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


SIGMA 9 (2013), 008, 22 pages      arXiv:1207.4596      https://doi.org/10.3842/SIGMA.2013.008
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

The Construction of Spin Foam Vertex Amplitudes

Eugenio Bianchi a and Frank Hellmann b
a) Perimeter Institute for Theoretical Physics, Canada
b) Max Planck Institute for Gravitational Physics (AEI), Germany

Received July 20, 2012, in final form January 27, 2013; Published online January 31, 2013

Abstract
Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine.

Key words: spin foam models; discrete quantum gravity; generalized lattice gauge theory.

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References

  1. Ambjørn J., Burda Z., Jurkiewicz J., Kristjansen C.F., Quantum gravity represented as dynamical triangulations, Acta Phys. Polon. B 23 (1992), 991-1030.
  2. Ambjørn J., Jurkiewicz J., Loll R., Causal dynamical triangulations and the quest for quantum gravity, arXiv:1004.0352.
  3. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  4. Baez J.C., Spin foam models, Classical Quantum Gravity 15 (1998), 1827-1858, gr-qc/9709052.
  5. Baez J.C., Christensen J.D., Egan G., Asymptotics of 10j symbols, Classical Quantum Gravity 19 (2002), 6489-6513, gr-qc/0208010.
  6. Bahr B., Measure theoretic regularisation of BF theory, in preparation.
  7. Bahr B., Dittrich B., Breaking and restoring of diffeomorphism symmetry in discrete gravity, arXiv:0909.5688.
  8. Bahr B., Dittrich B., (Broken) gauge symmetries and constraints in Regge calculus, Classical Quantum Gravity 26 (2009), 225011, 34 pages, arXiv:0905.1670.
  9. Bahr B., Dittrich B., Improved and perfect actions in discrete gravity, Phys. Rev. D 80 (2009), 124030, 15 pages, arXiv:0907.4323.
  10. Bahr B., Dittrich B., Regge calculus from a new angle, New J. Phys. 12 (2010), 033010, 10 pages, arXiv:0907.4325.
  11. Bahr B., Dittrich B., Steinhaus S., Perfect discretization of reparametrization invariant path integrals, Phys. Rev. D 83 (2011), 105026, 19 pages, arXiv:1101.4775.
  12. Bahr B., Hellmann F., Kamiński W., Kisielowski M., Lewandowski J., Operator spin foam models, Classical Quantum Gravity 28 (2011), 105003, 23 pages, arXiv:1010.4787.
  13. Baratin A., Flori C., Thiemann T., The Holst spin foam model via cubulations, New J. Phys. 14 (2012), 103054, 30 pages, arXiv:0812.4055.
  14. Baratin A., Girelli F., Oriti D., Diffeomorphisms in group field theories, Phys. Rev. D 83 (2011), 104051, 22 pages, arXiv:1101.0590.
  15. 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.
  16. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
  17. 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.
  18. Barrett J.W., Crane L., A Lorentzian signature model for quantum general relativity, Classical Quantum Gravity 17 (2000), 3101-3118, gr-qc/9904025.
  19. Barrett J.W., Crane L., Relativistic spin networks and quantum gravity, J. Math. Phys. 39 (1998), 3296-3302, gr-qc/9709028.
  20. 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.
  21. 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.
  22. 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.
  23. Barrett J.W., Naish-Guzman I., The Ponzano-Regge model, Classical Quantum Gravity 26 (2011), 155014, 48 pages, arXiv:0803.3319.
  24. Ben Geloun J., Ward-Takahashi identities for the colored Boulatov model, J. Phys. A: Math. Theor. 44 (2011), 415402, 30 pages, arXiv:1106.1847.
