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


SIGMA 8 (2012), 055, 79 pages      arXiv:1112.1961      https://doi.org/10.3842/SIGMA.2012.055
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Spin Foams and Canonical Quantization

Sergei Alexandrov a, b, Marc Geiller c and Karim Noui d, c
a) Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France
b) CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France
c) Laboratoire APC, Université Paris Diderot Paris 7, 75013 Paris, France
d) LMPT, Université François Rabelais, Parc de Grandmont, 37200 Tours, France

Received January 30, 2012, in final form August 12, 2012; Published online August 19, 2012

Abstract
This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possess in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.

Key words: spin foam models; loop quantum gravity; canonical quantization.

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References

  1. Alekseev A.Y., Integrability in the Hamiltonian Chern-Simons theory, St. Petersburg Math. J. 6 (1995), 241-253.
  2. Alekseev A.Y., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern-Simons theory. I, Comm. Math. Phys. 172 (1995), 317-358, hep-th/9403066.
  3. Alekseev A.Y., Grosse H., Schomerus V., Combinatorial quantization of the Hamiltonian Chern-Simons theory. II, Comm. Math. Phys. 174 (1996), 561-604, hep-th/9408097.
  4. Alekseev A.Y., Schomerus V., Representation theory of Chern-Simons observables, Duke Math. J. 85 (1996), 447-510, q-alg/9503016.
  5. 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.
  6. Alesci E., Rovelli C., Complete LQG propagator: difficulties with the Barrett-Crane vertex, Phys. Rev. D 76 (2007), 104012, 22 pages, arXiv:0708.0883.
  7. Alexandrov S., Choice of connection in loop quantum gravity, Phys. Rev. D 65 (2002), 024011, 7 pages, gr-qc/0107071.
  8. Alexandrov S., New vertices and canonical quantization, Phys. Rev. D 82 (2010), 024024, 9 pages, arXiv:1004.2260.
  9. Alexandrov S., Simplicity and closure constraints in spin foam models of gravity, Phys. Rev. D 78 (2008), 044033, 10 pages, arXiv:0802.3389.
  10. Alexandrov S., SO(4,C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter, Classical Quantum Gravity 17 (2000), 4255-4268, gr-qc/0005085.
  11. Alexandrov S., Spin foam model from canonical quantization, Phys. Rev. D 77 (2008), 024009, 15 pages, arXiv:0705.3892.
  12. Alexandrov S., Buffenoir E., Roche P., Plebanski theory and covariant canonical formulation, Classical Quantum Gravity 24 (2007), 2809-2824, gr-qc/0612071.
  13. Alexandrov S., Kádár Z., Timelike surfaces in Lorentz covariant loop gravity and spin foam models, Classical Quantum Gravity 22 (2005), 3491-3509, gr-qc/0501093.
  14. Alexandrov S., Krasnov K., Hamiltonian analysis of non-chiral Plebanski theory and its generalizations, Classical Quantum Gravity 26 (2009), 055005, 10 pages, arXiv:0809.4763.
  15. Alexandrov S., Livine E.R., SU(2) loop quantum gravity seen from covariant theory, Phys. Rev. D 67 (2003), 044009, 15 pages, gr-qc/0209105.
  16. Alexandrov S., Roche P., Critical overview of loops and foams, Phys. Rep. 506 (2011), 41-86, arXiv:1009.4475.
  17. Alexandrov S., Vassilevich D., Area spectrum in Lorentz covariant loop gravity, Phys. Rev. D 64 (2001), 044023, 7 pages, gr-qc/0103105.
  18. Alexandrov S.Y., Vassilevich D.V., Path integral for the Hilbert-Palatini and Ashtekar gravity, Phys. Rev. D 58 (1998), 124029, 13 pages, gr-qc/9806001.
  19. Ashtekar A., Fairhurst S., Willis J.L., Quantum gravity, shadow states and quantum mechanics, Classical Quantum Gravity 20 (2003), 1031-1061, gr-qc/0207106.
