat. J. Modern Phys. A 25 (2010), 2897-2916, arXiv:0912.4907.
  22. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2011), 155014, 48 pages, arXiv:0803.3319.
  23. Bergeron M., Semenoff G.W., Szabo R.J., Canonical BF-type topological field theory and fractional statistics of strings, Nuclear Phys. B 437 (1995), 695-721, hep-th/9407020.
  24. Blau M., Thompson G., A new class of topological field theories and the Ray-Singer torsion, Phys. Lett. B 228 (1989), 64-68.
  25. Blau M., Thompson G., Topological gauge theories of antisymmetric tensor fields, Ann. Physics 205 (1991), 130-172.
  26. Blohmann C., Fernandes M.C.B., Weinstein A., Groupoid symmetry and constraints in general relativity, arXiv:1003.2857.
  27. Bonzom V., Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex, Phys. Rev. D 84 (2011), 024009, 13 pages, arXiv:1101.1615.
  28. Bonzom V., Fleury P., Asymptotics of Wigner 3nj-symbols with small and large angular momenta: an elementary method, arXiv:1108.1569.
  29. Bonzom V., Freidel L., The Hamiltonian constraint in 3d Riemannian loop quantum gravity, Classical Quantum Gravity 28 (2011), 195006, 24 pages, arXiv:1101.3524.
  30. Bonzom V., Gurau R., Riello A., Rivasseau V., Critical behavior of colored tensor models in the large N limit, Nuclear Phys. B 853 (2011), 174-195, arXiv:1105.3122.
  31. Bonzom V., Livine E.R., Yet another recursion relation for the 6j-symbol, arXiv:1103.3415.
  32. Bonzom V., Livine E.R., Smerlak M., Speziale S., Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model, Nuclear Phys. B 804 (2008), 507-526, arXiv:0802.3983.
  33. Bonzom V., Livine E.R., Speziale S., Recurrence relations for spin foam vertices, Classical Quantum Gravity 27 (2010), 125002, 32 pages, arXiv:0911.2204.
  34. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, arXiv:1103.3961.
  35. Bonzom V., Smerlak M., Bubble divergences from cellular cohomology, Lett. Math. Phys. 93 (2010), 295-305, arXiv:1004.5196.
  36. Bonzom V., Smerlak M., Bubble divergences from twisted cohomology, arXiv:1008.1476.
  37. Bonzom V., Smerlak M., Gauge symmetries in spinfoam gravity: the case for 'cellular quantization', arXiv:1201.4996.
  38. Carfora M., Marzuoli A., Rasetti M., Quantum tetrahedra, J. Phys. Chem. A 113 (2009), 15376-15383, arXiv:1001.4402.
  39. Cattaneo A.S., Cotta-Ramusino P., Fröhlich J., Martellini M., Topological BF theories in 3 and 4 dimensions, J. Math. Phys. 36 (1995), 6137-6160, hep-th/9505027.
  40. Cattaneo A.S., Cotta-Ramusino P., Fucito F., Martellini M., Rinaldi M., Tanzini A., Zeni M., Four-dimensional Yang-Mills theory as a deformation of topological BF theory, Comm. Math. Phys. 197 (1998), 571-621, hep-th/9705123.
  41. Constantinidis C.P., Piguet O., Gieres F., Sarandy M.S., On the symmetries of BF models and their relation with gravity, J. High Energy Phys. 2002 (2002), no. 1, 017, 25 pages, hep-th/0111273.
  42. David F., A model of random surfaces with nontrivial critical behaviour, Nuclear Phys. B 257 (1985), 543-576.
  43. De Pietri R., Freidel L., so(4) Plebanski action and relativistic spin-foam model, Classical Quantum Gravity 16 (1999), 2187-2196, gr-qc/9804071.
  44. Di Francesco P., Ginsparg P., Zinn-Justin J., 2D gravity and random matrices, Phys. Rep. 254 (1995), no. 1-2, 133 pages, hep-th/9306153.
