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  • Cited by 23
  • Print publication year: 2012
  • Online publication date: August 2012

12 - The microscopic dynamics of quantum space as a group field theory

Summary

We provide a rather extended introduction to the group field theory approach to quantum gravity, and the main ideas behind it. We present in some detail the GFT quantization of 3D Riemannian gravity, and discuss briefly the current status of the 4-dimensional extensions of this construction. We also briefly report on some recent results, concerning both the mathematical definition of GFT models as bona fide field theories, and avenues towards extracting testable physics from them.

Introduction

The field of non-perturbative and background-independent quantum gravity has progressed considerably over the past few decades [78]. New research directions are being developed, new important developments are taking place in existing approaches, and some of these approaches are converging to one another. As a result, ideas and tools from one become relevant to another, and trigger further progress. The group field theory (GFT) formalism [39, 77, 79] nicely captures this convergence of approaches and ideas. It is a generalization of the much studied matrix models for 2D quantum gravity and string theory [28, 53]. At the same time, it generalizes it, as we are going to explain, by incorporating the insights coming from canonical loop quantum gravity and its covariant spin foam formulation of the dynamics, and so it became an important part of this approach to the quantization of 4D gravity [72, 74, 81, 85]. Furthermore, it is a point of convergence of the same loop quantum gravity approach and of simplicial quantum gravity approaches, like quantum Regge calculus [93] and dynamical triangulations [3, 79], in that the covariant dynamics of the first takes the form, as we are going to see, of simplicial path integrals.