  20. Ashtekar A., Husain V., Rovelli C., Samuel J., Smolin L., 2+1 quantum gravity as a toy model for the 3+1 theory, Classical Quantum Gravity 6 (1989), L185-L193.
  21. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  22. Ashtekar A., Lewandowski J., Quantum theory of geometry. I. Area operators, Classical Quantum Gravity 14 (1997), A55-A81, gr-qc/9602046.
  23. Ashtekar A., Lewandowski J., Quantum theory of geometry. II. Volume operators, Adv. Theor. Math. Phys. 1 (1997), 388-429, gr-qc/9711031.
  24. Ashtekar A., Loll R., New loop representations for 2+1 gravity, Classical Quantum Gravity 11 (1994), 2417-2434, gr-qc/9405031.
  25. Baez J.C., Christensen J.D., Egan G., Asymptotics of 10j symbols, Classical Quantum Gravity 19 (2002), 6489-6513, gr-qc/0208010.
  26. Bais F.A., Muller N.M., Schroers B.J., Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity, Nuclear Phys. B 640 (2002), 3-45, hep-th/0205021.
  27. Balachandran A.P., Marmo G., Skagerstam B.S., Stern A., Gauge symmetries and fibre bundles: applications to particle dynamics, Lecture Notes in Physics, Vol. 188, Springer-Verlag, Berlin, 1983.
  28. 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.
  29. Baratin A., Flori C., Thiemann T., The Holst spin foam model via cubulations, arXiv:0812.4055.
  30. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
  31. 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.
  32. Barrett J.W., Crane L., A Lorentzian signature model for quantum general relativity, Classical Quantum Gravity 17 (2000), 3101-3118, gr-qc/9904025.
  33. Barrett J.W., Crane L., Relativistic spin networks and quantum gravity, J. Math. Phys. 39 (1998), 3296-3302, gr-qc/9709028.
  34. Barrett J.W., Dowdall R.J., Fairbairn W.J., Gomes H., Hellmann F., Asymptotic analysis of the Engle-Pereira-Rovelli-Livine four-simplex amplitude, J. Math. Phys. 50 (2009), 112504, 30 pages, arXiv:0902.1170.
  35. 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.
  36. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2009), 155014, 48 pages, arXiv:0803.3319.
  37. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model and Reidemeister torsion, in On Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories, World Scientific Publishing, Singapore, 2006, 2782-2784, gr-qc/0612170.
  38. Barrett J.W., Steele C.M., Asymptotics of relativistic spin networks, Classical Quantum Gravity 20 (2003), 1341-1361, gr-qc/0209023.
  39. Barros e Sá N., Hamiltonian analysis of general relativity with the Immirzi parameter, Internat. J. Modern Phys. D 10 (2001), 261-272, gr-qc/0006013.
  40. Bianchi E., The length operator in loop quantum gravity, Nuclear Phys. B 807 (2009), 591-624, arXiv:0806.4710.
  41. Bojowald M., Perez A., Spin foam quantization and anomalies, Gen. Relativity Gravitation 42 (2010), 877-907, gr-qc/0303026.
  42. Bonzom V., From lattice BF gauge theory to area–angle Regge calculus, Classical Quantum Gravity 26 (2009), 155020, 25 pages, arXiv:0903.0267.
  43. 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.
  44. Bonzom V., Freidel L., The Hamiltonian constraint in 3d Riemannian loop quantum gravity, Classical Quantum Gravity 28 (2011), 195006, 24 pages, arXiv:1101.3524.
  45. Bonzom V., Laddha A., Lessons from toy-models for the dynamics of loop quantum gravity, SIGMA 8 (2012), 009, 50 pages, arXiv:1110.2157.
  46. Bonzom V., Smerlak M., Bubble divergences from cellular cohomology, Lett. Math. Phys. 93 (2010), 295-305, arXiv:1004.5196.