  45. Dittrich B., Eckert F.C., Martin-Benito M., Coarse graining methods for spin net and spin foam models, arXiv:1109.4927.
  46. Dittrich B., Höhn P.A., From covariant to canonical formulations of discrete gravity, Classical Quantum Gravity 27 (2010), 155001, 37 pages, arXiv:0912.1817.
  47. Dittrich B., Ryan J.P., Phase space descriptions for simplicial 4D geometries, Classical Quantum Gravity 28 (2011), 065006, 34 pages, arXiv:0807.2806.
  48. Dittrich B., Ryan J.P., Simplicity in simplicial phase space, Phys. Rev. D 82 (2010), 064026, 19 pages, arXiv:1006.4295.
  49. Dittrich B., Thiemann T., Testing the master constraint programme for loop quantum gravity. I. General framework, Classical Quantum Gravity 23 (2006), 1025-1065, gr-qc/0411138.
  50. Dittrich B., Thiemann T., Testing the master constraint programme for loop quantum gravity. II. Finite-dimensional systems, Classical Quantum Gravity 23 (2006), 1067-1088, gr-qc/0411139.
  51. Dittrich B., Thiemann T., Testing the master constraint programme for loop quantum gravity. III. SL(2,R) models, Classical Quantum Gravity 23 (2006), 1089-1120, gr-qc/0411140.
  52. Dittrich B., Thiemann T., Testing the master constraint programme for loop quantum gravity. IV. Free field theories, Classical Quantum Gravity 23 (2006), 1121-1142, gr-qc/0411141.
  53. Dittrich B., Thiemann T., Testing the master constraint programme for loop quantum gravity. V. Interacting field theories, Classical Quantum Gravity 23 (2006), 1143-1162, gr-qc/0411142.
  54. Dupuis M., Livine E.R., Pushing the asymptotics of the 6j-symbol further, Phys. Rev. D 80 (2009), 024035, 14 pages, arXiv:0905.4188.
  55. Dupuis M., Livine E.R., The 6j-symbol: recursion, correlations and asymptotics, Classical Quantum Gravity 27 (2010), 135003, 15 pages, arXiv:0910.2425.
  56. Fairbairn W.J., Perez A., Extended matter coupled to BF theory, Phys. Rev. D 78 (2008), 024013, 21 pages, arXiv:0709.4235.
  57. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  58. Freidel L., Krasnov K., Puzio R., BF description of higher-dimensional gravity theories, Adv. Theor. Math. Phys. 3 (1999), 1289-1324, hep-th/9901069.
  59. Freidel L., Louapre D., Ponzano-Regge model revisited. I. Gauge fixing, observables and interacting spinning particles, Classical Quantum Gravity 21 (2004), 5685-5726, arXiv:hep-th/0401076.
  60. Freidel L., Speziale S., On the relations between gravity and BF theories, arXiv:1201.4247.
  61. Freidel L., Speziale S., Twisted geometries: a geometric parametrisation of SU(2) phase space, Phys. Rev. D 82 (2010), 084040, 16 pages, arXiv:1001.2748.
  62. Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, hep-th/0501191.
  63. Frohman C., Kania-Bartoszynska J., Dubois' torsion, A-polynomial and quantum invariants, arXiv:1101.2695.
  64. Gambini R., Lewandowski J., Marolf D., Pullin J., On the consistency of the constraint algebra in spin network quantum gravity, Internat. J. Modern Phys. D 7 (1998), 97-109, arXiv:gr-qc/9710018.
  65. Gieres F., Grimstrup J.M., Nieder H., Pisar T., Schweda M., Topological field theories and their symmetries within the Batalin-Vilkovisky framework, Phys. Rev. D 66 (2002), 025027, 14 pages, hep-th/0111258.
  66. Giulini D., Marolf D., On the generality of refined algebraic quantization, Classical Quantum Gravity 16 (1999), 2479-2488, gr-qc/9812024.
