Abstract
We discuss the quantum Poincaré symmetries of the ϱ-Minkowski spacetime, a space characterised by an angular form of noncommutativity. We show that it is possible to give them both a bicrossproduct and a Drinfel’d twist structure. We also obtain a new noncommutative ⋆-product, which is cyclic with respect to the standard integral measure.
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S. Majid and H. Ruegg, Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B 334 (1994) 348 [hep-th/9405107] [INSPIRE].
V.G. Drinfeld, Constant quasiclassical solutions of the Yang-Baxter quantum equation, Dokl. Akad. Nauk SSSR 273 (1983) 531.
T. Jurić, S. Meljanac and A. Samsarov, Light-like κ-deformations and scalar field theory via Drinfeld twist, J. Phys. Conf. Ser. 634 (2015) 012005 [arXiv:1506.02475] [INSPIRE].
J. Lukierski, H. Ruegg and W.J. Zakrzewski, Classical quantum mechanics of free κ relativistic systems, Annals Phys. 243 (1995) 90 [hep-th/9312153] [INSPIRE].
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, Q deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and κ-deformed field theory, Phys. Lett. B 293 (1992) 344 [INSPIRE].
E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, Three dimensional quantum groups from contraction of SU(2)q, J. Math. Phys. 31 (1990) 2548 [INSPIRE].
P.P. Kulish, V.D. Lyakhovsky and A.I. Mudrov, Extended Jordanian twists for Lie algebras, J. Math. Phys. 40 (1999) 4569 [math/9806014] [INSPIRE].
M. Gerstenhaber, A. Giaquinto and S.D. Schack, Quantum symmetry, Lect. Notes Math. 1510 (1992) 9.
V.N. Tolstoy, Chains of extended Jordanian twists for Lie superalgebras, math/0402433.
A. Borowiec and A. Pachol, κ-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009) 045012 [arXiv:0812.0576] [INSPIRE].
N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990) 331 [INSPIRE].
J.-G. Bu et al., κ-deformed spacetime from twist, Phys. Lett. B 665 (2008) 95 [hep-th/0611175] [INSPIRE].
T.R. Govindarajan et al., Twisted statistics in κ-Minkowski spacetime, Phys. Rev. D 77 (2008) 105010 [arXiv:0802.1576] [INSPIRE].
P. Kosinski and P. Maslanka, Lorentz-invariant interpretation of noncommutative space-time: global version, hep-th/0408100 [INSPIRE].
J. Lukierski and M. Woronowicz, New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries, Phys. Lett. B 633 (2006) 116 [hep-th/0508083] [INSPIRE].
R. Oeckl, Untwisting noncommutative Rd and the equivalence of quantum field theories, Nucl. Phys. B 581 (2000) 559 [hep-th/0003018] [INSPIRE].
J. Wess, Deformed coordinate spaces: derivatives, in the proceedings of the 1st Balkan workshop on mathematical, theoretical and phenomenological challenges beyond the Standard Model: perspectives of Balkans collaboration, (2003), p. 122 [https://doi.org/10.1142/9789812702166_0010] [hep-th/0408080] [INSPIRE].
M. Chaichian, P.P. Kulish, K. Nishijima and A. Tureanu, On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT, Phys. Lett. B 604 (2004) 98 [hep-th/0408069] [INSPIRE].
M. Chaichian, P. Presnajder and A. Tureanu, New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications, Phys. Rev. Lett. 94 (2005) 151602 [hep-th/0409096] [INSPIRE].
M. Dimitrijević Ćirić, N. Konjik and A. Samsarov, Noncommutative scalar field in the nonextremal Reissner-Nordström background: quasinormal mode spectrum, Phys. Rev. D 101 (2020) 116009 [arXiv:1904.04053] [INSPIRE].
F. Lizzi and P. Vitale, Time discretization from noncommutativity, Phys. Lett. B 818 (2021) 136372 [arXiv:2101.06633] [INSPIRE].
F. Lizzi, L. Scala and P. Vitale, Localization and observers in ϱ-Minkowski spacetime, Phys. Rev. D 106 (2022) 025023 [arXiv:2205.10862] [INSPIRE].
M. Dimitrijevic Ciric et al., Noncommutative field theory from angular twist, Phys. Rev. D 98 (2018) 085011 [arXiv:1806.06678] [INSPIRE].