  47. Bonzom V., Smerlak M., Bubble divergences from twisted cohomology, Comm. Math. Phys. 312 (2012), 399-426, arXiv:1008.1476.
  48. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincaré 13 (2012), 185-208, arXiv:1103.3961.
  49. Buffenoir E., Henneaux M., Noui K., Roche P., Hamiltonian analysis of Plebanski theory, Classical Quantum Gravity 21 (2004), 5203-5220, gr-qc/0404041.
  50. Buffenoir E., Noui K., Roche P., Hamiltonian quantization of Chern-Simons theory with SL(2,C) group, Classical Quantum Gravity 19 (2002), 4953-5015, hep-th/0202121.
  51. Capovilla R., Dell J., Jacobson T., Mason L., Self-dual 2-forms and gravity, Classical Quantum Gravity 8 (1991), 41-57.
  52. Carlip S., Quantum gravity in 2+1 dimensions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1998.
  53. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  54. Cianfrani F., Montani G., Towards loop quantum gravity without the time gauge, Phys. Rev. Lett. 102 (2009), 091301, 4 pages, arXiv:0811.1916.
  55. Conrady F., Freidel L., Semiclassical limit of 4-dimensional spin foam models, Phys. Rev. D 78 (2008), 104023, 18 pages, arXiv:0809.2280.
  56. Conrady F., Hnybida J., A spin foam model for general Lorentzian 4-geometries, Classical Quantum Gravity 27 (2010), 185011, 23 pages, arXiv:1002.1959.
  57. Crane L., Perez A., Rovelli C., Perturbative finiteness in spin-foam quantum gravity, Phys. Rev. Lett. 87 (2001), 181301, 4 pages.
  58. De Pietri R., Freidel L., so(4) Pleba\'nski action and relativistic spin-foam model, Classical Quantum Gravity 16 (1999), 2187-2196, gr-qc/9804071.
  59. de Sousa Gerbert P., On spin and (quantum) gravity in 2+1 dimensions, Nuclear Phys. B 346 (1990), 440-472.
  60. Deser S., Jackiw R., Three-dimensional cosmological gravity: dynamics of constant curvature, Ann. Physics 153 (1984), 405-416.
  61. Deser S., Jackiw R., 't Hooft G., Three-dimensional Einstein gravity: dynamics of flat space, Ann. Physics 152 (1984), 220-235.
  62. Ding Y., Rovelli C., The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory, Classical Quantum Gravity 27 (2010), 205003, 11 pages, arXiv:1006.1294.
  63. Dupuis M., Livine E.R., Lifting SU(2) spin networks to projected spin networks, Phys. Rev. D 82 (2010), 064044, 11 pages, arXiv:1008.4093.
  64. Elitzur S., Moore G., Schwimmer A., Seiberg N., Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nuclear Phys. B 326 (1989), 108-134.
  65. Engle J., Han M., Thiemann T., Canonical path integral measures for Holst and Plebanski gravity. I. Reduced phase space derivation, Classical Quantum Gravity 27 (2010), 245014, 29 pages, arXiv:0911.3433.
  66. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  67. Engle J., Pereira R., Rovelli C., Flipped spinfoam vertex and loop gravity, Nuclear Phys. B 798 (2008), 251-290, arXiv:0708.1236.
  68. Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
  69. Fairbairn W.J., Meusburger C., Quantum deformation of two four-dimensional spin foam models, J. Math. Phys. 53 (2012), 022501, 37 pages, arXiv:1012.4784.
  70. Fock V.V., Rosly A.A., Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 67-86, math.QA/9802054.
  71. Foxon T.J., Spin networks, Turaev-Viro theory and the loop representation, Classical Quantum Gravity 12 (1995), 951-964, gr-qc/9408013.