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References
[1] S., Alexandrov, Phys. Rev.D 78, 044033 (2008) [arXiv: 0802.3389 [gr-qc]].
[2] J., Ambjørn, B., Durhuus, T., Jonsson, Mod. Phys. Lett. A6, 1133–46 (1991).
[3] J., Ambjørn, J., Jurkiewicz, R., Loll, Phys. Rev.D 72, 064014 (2005) [arXiv: hep-th/0505154].
[4] G., Amelino-Camelia, Lect. Notes Phys. 669, 59–100 (2004) [arXiv: gr-qc/0412136].
[5] P., Anspinwall, B., Greene, D., Morrison, Nucl. Phys.B 416, 414–80 (1994) hep-th/9309097.
[6] J. C., Baez, J. W., Barrett, Adv. Theor. Math. Phys. 3, 815 (1999) gr-qc/9903060.
[7] T., Banks, Nucl. Phys. B 309, 493 (1988).
[8] A., Baratin, C., Flori, T., Thiemann [arXiv: 0812.4055 [gr-qc]].
[9] A., Baratin, B., Dittrich, D., Oriti, J., Tambornino (2010), [arXiv:1004.3450 [hep-th]].
[10] A., Baratin, F., Girelli, D., Oriti (2010), [arXiv:1101.0590 [hep-th]].
[11] A., Baratin, D., Oriti [arXiv: 1002.4723 [hep-th]].
[12] A., Baratin, D., Oriti (2010), to appear.
[13] A., Baratin, D., Oriti (2010), Phys. Rev. Lett. 105 221302 (2010).
[14] A., Barbieri, Nucl. Phys.B 518, 714 (1998) gr-qc/9707010.
[15] C., Barcelo, S., Liberati, M., Visser, Living Rev. Rel. 8, 12 (2005) [arXiv: gr-qc/0505065].
[16] J. W., Barrett, L., Crane, J. Math. Phys. 39, 3296 (1998), gr-qc/9709028.
[17] J., Barrett, R., Dowdall, W., Fairbairn, H., Gomes, F., Hellman, J. Math. Phys. 50, 112504 (2009), [arXiv:0902.1170 [gr-qc]].
[18] J., Barrett, R., Dowdall, W., Fairbairn, F., Hellman, R., Pereira, Class. Quant. Grav. 27, 165009 (2010) [arXiv: 0907.2440 [gr-qc]].
[19] J., Barrett, I., Naish-Guzman, Class. Quant. Grav. 26, 155014 (2009) [arXiv: 0803.3319 [gr-qc]].
[20] J., Ben Geloun, J., Magnen, V., RivasseauEuro. Phys. J. C70, 1119–30 (2010) [arXiv: 0911.1719 [hep-th]].
[21] J., Ben Geloun, T., Krajewski, J., Magnen, V., RivasseauClass. Quant. Grav. 27, 155012 (2010) [arXiv: 1002.3592 [hep-th]].
[22] M., Bojowald, Living Rev. Rel. 11, 4 (2008).
[23] V., Bonzom, E., LivinePhys. Rev. D79, 064034 (2009) [arXiv:0812.3456].
[24] D., V. Boulatov, Mod. Phys. Lett. A7, 1629–46 (1992) [arXiv:hep-th/9202074].
[25] S., Coleman, Nucl. Phys.B 310, 643 (1988).
[26] F., Conrady, L., Freidel, Phys. Rev.D 78, 104023 (2008) [arXiv: 0809.2280].
[27] F., Conrady, L., Freidel, Class. Quant. Grav. 25, 245010 (2008) [arXiv: 0806.4640].
[28] F., David, Nucl. Phys.B 257, 45 (1985).
[29] R., De Pietri, L., Freidel, Class. Quant. Grav. 16, 2187 (1999) gr-qc/9804071.
[30] R., De Pietri, L., Freidel, K., Krasnov, C., Rovelli, Nucl. Phys.B 574, 785 (2000) [arXiv: hep-th/9907154].
[31] B., Dittrich, [arXiv:0810.3594[gr-qc]].
[32] B., Dittrich, J., RyanPhys. Rev. D82, 064026 (2010) [arXiv:0807.2806].
[33] F., Dowker, R., Sorkin, Class. Quant. Grav. 15, 1153–67 (1998) gr-qc/9609064.
[34] F., Dowker, in The Future of Theoretical Physics and Cosmology, 436–52, Cambridge University Press (2002), gr-qc/0206020.
[35] J., Engle, R., Pereira, C., Rovelli, Phys. Rev. Lett. 99, 161301 (2007) [arXiv: 0705.2388].
[36] J., Engle, R., Pereira, C., Rovelli, Nucl. Phys.B 798, 251 (2008) [arXiv: 0708.1236].
[37] J., Engle, E., Livine, R., Pereira, C., Rovelli, Nucl. Phys.B 799, 136 (2008) [arXiv:0711.0146].
[38] W., Fairbairn, E., Livine, Class. Quant. Grav. 24, 5277 (2007) [arXiv: gr-qc/0702125].
[39] L., Freidel, Int. J. Phys. 44, 1769–83, (2005) [arXiv: hep-th/0505016].
[40] L., Freidel, R., Gurau, D., Oriti, Phys. Rev.D 80, 044007 (2009) [arXiv: 0905.3772].
[41] L., Freidel, J., Kowalski-Glikman, S., Nowak (2007), arXiv:0706.3658 [hep-th].
[42] L., Freidel, K., Krasnov, Class. Quant. Grav. 25, 125018 (2008) [arXiv: 0708.1595].
[43] L., Freidel, E., Livine, Class. Quant. Grav. 23, 2021 (2006) [arXiv: hep-th/0502106].
[44] L., Freidel, E., Livine, C., Rovelli, Class. Quant. Grav. 20, 1463–78 (2003) [arXiv: gr-qc/0212077].