F. Lizzi, M. Manfredonia and F. Mercati, The momentum spaces of κ-Minkowski noncommutative spacetime, Nucl. Phys. B 958 (2020) 115117 [arXiv:2001.08756] [INSPIRE].
A. Sitarz, Noncommutative differential calculus on the κ-Minkowski space, Phys. Lett. B 349 (1995) 42 [hep-th/9409014] [INSPIRE].
J. Kowalski-Glikman and S. Nowak, Noncommutative space-time of doubly special relativity theories, Int. J. Mod. Phys. D 12 (2003) 299 [hep-th/0204245] [INSPIRE].
M. Dimitrijevic et al., Deformed field theory on κ space-time, Eur. Phys. J. C 31 (2003) 129 [hep-th/0307149] [INSPIRE].
G. Amelino-Camelia et al., A no-pure-boost uncertainty principle from spacetime noncommutativity, Phys. Lett. B 671 (2009) 298 [arXiv:0707.1863] [INSPIRE].
M. Arzano, J. Kowalski-Glikman and A. Walkus, Lorentz invariant field theory on κ-Minkowski space, Class. Quant. Grav. 27 (2010) 025012 [arXiv:0908.1974] [INSPIRE].
F. Lizzi, M. Manfredonia, F. Mercati and T. Poulain, Localization and reference frames in κ-Minkowski spacetime, Phys. Rev. D 99 (2019) 085003 [arXiv:1811.08409] [INSPIRE].
A. Ballesteros, G. Gubitosi, I. Gutierrez-Sagredo and F. Mercati, Fuzzy worldlines with κ-Poincaré symmetries, JHEP 12 (2021) 080 [arXiv:2109.09699] [INSPIRE].
M. Arzano and J. Kowalski-Glikman, A group theoretic description of the κ-Poincaré Hopf algebra, Phys. Lett. B 835 (2022) 137535 [arXiv:2204.09394] [INSPIRE].
G. Amelino-Camelia, Quantum-spacetime phenomenology, Living Rev. Rel. 16 (2013) 5 [arXiv:0806.0339] [INSPIRE].
J. Lukierski and H. Ruegg, Quantum κ-Poincaré in any dimension, Phys. Lett. B 329 (1994) 189 [hep-th/9310117] [INSPIRE].
J. Kowalski-Glikman and S. Nowak, Doubly special relativity theories as different bases of κ-Poincaré algebra, Phys. Lett. B 539 (2002) 126 [hep-th/0203040] [INSPIRE].
G. Gubitosi and F. Mercati, Relative locality in κ-Poincaré, Class. Quant. Grav. 30 (2013) 145002 [arXiv:1106.5710] [INSPIRE].
G. Gubitosi and S. Heefer, Relativistic compatibility of the interacting κ-Poincaré model and implications for the relative locality framework, Phys. Rev. D 99 (2019) 086019 [arXiv:1903.04593] [INSPIRE].
S. Gutt, An explicit *-product on the cotangent bundle of a Lie group, Lett. Math. Phys. 7 (1983) 249.
J.M. Gracia-Bondia, F. Lizzi, G. Marmo and P. Vitale, Infinitely many star products to play with, JHEP 04 (2002) 026 [hep-th/0112092] [INSPIRE].
G. Amelino-Camelia, L. Barcaroli, S. Bianco and L. Pensato, Planck-scale dual-curvature lensing and spacetime noncommutativity, Adv. High Energy Phys. 2017 (2017) 6075920 [arXiv:1708.02429] [INSPIRE].
M.D. Ćirić, N. Konjik and A. Samsarov, Noncommutative scalar quasinormal modes of the Reissner-Nordström black hole, Class. Quant. Grav. 35 (2018) 175005 [arXiv:1708.04066] [INSPIRE].
M.D. Ćirić, N. Konjik and A. Samsarov, Search for footprints of quantum spacetime in black hole QNM spectrum, arXiv:1910.13342 [INSPIRE].
G. Amelino-Camelia, L. Barcaroli and N. Loret, Modeling transverse relative locality, Int. J. Theor. Phys. 51 (2012) 3359 [arXiv:1107.3334] [INSPIRE].
K. Hersent and J.-C. Wallet, Field theories on ρ-deformed Minkowski space-time, JHEP 07 (2023) 031 [arXiv:2304.05787] [INSPIRE].
M. Kurkov and P. Vitale, Four-dimensional noncommutative deformations of U(1) gauge theory and L∞ bootstrap, JHEP 01 (2022) 032 [arXiv:2108.04856] [INSPIRE].
