Continuous point symmetries in group field theories. (English) Zbl 1362.81066
Summary: We discuss the notion of symmetries in non-local field theories characterized by integro-differential equations of motion, from a geometric perspective. We then focus on group field theory (GFT) models of quantum gravity and provide a general analysis of their continuous point symmetry transformations, including the generalized conservation laws following from them.
MSC:
| 81T10 | Model quantum field theories |
| 70S20 | More general nonquantum field theories in mechanics of particles and systems |
| 70S05 | Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems |
| 70S10 | Symmetries and conservation laws in mechanics of particles and systems |
| 83C45 | Quantization of the gravitational field |
| 82C10 | Quantum dynamics and nonequilibrium statistical mechanics (general) |
References:
| [1] | Oriti D 2006 The group field theory approach to quantum gravity (arXiv:gr-qc/0607032) · Zbl 1223.83025 |
| [2] | Oriti D 2009 The group field theory approach to quantum gravity: some recent results AIP Conf. Proc.1196 209-18 · doi:10.1063/1.3284386 |
| [3] | Oriti D 2011 The microscopic dynamics of quantum space as a group field theory Proc., Foundations of Space and Time: Reflections on Quantum Gravity(Cape Town, South Africa,) pp 257-320 · Zbl 1269.83008 |
| [4] | Oriti D 2016 Group field theory as the 2nd quantization of loop quantum gravity Class. Quantum Grav.33 085005 · Zbl 1338.83082 · doi:10.1088/0264-9381/33/8/085005 |
| [5] | Krajewski T 2011 Group field theories PoSQGQGS2011 005 |
| [6] | Oriti D 2014 Group field theory and loop quantum gravity (arXiv:1408.7112) · Zbl 1195.83045 |
| [7] | Gross M 1992 Tensor models and simplicial quantum gravity in >2D Nucl. Phys. Proc. Suppl.25A 144-9 · Zbl 0957.83511 · doi:10.1016/S0920-5632(05)80015-5 |
| [8] | Gurau R and Ryan J P 2012 Colored tensor models—a review SIGMA8 020 · Zbl 1242.05094 |
| [9] | Rivasseau V 2011 Quantum gravity and renormalization: the tensor track AIP Conf. Proc.1444 18-29 |
| [10] | Carrozza S 2013 Tensorial methods and renormalization in group field theories PhD Thesis Orsay, LPT |
| [11] | Kegeles A and Oriti D 2016 Generalized conservation laws in non-local field theories J. Phys. A: Math. Theor.49 135401 · Zbl 1368.70039 · doi:10.1088/1751-8113/49/13/135401 |
| [12] | Zawistowski Z J 2001 Symmetries of integro-differential equations eConf C 0107094 263-70 · Zbl 1038.58050 |
| [13] | Ibragimov N H, Kovalev V and Pustovalov V 2002 Symmetries of integro-differential equations: a survey of methods illustrated by the benny equations Nonlinear Dyn.28 135-53 · Zbl 1023.45008 · doi:10.1023/A:1015061100660 |
| [14] | Meleshko S V, Grigoriev Y N, Ibragimov N K and Kovalev V F 2010 Symmetries of Integro-Differential Equations: with Applications in Mechanics and Plasma Physics (New York: Springer) · Zbl 1203.45006 |
| [15] | Ben Geloun J 2012 Classical group field theory J. Math. Phys.53 022901 · Zbl 1274.83048 · doi:10.1063/1.3682651 |
| [16] | Baratin A and Oriti D 2010 Group field theory with non-commutative metric variables Phys. Rev. Lett.105 221302 · doi:10.1103/PhysRevLett.105.221302 |
| [17] | Baratin A and Oriti D 2012 Group field theory and simplicial gravity path integrals: a model for Holst-Plebanski gravity Phys. Rev. D 85 044003 · doi:10.1103/PhysRevD.85.044003 |