  67. Gross M., Tensor models and simplicial quantum gravity in >2-D, Nuclear Phys. B Proc. Suppl. 25A (1992), 144-149.
  68. Gurau R., The complete 1/N expansion of colored tensor models in arbitrary dimension, arXiv:1102.5759.
  69. Gurau R., Ryan J.P., Colored tensor models - a review, arXiv:1109.4812.
  70. Hájícek P., Isham C.J., The symplectic geometry of a parametrized scalar field on a curved background, J. Math. Phys. 37 (1996), 3505-3521, gr-qc/9510028.
  71. Han M., Thiemann T., On the relation between operator constraint, master constraint, reduced phase space and path integral quantization, Classical Quantum Gravity 27 (2010), 225019, 46 pages, arXiv:0911.3428.
  72. Horowitz G.T., Exactly soluble diffeomorphism invariant theories, Comm. Math. Phys. 125 (1989), 417-437.
  73. Jeffrey L.C., Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992), 563-604.
  74. Kazakov V.A., Bilocal regularization of models of random surfaces, Phys. Lett. B 150 (1985), 282-284.
  75. Kitaev A.Y., Fault-tolerant quantum computation by anyons, Ann. Physics 303 (2003), 2-30, quant-ph/9707021.
  76. Kuchar K., Parametrized scalar field on R×S1: dynamical pictures, spacetime diffeomorphisms, and conformal isometries, Phys. Rev. D 39 (1989), 1579-1593.
  77. Laddha A., Varadarajan M., Hamiltonian constraint in polymer parametrized field theory, Phys. Rev D 83 (2011), 025019, 27 pages, arXiv:1011.2463.
  78. Laddha A., Varadarajan M., Polymer quantization of the free scalar field and its classical limit, Classical Quantum Gravity 27 (2010), 175010, 45 pages, arXiv:1001.3505.
  79. Laddha A., Varadarajan M., The diffeomorphism constraint operator in loop quantum gravity, Classical Quantum Gravity 28 (2011), 195010, 29 pages, arXiv:1105.0636.
  80. Levin M.A., Wen X.G., String-net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005), 045110, 21 pages, cond-mat/0404617.
  81. Lewandowski J., Marolf D., Loop constraints: a habitat and their algebra, Internat. J. Modern Phys. D 7 (1998), 299-330, gr-qc/9710016.
  82. Lewandowski J., Okolów A., Sahlmann H., Thiemann T., Uniqueness of diffeomorphism invariant states on holonomy-flux algebras, Comm. Math. Phys. 267 (2006), 703-733, gr-qc/0504147.
  83. Livine E.R., Speziale S., New spinfoam vertex for quantum gravity, Phys. Rev. D 76 (2007), 084028, 14 pages, arXiv:0705.0674.
  84. Livine E.R., Tambornino J., Spinor representation for loop quantum gravity, J. Math. Phys. 53 (2012), 012503, 33 pages, arXiv:1105.3385.
  85. Lucchesi C., Piguet O., Sorella S.P., Renormalization and finiteness of topological BF theories, Nuclear Phys. B 395 (1993), 325-353, hep-th/9208047.
  86. Maggiore N., Sorella S.P., Perturbation theory for antisymmetric tensor fields in four dimensions, Internat. J. Modern Phys. A 8 (1993), 929-945, hep-th/9204044.
  87. Nicolai H., Peeters K., Zamaklar M., Loop quantum gravity: an outside view, Classical Quantum Gravity 22 (2005), R193-R247, hep-th/0501114.
  88. 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.
  89. Noui K., Perez A., Pranzetti D., Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity, arXiv:1105.0439.
  90. Ooguri H., Partition functions and topology-changing amplitudes in the three-dimensional lattice gravity of Ponzano and Regge, Nuclear Phys. B 382 (1992), 276-304, hep-th/9112072.
  91. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  92. Oriti D., The group field theory approach to quantum gravity: some recent results, arXiv:0912.2441.
  93. Perez A., Introduction to loop quantum gravity and spin foams, gr-qc/0409061.
  94. Perez A., Regularization ambiguities in loop quantum gravity, Phys. Rev. D 73 (2006), 044007, 18 pages, gr-qc/0509118.