  25. Ben Geloun J., Bonzom V., Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function, Internat. J. Theoret. Phys. 50 (2011), 2819-2841, arXiv:1101.4294.
  26. Ben Geloun J., Gurau R., Rivasseau V., EPRL/FK group field theory, Europhys. Lett. 92 (2010), 60008, 6 pages, arXiv:1008.0354.
  27. Ben Geloun J., Krajewski T., Magnen J., Rivasseau V., Linearized group field theory and power-counting theorems, Classical Quantum Gravity 27 (2010), 155012, 14 pages, arXiv:1002.3592.
  28. Ben Geloun J., Magnen J., Rivasseau V., Bosonic colored group field theory, Eur. Phys. J. C Part. Fields 70 (2010), 1119-1130, arXiv:0911.1719.
  29. Bianchi E., Ding Y., Lorentzian spinfoam propagator, Phys. Rev. D 86 (2012), 104040, 11 pages, arXiv:1109.6538.
  30. Bianchi E., Doná P., Speziale S., Polyhedra in loop quantum gravity, Phys. Rev. D 83 (2011), 044035, 17 pages, arXiv:1009.3402.
  31. Bonzom V., Spin foam models for quantum gravity from lattice path integrals, Phys. Rev. D 80 (2009), 064028, 15 pages, arXiv:0905.1501.
  32. Bonzom V., Smerlak M., Bubble divergences from cellular cohomology, Lett. Math. Phys. 93 (2010), 295-305, arXiv:1004.5196.
  33. Bonzom V., Smerlak M., Bubble divergences from twisted cohomology, Comm. Math. Phys. 312 (2012), 399-426, arXiv:1008.1476.
  34. Bonzom V., Smerlak M., Bubble divergences: sorting out topology from cell structure, Ann. Henri Poincaré 13 (2012), 185-208, arXiv:1103.3961.
  35. Bonzom V., Smerlak M., Gauge symmetries in spin-foam gravity: the case for "cellular quantization", Phys. Rev. Lett. 108 (2012), 241303, 5 pages, arXiv:1201.4996.
  36. Borja E.F., Diaz-Polo J., Garay I., U(N) and holomorphic methods for LQG and spin foams, arXiv:1110.4578.
  37. Borja E.F., Freidel L., Garay I., Livine E.R., U(N) tools for loop quantum gravity: the return of the spinor, Classical Quantum Gravity 28 (2011), 055005, 28 pages, arXiv:1010.5451.
  38. Boulatov D.V., A model of three-dimensional lattice gravity, Modern Phys. Lett. A 7 (1992), 1629-1646, hep-th/9202074.
  39. Capovilla R., Dell J., Jacobson T., Mason L., Self-dual 2-forms and gravity, Classical Quantum Gravity 8 (1991), 41-57.
  40. Capovilla R., Jacobson T., Dell J., General relativity without the metric, Phys. Rev. Lett. 63 (1989), 2325-2328.
  41. Carrozza S., Oriti D., Rivasseau V., Renormalization of tensorial group field theories: Abelian U(1) models in four dimensions, arXiv:1207.6734.
  42. Crane L., Perez A., Rovelli C., A finiteness proof for the Lorentzian state sum spin foam model for quantum general relativity, gr-qc/0104057.
  43. De Pietri R., Freidel L., Krasnov K., Rovelli C., Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space, Nuclear Phys. B 574 (2000), 785-806, hep-th/9907154.
  44. Ding Y., Han M., Rovelli C., Generalized spinfoams, Phys. Rev. D 83 (2011), 124020, 17 pages, arXiv:1011.2149.
  45. 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.
  46. Dittrich B., Ryan J.P., Phase space descriptions for simplicial 4D geometries, Classical Quantum Gravity 28 (2011), 065006, 34 pages, arXiv:0807.2806.
  47. Dittrich B., Ryan J.P., Simplicity in simplicial phase space, Phys. Rev. D 82 (2010), 064026, 19 pages, arXiv:1006.4295.
  48. Dittrich B., Speziale S., Area-angle variables for general relativity, New J. Phys. 10 (2008), 083006, 12 pages, arXiv:0802.0864.
  49. Dupuis M., Freidel L., Livine E.R., Speziale S., Holomorphic Lorentzian simplicity constraints, J. Math. Phys. 53 (2012), 032502, 18 pages, arXiv:1107.5274.