G. Gubitosi, F. Lizzi, J.J. Relancio and P. Vitale, Double quantization, Phys. Rev. D 105 (2022) 126013 [arXiv:2112.11401] [INSPIRE].
S. Majid, Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130 (1990) 17.
S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math. 141 (1990) 311.
W.M. Singer, Extension theory for connected Hopf algebras, J. Algebra 21 (1972) 1.
G. Amelino-Camelia and S. Majid, Waves on noncommutative space-time and gamma-ray bursts, Int. J. Mod. Phys. A 15 (2000) 4301 [hep-th/9907110] [INSPIRE].
L. Freidel, J. Kowalski-Glikman and S. Nowak, Field theory on κ-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry, Int. J. Mod. Phys. A 23 (2008) 2687 [arXiv:0706.3658] [INSPIRE].
J. Kowalski-Glikman, Living in curved momentum space, Int. J. Mod. Phys. A 28 (2013) 1330014 [arXiv:1303.0195] [INSPIRE].
G. Felder and B. Shoikhet, Deformation quantization with traces, math/0002057 [INSPIRE].
S. Galluccio, F. Lizzi and P. Vitale, Twisted noncommutative field theory with the Wick-Voros and Moyal products, Phys. Rev. D 78 (2008) 085007 [arXiv:0810.2095] [INSPIRE].
S. Galluccio, F. Lizzi and P. Vitale, Translation invariance, commutation relations and ultraviolet/infrared mixing, JHEP 09 (2009) 054 [arXiv:0907.3640] [INSPIRE].
V.G. Kupriyanov and P. Vitale, Noncommutative Rd via closed star product, JHEP 08 (2015) 024 [arXiv:1502.06544] [INSPIRE].
G. Amelino-Camelia et al., Quantum-gravity-induced dual lensing and IceCube neutrinos, Int. J. Mod. Phys. D 26 (2017) 1750076 [arXiv:1609.03982] [INSPIRE].
P. Kosiński and P. Maślanka, The κ-Weyl group and its algebra, in From field theory to quantum groups, World Scientific, Singapore (1996), p. 41 [https://doi.org/10.1142/9789812830425_0003].
P. Zaugg, The quantum Poincaré group from quantum group contraction, J. Phys. 28 (1995) 2589 [hep-th/9409100] [INSPIRE].
A. Agostini, Fields and symmetries in κ-Minkowski noncommutative spacetime, Ph.D. thesis, Università di Napoli, Naples, Italy (2003) [hep-th/0312305] [INSPIRE].
V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, U.K. (1994) [INSPIRE].
S. Zakrzewski, Quantum Poincaré group related to the κ-Poincaré algebra, J. Phys. A 27 (1994) 2075.
J. Lukierski and H. Ruegg, Quantum κ-Poincaré in any dimension, Phys. Lett. B 329 (1994) 189 [hep-th/9310117] [INSPIRE].
Acknowledgments
We acknowledge support from the INFN Iniziativa Specifica GeoSymQFT (F.L., P.V.) and Quagrap (G.F., G.G.). F.L. acknowledges financial support from the State Agency for Research of the Spanish Ministry of Science and Innovation through the “Unit of Ex- cellence Maria de Maeztu 2020–2023” award to the Institute of Cosmos Sciences (Grant No. CEX2019-000918-M) and from Grants No. PID2019-105614 GB-C21 and No. 2017-SGR-929. The research of G.F, G.G, F.L. and P.V. was carried out in the frame of Pro- gramme STAR Plus, financially supported by UniNA and Compagnia di San Paolo. L.S. acknowledges financial support from the doctoral school of the University of Wrocław and the SONATA BIS grant 2021/42/E/ST2/00304 from the National Science Centre (NCN), Poland. G.G. acknowledges financial support by MIUR, PRIN 2017 grant 20179ZF5KS. This work contributes to the European Union COST Action CA18108 Quantum gravity phenomenology in the multi-messenger approach. F.L. and P.V. acknowledge financial sup- port by ICSC — Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union — NextGenerationEU.
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Fabiano, G., Gubitosi, G., Lizzi, F. et al. Bicrossproduct vs. twist quantum symmetries in noncommutative geometries: the case of ϱ-Minkowski. J. High Energ. Phys. 2023, 220 (2023). https://doi.org/10.1007/JHEP08(2023)220
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DOI: https://doi.org/10.1007/JHEP08(2023)220