| [18] | Baratin A and Oriti D 2011 Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model New J. Phys.13 125011 · Zbl 1448.83034 · doi:10.1088/1367-2630/13/12/125011 |
| [19] | Baez J C 1998 Spin foam models Class. Quantum Grav.15 1827-58 · Zbl 0932.83014 · doi:10.1088/0264-9381/15/7/004 |
| [20] | Alexandrov S and Roche P 2011 Critical overview of loops and foams Phys. Rep.506 41-86 · doi:10.1016/j.physrep.2011.05.002 |
| [21] | Ben Geloun J, Gurau R and Rivasseau V 2010 EPRL/FK group field theory Europhys. Lett.92 60008 · doi:10.1209/0295-5075/92/60008 |
| [22] | Alexandrov S, Geiller M and Noui K 2012 Spin foams and canonical quantization SIGMA8 055 · Zbl 1270.83018 |
| [23] | Perez A 2013 The spin foam approach to quantum gravity Living Rev. Relativ.16 3 · Zbl 1320.83008 · doi:10.12942/lrr-2013-3 |
| [24] | Rovelli C 2008 Loop quantum gravity Living Rev. Relativ.11 · Zbl 1316.83029 · doi:10.12942/lrr-2008-5 |
| [25] | Chiou D-W 2014 Loop quantum gravity Int. J. Mod. Phys. D 24 1530005 · Zbl 1312.83001 · doi:10.1142/S0218271815300050 |
| [26] | Gielen S, Oriti D and Sindoni L 2014 Homogeneous cosmologies as group field theory condensates J. High Energy Phys.JHEP06(2014) 013 · Zbl 1333.81187 · doi:10.1007/JHEP06(2014)013 |
| [27] | Gielen S and Oriti D 2014 Quantum cosmology from quantum gravity condensates: cosmological variables and lattice-refined dynamics New J. Phys.16 123004 · Zbl 1451.85010 · doi:10.1088/1367-2630/16/12/123004 |
| [28] | Gielen S 2014 Quantum cosmology of (loop) quantum gravity condensates: an example Class. Quantum Grav.31 155009 · Zbl 1296.83019 · doi:10.1088/0264-9381/31/15/155009 |
| [29] | Oriti D, Sindoni L and Wilson-Ewing E 2016 Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates Class. Quantum Grav.33 24001 · Zbl 1351.83073 · doi:10.1088/0264-9381/33/22/224001 |
| [30] | Ben Geloun J and Bonzom V 2011 Radiative corrections in the Boulatov-Ooguri tensor model: the 2-point function Int. J. Theor. Phys.50 2819-41 · Zbl 1236.81164 · doi:10.1007/s10773-011-0782-2 |
| [31] | Ben Geloun J 2013 On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov-Ooguri models J. Phys. A: Math. Theor.46 402002 · Zbl 1277.83040 · doi:10.1088/1751-8113/46/40/402002 |
| [32] | Gurau R 2011 Colored group field theory Commun. Math. Phys.304 69-93 · Zbl 1214.81170 · doi:10.1007/s00220-011-1226-9 |
| [33] | Gurau R 2010 Lost in translation: topological singularities in group field theory Class. Quantum Grav.27 235023 · Zbl 1205.83022 · doi:10.1088/0264-9381/27/23/235023 |
| [34] | Boulatov D V 1992 A model of three-dimensional lattice gravity Mod. Phys. Lett. A 7 1629-46 · Zbl 1020.83539 · doi:10.1142/S0217732392001324 |
| [35] | Bonzom V, Gurau R and Rivasseau V 2012 Random tensor models in the large N limit: uncoloring the colored tensor models Phys. Rev. D 85 084037 · doi:10.1103/PhysRevD.85.084037 |
| [36] | Benedetti D, Ben Geloun J and Oriti D 2015 Functional renormalisation group approach for tensorial group field theory: a rank-3 model J. High Energy Phys.JHEP03(2015) 084 · Zbl 1388.83088 · doi:10.1007/JHEP03(2015)084 |
| [37] | Freidel L, Krasnov K and Puzio R 1999 BF description of higher dimensional gravity theories Adv. Theor. Math. Phys.3 1289-324 · Zbl 0997.83020 · doi:10.4310/ATMP.1999.v3.n5.a3 |