  72. Freidel L., Group field theory: an overview, Internat. J. Theoret. Phys. 44 (2005), 1769-1783, hep-th/0505016.
  73. Freidel L., Geiller M., Ziprick J., Continuous formulation of the loop quantum gravity phase space, arXiv:1110.4833.
  74. Freidel L., Gurau R., Oriti D., Group field theory renormalization in the 3D case: power counting of divergences, Phys. Rev. D 80 (2009), 044007, 20 pages, arXiv:0905.3772.
  75. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  76. Freidel L., Krasnov K., Puzio R., BF description of higher-dimensional gravity theories, Adv. Theor. Math. Phys. 3 (1999), 1289-1324, hep-th/9901069.
  77. 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.
  78. Freidel L., Livine E.R., Ponzano-Regge model revisited. III. Feynman diagrams and effective field theory, Classical Quantum Gravity 23 (2006), 2021-2061, hep-th/0502106.
  79. Freidel L., Livine E.R., Spin networks for noncompact groups, J. Math. Phys. 44 (2003), 1322-1356, hep-th/0205268.
  80. Freidel L., Livine E.R., Rovelli C., Spectra of length and area in (2+1) Lorentzian loop quantum gravity, Classical Quantum Gravity 20 (2003), 1463-1478, gr-qc/0212077.
  81. Freidel L., Louapre D., Asymptotics of 6j and 10j symbols, Classical Quantum Gravity 20 (2003), 1267-1294, hep-th/0209134.
  82. Freidel L., Louapre D., Nonperturbative summation over 3D discrete topologies, Phys. Rev. D 68 (2003), 104004, 16 pages, hep-th/0211026.
  83. Freidel L., Louapre D., Ponzano-Regge model revisited. I. Gauge fixing, observables and interacting spinning particles, Classical Quantum Gravity 21 (2004), 5685-5726, hep-th/0401076.
  84. Freidel L., Louapre D., Ponzano-Regge model revisited. II. Equivalence with Chern-Simons, gr-qc/0410141.
  85. Freyd P., Yetter D., Hoste J., Lickorish W.B.R., Millett K., Ocneanu A., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 239-246.
  86. Geiller M., Lachièze-Rey M., Noui K., A new look at Lorentz-covariant loop quantum gravity, Phys. Rev. D 84 (2011), 044002, 19 pages, arXiv:1105.4194.
  87. Geiller M., Lachièze-Rey M., Noui K., Sardelli F., A Lorentz-covariant connection for canonical gravity, SIGMA 7 (2011), 083, 10 pages, arXiv:1103.4057.
  88. Geiller M., Noui K., Testing the imposition of the spin foam simplicity constraints, Classical Quantum Gravity 29 (2012), 135008, 28 pages, arXiv:1112.1965.
  89. Giulini D., On the configuration space topology in general relativity, Helv. Phys. Acta 68 (1995), 86-111, gr-qc/9301020.
  90. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
  91. Gott III J.R., Closed timelike curves produced by pairs of moving cosmic strings: exact solutions, Phys. Rev. Lett. 66 (1991), 1126-1129.
  92. Grot N., Rovelli C., Moduli-space structure of knots with intersections, J. Math. Phys. 37 (1996), 3014-3021, gr-qc/9604010.
  93. Han M., 4-dimensional spin-foam model with quantum Lorentz group, J. Math. Phys. 52 (2011), 072501, 22 pages, arXiv:1012.4216.
  94. Han M., Canonical path-integral measures for Holst and Plebanski gravity. II. Gauge invariance and physical inner product, Classical Quantum Gravity 27 (2010), 245015, 39 pages, arXiv:0911.3436.
  95. Henneaux M., Slavnov A.A., A note on the path integral for systems with primary and secondary second class constraints, Phys. Lett. B 338 (1994), 47-50, hep-th/9406161.
  96. Holst S., Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996), 5966-5969, gr-qc/9511026.
  97. Horowitz G.T., Exactly soluble diffeomorphism invariant theories, Comm. Math. Phys. 125 (1989), 417-437.
  98. Jones V.F.R., A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), 219-225.
  99. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  100. Kaminski W., Kisielowski M., Lewandowski J., The EPRL intertwiners and corrected partition function, Classical Quantum Gravity 27 (2010), 165020, 15 pages, arXiv:0912.0540.