  50. Dupuis M., Livine E.R., Holomorphic simplicity constraints for 4D Riemannian spinfoam models, J. Phys. Conf. Ser. 360 (2012), 012046, 4 pages, arXiv:1111.1125.
  51. Dupuis M., Livine E.R., Holomorphic simplicity constraints for 4D spinfoam models, Classical Quantum Gravity 28 (2011), 215022, 32 pages, arXiv:1104.3683.
  52. Dupuis M., Livine E.R., Revisiting the simplicity constraints and coherent intertwiners, Classical Quantum Gravity 28 (2011), 085001, 36 pages, arXiv:1006.5666.
  53. Dupuis M., Ryan J.P., Speziale S., Discrete gravity models and loop quantum gravity: a short review, SIGMA 8 (2012), 052, 31 pages, arXiv:1204.5394.
  54. Dupuis M., Speziale S., Tambornino J., Spinors and twistors in loop gravity and spin foams, arXiv:1201.2120.
  55. Engle J., A proposed proper EPRL vertex amplitude, arXiv:1111.2865.
  56. Engle J., A spin-foam vertex amplitude with the correct semiclassical limit, arXiv:1201.2187.
  57. Engle J., Livine E., Pereira R., Rovelli C., LQG vertex with finite Immirzi parameter, Nuclear Phys. B 799 (2008), 136-149, arXiv:0711.0146.
  58. Engle J., Pereira R., Regularization and finiteness of the Lorentzian loop quantum gravity vertices, Phys. Rev. D 79 (2009), 084034, 10 pages, arXiv:0805.4696.
  59. Engle J., Pereira R., Rovelli C., Flipped spinfoam vertex and loop gravity, Nuclear Phys. B 798 (2008), 251-290, arXiv:0708.1236.
  60. Engle J., Pereira R., Rovelli C., Loop-quantum-gravity vertex amplitude, Phys. Rev. Lett. 99 (2007), 161301, 4 pages, arXiv:0705.2388.
  61. 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.
  62. Freidel L., Krasnov K., A new spin foam model for 4D gravity, Classical Quantum Gravity 25 (2008), 125018, 36 pages, arXiv:0708.1595.
  63. Freidel L., Krasnov K., Spin foam models and the classical action principle, Adv. Theor. Math. Phys. 2 (1999), 1183-1247, hep-th/9807092.
  64. 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.
  65. Freidel L., Livine E.R., The fine structure of SU(2) intertwiners from U(N) representations, J. Math. Phys. 51 (2010), 082502, 19 pages, arXiv:0911.3553.
  66. Freidel L., Livine E.R., U(N) coherent states for loop quantum gravity, J. Math. Phys. 52 (2011), 052502, 21 pages, arXiv:1005.2090.
  67. 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.
  68. Freidel L., Speziale S., On the relations between gravity and BF theories, SIGMA 8 (2012), 032, 15 pages, arXiv:1201.4247.
  69. Han M., 4-dimensional spin-foam model with quantum Lorentz group, J. Math. Phys. 52 (2011), 072501, 22 pages, arXiv:1012.4216.
  70. Hellmann F., State sums and geometry, Ph.D. thesis, University of Nottingham, 2010, arXiv:1102.1688.
  71. Hellmann F., Kamiński W., Geometric asymptotics for spin foam lattice gauge gravity on arbitrary triangulations, arXiv:1210.5276.
  72. Holst S., Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996), 5966-5969, gr-qc/9511026.
  73. Horowitz G.T., Exactly soluble diffeomorphism invariant theories, Comm. Math. Phys. 125 (1989), 417-437.
  74. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  75. Kamiński W., Kisielowski M., Lewandowski J., Spin-foams for all loop quantum gravity, Classical Quantum Gravity 27 (2010), 095006, 24 pages, arXiv:0909.0939.
  76. Kamiński W., Kisielowski M., Lewandowski J., The EPRL intertwiners and corrected partition function, Classical Quantum Gravity 27 (2010), 165020, 15 pages, arXiv:0912.0540.
  77. Kamiński W., Kisielowski M., Lewandowski J., The kernel and the injectivity of the EPRL map, Classical Quantum Gravity 29 (2012), 085001, 20 pages, arXiv:1109.5023.