| [38] | Barrett J W and Crane L 1998 Relativistic spin networks and quantum gravity J. Math. Phys.39 3296-302 · Zbl 0967.83006 · doi:10.1063/1.532254 |
| [39] | Olver P J 2000 Applications of Lie Groups to Differential Equations (New York: Springer) · Zbl 0937.58026 |
| [40] | Lie S 1880 Theorie der transformationsgruppen i Math. Ann.16 441-528 · JFM 12.0292.01 · doi:10.1007/BF01446218 |
| [41] | Noether E 1971 Invariant variation problems Transp. Theory Stat. Phys.1 186-207 · Zbl 0292.49008 · doi:10.1080/00411457108231446 |
| [42] | Fushchich W and Kornyak V V 1989 Computer algebra application for determining lie and lie-bäcklund symmetries of differential equations J. Symb. Comput.7 611-9 · Zbl 0691.35004 · doi:10.1016/S0747-7171(89)80043-4 |
| [43] | Akhtarshenas S J 2010 Differential geometry on SU(N): left and right invariant vector fields and one-forms (arXiv:1003.2708) |
| [44] | Gielen S, Oriti D and Sindoni L 2013 Cosmology from group field theory Formalism for quantum gravity Phys. Rev. Lett.111 031301 · doi:10.1103/PhysRevLett.111.031301 |
| [45] | Gielen S 2015 Perturbing a quantum gravity condensate Phys. Rev. D 91 043526 · doi:10.1103/PhysRevD.91.043526 |
| [46] | Gielen S 2015 Identifying cosmological perturbations in group field theory condensates J. High Energy Phys.JHEP08(2015) 010 · Zbl 1388.83916 · doi:10.1007/JHEP08(2015)010 |
| [47] | Oriti D, Sindoni L and Wilson-Ewing E 2017 Class. Quant. Grav.34 04LT01 · Zbl 1358.83103 · doi:10.1088/1361-6382/aa549a |
| [48] | de Cesare M, Pithis A G A and Sakellariadou M 2016 Phys. Rev.94 064051 · doi:10.1103/PhysRevD.94.064051 |
| [49] | Dittrich B 2007 Partial and complete observables for Hamiltonian constrained systems Gen. Relativ. Gravit.39 1891-927 · Zbl 1145.83015 · doi:10.1007/s10714-007-0495-2 |
| [50] | Dittrich B 2006 Partial and complete observables for canonical general relativity Class. Quantum Grav.23 6155-84 · Zbl 1111.83015 · doi:10.1088/0264-9381/23/22/006 |
| [51] | Baratin A, Dittrich B, Oriti D and Tambornino J 2011 Non-commutative flux representation for loop quantum gravity Class. Quantum Grav.28 175011 · Zbl 1223.83051 · doi:10.1088/0264-9381/28/17/175011 |
| [52] | Gielen S and Sindoni L 2016 SIGMA12 082 · Zbl 1347.83015 · doi:10.3842/SIGMA.2016.082 |
| [53] | Ben Geloun J, Martini R and Oriti D 2015 Functional renormalization group analysis of a tensorial group field theory on R3 Europhys. Lett.112 31001 · doi:10.1209/0295-5075/112/31001 |
| [54] | Ben Geloun J, Martini R and Oriti D 2016 Functional renormalisation group analysis of tensorial group field theories on Rd Phys. Rev. D 94 024017 · doi:10.1103/PhysRevD.94.024017 |
| [55] | Ben Geloun J and Rivasseau V 2013 A renormalizable 4-dimensional tensor field theory Commun. Math. Phys.318 69-109 · Zbl 1261.83016 · doi:10.1007/s00220-012-1549-1 |
| [56] | Carrozza S, Oriti D and Rivasseau V 2014 Renormalization of tensorial group field theories: Abelian U(1) models in four dimensions Commun. Math. Phys.327 603-41 · Zbl 1291.83102 · doi:10.1007/s00220-014-1954-8 |
| [57] | Ben Geloun J 2016 Renormalizable tensor field theories 18th Int. Congress on Mathematical Physics(Santiago de Chile, Chile, 27 July-1 August 2015) |
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