  101. Koornwinder T.H., Bais F.A., Muller N.M., Tensor product representations of the quantum double of a compact group, Comm. Math. Phys. 198 (1998), 157-186, q-alg/9712042.
  102. Koornwinder T.H., Muller N.M., The quantum double of a (locally) compact group, J. Lie Theory 7 (1997), 101-120, q-alg/9605044.
  103. Labastida J.M.F., Knot invariants and Chern-Simons theory, in European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., Vol. 202, Birkhäuser, Basel, 2001, 467-477, hep-th/0007152.
  104. Labastida J.M.F., Ramallo A.V., Operator formalism for Chern-Simons theories, Phys. Lett. B 227 (1989), 92-102.
  105. Liu L., Montesinos M., Perez A., Topological limit of gravity admitting an SU(2) connection formulation, Phys. Rev. D 81 (2010), 064033, 9 pages, arXiv:0906.4524.
  106. Livine E.R., Projected spin networks for Lorentz connection: linking spin foams and loop gravity, Classical Quantum Gravity 19 (2002), 5525-5541, gr-qc/0207084.
  107. Livine E.R., Ryan J.P., A note on B-observables in Ponzano-Regge 3D quantum gravity, Classical Quantum Gravity 26 (2009), 035013, 19 pages, arXiv:0808.0025.
  108. Livine E.R., Speziale S., Consistently solving the simplicity constraints for spinfoam quantum gravity, Europhys. Lett. 81 (2008), 50004, 6 pages, arXiv:0708.1915.
  109. Magnen J., Noui K., Rivasseau V., Smerlak M., Scaling behavior of three-dimensional group field theory, Classical Quantum Gravity 26 (2009), 185012, 25 pages, arXiv:0906.5477.
  110. Marolf D.M., Loop representations for 2+1 gravity on a torus, Classical Quantum Gravity 10 (1993), 2625-2647, gr-qc/9303019.
  111. Matschull H.J., The phase space structure of multi-particle models in 2+1 gravity, Classical Quantum Gravity 18 (2001), 3497-3560, gr-qc/0103084.
  112. Meusburger C., Noui K., Combinatorial quantisation of the Euclidean torus universe, Nuclear Phys. B 841 (2010), 463-505, arXiv:1007.4615.
  113. Meusburger C., Schroers B.J., Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity, Classical Quantum Gravity 22 (2005), 3689-3724, gr-qc/0505071.
  114. Meusburger C., Schroers B.J., Mapping class group actions in Chern-Simons theory with gauge group G×g*, Nuclear Phys. B 706 (2005), 569-597, hep-th/0312049.
  115. Meusburger C., Schroers B.J., Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity, Classical Quantum Gravity 20 (2003), 2193-2233, gr-qc/0301108.
  116. Meusburger C., Schroers B.J., The quantisation of Poisson structures arising in Chern-Simons theory with gauge group G×g*, Adv. Theor. Math. Phys. 7 (2003), 1003-1043, hep-th/0310218.
  117. Mikovic A., Vojinovic M., Effective action for EPRL/FK spin foam models, J. Phys. Conf. Ser. 360 (2012), 012049, 4 pages, arXiv:1110.6114.
  118. Noui K., Three-dimensional loop quantum gravity: particles and the quantum double, J. Math. Phys. 47 (2006), 102501, 30 pages, gr-qc/0612144.
  119. Noui K., Three-dimensional loop quantum gravity: towards a self-gravitating quantum field theory, Classical Quantum Gravity 24 (2007), 329-360, gr-qc/0612145.
  120. Noui K., Perez A., Three-dimensional loop quantum gravity: coupling to point particles, Classical Quantum Gravity 22 (2005), 4489-4513, gr-qc/0402111.