  78. Krajewski T., Magnen J., Rivasseau V., Tanasa A., Vitale P., Quantum corrections in the group field theory formulation of the Engle-Pereira-Rovelli-Livine and Freidel-Krasnov models, Phys. Rev. D 82 (2010), 124069, 20 pages, arXiv:1007.3150.
  79. Krasnov K., Plebański formulation of general relativity: a practical introduction, Gen. Relativity Gravitation 43 (2011), 1-15, arXiv:0904.0423.
  80. Livine E.R., Matrix models as non-commutative field theories on R3, Classical Quantum Gravity 26 (2009), 195014, 19 pages, arXiv:0811.1462.
  81. Livine E.R., Oriti D., Causality in spin foam models for quantum gravity, gr-qc/0302018.
  82. Livine E.R., Oriti D., Implementing causality in the spin foam quantum geometry, Nuclear Phys. B 663 (2003), 231-279, gr-qc/0210064.
  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., Speziale S., Solving the simplicity constraints for spinfoam quantum gravity, Europhys. Lett. 81 (2008), 50004, 6 pages, arXiv:0708.1915.
  85. Livine E.R., Speziale S., Tambornino J., Twistor networks and covariant twisted geometries, Phys. Rev. D 85 (2012), 064002, 12 pages, arXiv:1108.0369.
  86. Livine E.R., Tambornino J., Loop gravity in terms of spinors, J. Phys. Conf. Ser. 360 (2012), 012023, 5 pages, arXiv:1109.3572.
  87. Livine E.R., Tambornino J., Spinor representation for loop quantum gravity, J. Math. Phys. 53 (2012), 012503, 33 pages, arXiv:1105.3385.
  88. Markopoulou F., Smolin L., Causal evolution of spin networks, Nuclear Phys. B 508 (1997), 409-430, gr-qc/9702025.
  89. Oeckl R., Discrete gauge theory. From lattices to TQFT, Imperial College Press, London, 2005.
  90. Ooguri H., Topological lattice models in four dimensions, Modern Phys. Lett. A 7 (1992), 2799-2810, hep-th/9205090.
  91. Pereira R., Lorentzian loop quantum gravity vertex amplitude, Classical Quantum Gravity 25 (2008), 085013, 8 pages, arXiv:0710.5043.
  92. Perez A., The spin foam approach to quantum gravity, arXiv:1205.2019.
  93. Plebański J.F., On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977), 2511-2520.
  94. 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.
  95. Reisenberger M.P., A lattice worldsheet sum for 4-d Euclidean general relativity, gr-qc/9711052.
  96. Reisenberger M.P., Classical Euclidean general relativity from "left-handed area = right-handed area", Classical Quantum Gravity 16 (1999), 1357-1371, gr-qc/9804061.
  97. Reisenberger M.P., Rovelli C., "Sum over surfaces" form of loop quantum gravity, Phys. Rev. D 56 (1997), 3490-3508, gr-qc/9612035.
  98. Rivasseau V., Towards renormalizing group field theory, PoS Proc. Sci. (2010), PoS(CNCFG2010), 004, 21 pages, arXiv:1103.1900.
  99. Rocek M., Williams R.M., Quantum Regge calculus, Phys. Lett. B 104 (1981), 31-37.
  100. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  101. Rovelli C., Wilson-Ewing E., Discrete symmetries in covariant loop quantum gravity, Phys. Rev. D 86 (2012), 064002, 7 pages, arXiv:1205.0733.
  102. Rühl W., The Lorentz group and harmonic analysis, W.A. Benjamin, Inc., New York, 1970.
  103. Tanasa A., Multi-orientable group field theory, J. Phys. A: Math. Theor. 45 (2012), 165401, 19 pages, arXiv:1109.0694.
  104. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  105. Williams R.M., Quantum Regge calculus, in Approaches to Quantum Gravity, Editor D. Oriti, Cambridge University Press, Cambridge, 2009, 360-377.


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