[45] L., Freidel, D., Louapre, Phys. Rev.D 68, 104004 (2003) [arXiv: hep-th/0211026].
[46] L., Freidel, D., Louapre, Nucl. Phys.B 662, 279–98, 2003 [arXiv: gr-qc/0212001].
[47] L., Freidel, D., Louapre, Class. Quant. Grav. 21, 5685–726 (2004) [arXiv: hep-th/0401076].
[48] L., Freidel, D., Louapre [arXiv: gr-qc/0410141].
[49] L., Freidel, S., Majid, Class. Quant. Grav. 25, 045006 (2008) [arXiv:hep-th/0601004].
[50] L., Freidel, A., Starodubtsev (2005) arXiv:hep-th/0501191.
[51] M., Gaul, C., Rovelli, Lect. Notes Phys. 541, 277 (2000) gr-qc/9910079.
[52] S., Giddings, A., Strominger, Nucl. Phys.B 321, 481 (1989).
[53] P., Ginsparg, “Matrix models of 2-d gravity”, [arXiv: hep-th/9112013].
[54] F., Girelli, E., Livine, D., Oriti, Phys. Rev.D 81, 024015 (2010) [arXiv: 0903.3475 [gr-qc]].
[55] D., Giulini, Gen. Rel. Grav. 41, 785–815 (2009) [arXiv:0902.3923].
[56] M., Gross, Nucl. Phys. Proc. Suppl. 25A, 144–149 (1992).
[57] R., Gurau [arXiv:0907.2582 [hep-th]].
[58] R., Gurau [arXiv:0911.1945 [hep-th]].
[59] G., Horowitz, Class. Quant. Grav. 8, 587–602 (1991).
[60] B. L., Hu, Int. J. Theor. Phys. 44 (2005) 1785–806 [arXiv:gr-qc/0503067].
[61] C., Isham, gr-qc/9510063.
[62] E., Joung, J., Mourad, K., Noui, J. Math. Phys. 50, 052503 (2009) [arXiv:0806.4121 [hep-th]].
[63] A., Klimyk, N., Vilenkin, Representations of Lie Groups and Special Functions, Springer Ed. (1995).
[64] J., Kowalski-Glikman, A., Starodubtsev, Phys. Rev.D 78, 084039 (2008), arXiv:0808.2613.
[65] K., Kuchar, in Winnipeg 1991, Proceedings, General Relativity and Relativistic Astrophysics, pp. 211–314.
[66] E., LivineClass. Quant. Grav. 26, 195014 (2009) [arXiv:0811.1462 [gr-qc]].
[67] E., Livine, S., Speziale, Europhys. Lett. 81, 50004 (2008) [arXiv:0708.1915 [gr-qc]].
[68] J., Magnen, K., Noui, V., Rivasseau, M., Smerlak, Class. Quant. Grav. 26, 185012 (2009) [arXiv:0906.5477].
[69] S., Majid, Foundations of Quantum Group Theory, Cambridge University Press (1995).
[70] M., McGuigan, Phys. Rev.D 38, 3031 (1988).
[71] H., Ooguri, Mod. Phys. Lett. A7, 2799 (1992) hep-th/9205090.
[72] D., Oriti, Rept. Prog. Phys. 64, 1489 (2001) [arXiv: gr-qc/0106091].
[73] D., Oriti, Phys. Lett.B 532, 363–72 (2002) [arXiv: gr-qc/0201077].
[74] D., Oriti, PhD thesis, University of Cambridge (2003) [arXiv: gr-qc/0311066].
[75] D., Oriti, in Quantum Gravity, B., Fauser, J., Tolksdorf, E., Zeidler (eds.), Birkhaeuser, Basel (2007) [arXiv: gr-qc/0512103].
[76] D., Oriti [arXiv:gr-qc/0607032].
[77] D., Oriti, Proceedings of Science [arXiv:0710.3276].
[78] D., Oriti (ed.), Approaches to Quantum Gravity, Cambridge University Press, Cambridge (2009).
[79] D., Oriti, J., Ryan, Class. Quant. Grav. 23, 6543 (2006) [arXiv: gr-qc/0602010].
[80] A., Perelomov, Generalized Coherent States and their Applications, Springer, Berlin (1986).
[81] A., Perez, Class. Quant. Grav. 20, R43 (2003) [arXiv: gr-qc/0301113].
[82] A., Perez, C., Rovelli, Nucl. Phys.B 599, 255 (2001) [arXiv: gr-qc/0006107].
[83] M. P., Reisenberger, gr-qc/9804061.
[84] C., Rovelli, in the Proceedings of the 9th Marcel Grossmann Meeting, Rome, Italy (2000), V. G., Gurzadyan et al. (eds), Singapore, World Scientific, gr-qc/0006061.
[85] C., Rovelli, Quantum Gravity, Cambridge University Press, Cambridge (2006).
[86] L., Smolin, in D., Rickles (ed.), The Structural Foundations of Quantum Gravity, pp. 196–239, hep-th/0507235.
[87] R., Sorkin, Int. J. Theor. Phys. 30, 923–48 (1991).
[88] R., Sorkin, Int. J. Theor. Phys. 36, 2759–81 (1997) gr-qc/9706002.
[89] C., Teitelboim, Phys. Rev.D 25, 3159 (1982).
[90] T., Thiemann, Modern Canonical Quantum general Relativity, Cambridge University Press, Cambridge (2007).
[91] V., Turaev, O., Viro, Topology 31, 865 (1992).
[92] G. E., Volovik, Proceedings of MG11, session “Analog Models of and for General Relativity”, arXiv:gr-qc/0612134.
[93] R., Williams, in [78].