  95. Perez A., Pranzetti D., On the regularization of the constraint algebra of quantum gravity in 2+1 dimensions with a nonvanishing cosmological constant, Classical Quantum Gravity 27 (2010), 145009, 20 pages, arXiv:1001.3292.
  96. Ponzano G., Regge T., Semi-classical limit of Racah coefficients, in Spectroscopic and Group Theoretical Methods in Physics, Editor F. Bloch, North-Holland, Amsterdam, 1968, 1-58.
  97. Roberts J., Classical 6j-symbols and the tetrahedron, Geom. Topol. 3 (1999), 21-66, math-ph/9812013.
  98. Rovelli C., A new look at loop quantum gravity, Classical Quantum Gravity 28 (2011), 114005, 24 pages, arXiv:1004.1780.
  99. Rovelli C., Discretizing parametrized systems: the magic of Ditt-invariance, arXiv:1107.2310.
  100. Rovelli C., Speziale S., On the geometry of loop quantum gravity on a graph, Phys. Rev. D 82 (2010), 044018, 6 pages, arXiv:1005.2927.
  101. Rozansky L., A large k asymptotics of Witten's invariant of Seifert manifolds, Comm. Math. Phys. 171 (1995), 279-322, hep-th/9303099.
  102. Sasakura N., Tensor model for gravity and orientability of manifold, Modern Phys. Lett. A 6 (1991), 2613-2623.
  103. Schulten K., Gordon R.G., Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta, J. Math. Phys. 16 (1975), 1971-1988.
  104. Smolin L., The classical limit and the form of the Hamiltonian constraint in nonperturbative quantum general relativity, gr-qc/9609034.
  105. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, gr-qc/0110034.
  106. Thiemann T., Quantum spin dynamics (QSD), Classical Quantum Gravity 15 (1998), 839-873, gr-qc/9606089.
  107. Thiemann T., Quantum spin dynamics (QSD). II. The kernel of the Wheeler-DeWitt constraint operator, Classical Quantum Gravity 15 (1998), 875-905, gr-qc/9606090.
  108. Thiemann T., Quantum spin dynamics (QSD). III. Quantum constraint algebra and physical scalar product in quantum general relativity, Classical Quantum Gravity 15 (1998), 1207-1247, gr-qc/9705017.
  109. Thiemann T., Quantum spin dynamics (QSD). IV. 2+1 Euclidean quantum gravity as a model to test 3+1 Lorentzian quantum gravity, Classical Quantum Gravity 15 (1998), 1249-1280, gr-qc/9705018.
  110. Thiemann T., Quantum spin dynamics. VIII. The master constraint, Classical Quantum Gravity 23 (2006), 2249-2265, gr-qc/0510011.
  111. Thiemann T., The Phoenix Project: master constraint programme for loop quantum gravity, Classical Quantum Gravity 23 (2006), 2211-2247, gr-qc/0305080.
  112. Varshalovich D.A., Moskalev A.N., Khersonskii V.K., Quantum theory of angular momentum, World Scientific Publishing Co. Inc., Teaneck, NJ, 1988.
  113. Witten E., 2+1 dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
  114. Witten E., On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), 153-209.
  115. Witten E., Topology-changing amplitudes in (2+1)-dimensional gravity, Nuclear Phys. B 323 (1989), 113-140.
  116. Yu L., Asymptotic limits of the Wigner 15j-symbol with small quantum numbers, arXiv:1104.3641.
  117. Yu L., Semiclassical analysis of the Wigner 12j symbol with one small angular momentum, Phys. Rev. A 84 (2011), 022101, 13 pages, arXiv:1104.3275.
  118. Yu L., Littlejohn R.G., Semiclassical analysis of the Wigner 9j symbol with small and large angular momenta, Phys. Rev. A 83 (2011), 052114, 16 pages, arXiv:1104.1499.
  119. Yutsis A.P., Levinson I.B., Vanagas V.V., Mathematical apparatus of the theory of angular momentum, Israel Program for Scientific Translations, Jerusalem, 1962.

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