  121. 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.
  122. Noui K., Perez A., Pranzetti D., Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravityr, J. High Energy Phys. 2011 (2011), no. 10, 036, 22 pages, arXiv:1105.0439.
  123. Noui K., Roche P., Cosmological deformation of Lorentzian spin foam models, Classical Quantum Gravity 20 (2003), 3175-3213, gr-qc/0211109.
  124. Oriti D., The group field theory approach to quantum gravity: some recent results, arXiv:0912.2441.
  125. Perez A., Finiteness of a spinfoam model for Euclidean quantum general relativity, Nuclear Phys. B 599 (2001), 427-434, gr-qc/0011058.
  126. Perez A., Introduction to loop quantum gravity and spin foams, gr-qc/0409061.
  127. 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.
  128. Perez A., Rovelli C., (3+1)-dimensional spin foam model of quantum gravity with spacelike and timelike components, Phys. Rev. D 64 (2001), 064002, 12 pages, gr-qc/0011037.
  129. Perez A., Rovelli C., A spin foam model without bubble divergences, Nuclear Phys. B 599 (2001), 255-282, gr-qc/0006107.
  130. Perez A., Rovelli C., Spin foam model for Lorentzian general relativity, Phys. Rev. D 63 (2001), 041501, 5 pages, gr-qc/0009021.
  131. Plebanski J.F., On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977), 2511-2520.
  132. Ponzano G., Regge T., Semiclassical limit of Racah coefficients, in Spectroscopy and Group Theoretical Methods in Physics, Editors F. Block et al., North Holland, Amsterdam, 1968, 1-58.
  133. Reisenberger M.P., On relativistic spin network vertices, J. Math. Phys. 40 (1999), 2046-2054, gr-qc/9809067.
  134. Reisenberger M.P., Rovelli C., "Sum over surfaces" form of loop quantum gravity, Phys. Rev. D 56 (1997), 3490-3508, gr-qc/9612035.
  135. Reshetikhin N., Turaev V.G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
  136. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  137. Rovelli C., Smolin L., Discreteness of area and volume in quantum gravity, Nuclear Phys. B 442 (1995), 593-619, gr-qc/9411005.
  138. Rovelli C., Speziale S., Lorentz covariance of loop quantum gravity, Phys. Rev. D 83 (2011), 104029, 6 pages, arXiv:1012.1739.
  139. Sahlmann H., Thiemann T., Chern-Simons theory, Stokes' theorem, and the Duflo map, J. Geom. Phys. 61 (2011), 1104-1121, arXiv:1101.1690.
  140. Samuel J., Is Barbero's Hamiltonian formulation a gauge theory of Lorentzian gravity?, Classical Quantum Gravity 17 (2000), L141-L148, gr-qc/0005095.
  141. Thiemann T., A length operator for canonical quantum gravity, J. Math. Phys. 39 (1998), 3372-3392, gr-qc/9606092.
  142. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  143. 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.
  144. Thiemann T., Quantum spin dynamics (QSD). VII. Symplectic structures and continuum lattice formulations of gauge field theories, Classical Quantum Gravity 18 (2001), 3293-3338, hep-th/0005232.
  145. Turaev V., Virelizier A., On two approaches to 3-dimensional TQFTs, arXiv:1006.3501.
  146. Turaev V.G., Viro O.Y., State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992), 865-902.
  147. Vassiliev V.A., Cohomology of knot spaces, in Theory of Singularities and its Applications, Adv. Soviet Math., Vol. 1, Amer. Math. Soc., Providence, RI, 1990, 23-69.
  148. Witten E., 2+1-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
  149. Witten E., Analytic continuation of Chern-Simons theory, in Chern-Simons Gauge Theory: 20 Years After, AMS/IP Stud. Adv. Math., Vol. 50, Amer. Math. Soc., Providence, RI, 2011, 347-446, arXiv:1001.2933.
  150. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.

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