Abstract
We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator.
This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The corresponding quantum Poincaré-Weyl Lie algebra of in-finitesimal translations, rotations and dilatations is obtained. The d’Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, Tests of quantum gravity from observations of gamma-ray bursts, Nature 393 (1998) 763 [astro-ph/9712103] [INSPIRE].
C.J. Hogan, Interferometers as probes of Planckian quantum geometry, Phys. Rev. D 85 (2012) 064007 [arXiv:1002.4880] [INSPIRE].
I. Ruo Berchera, I.P. Degiovanni, S. Olivares and M. Genovese, Quantum light in coupled interferometers for quantum gravity tests, Phys. Rev. Lett. 110 (2013) 213601 [arXiv:1304.7912] [INSPIRE].
S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quant. Grav. 30 (2013) 133001 [arXiv:1304.5795] [INSPIRE].
G. Amelino-Camelia, Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale, Int. J. Mod. Phys. D 11 (2002) 35 [gr-qc/0012051] [INSPIRE].
G. Amelino-Camelia, Testable scenario for relativity with minimum length, Phys. Lett. B 510 (2001) 255 [hep-th/0012238] [INSPIRE].
J. Magueijo and L. Smolin, Lorentz invariance with an invariant energy scale, Phys. Rev. Lett. 88 (2002) 190403 [hep-th/0112090] [INSPIRE].
J. Magueijo and L. Smolin, Generalized Lorentz invariance with an invariant energy scale, Phys. Rev. D 67 (2003) 044017 [gr-qc/0207085] [INSPIRE].
J. Kowalski-Glikman, Observer independent quantum of mass, Phys. Lett. A 286 (2001) 391 [hep-th/0102098] [INSPIRE].
S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. Rel. 16 (2013) 2 [arXiv:1203.6191] [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].
J. Wess, Deformed coordinate spaces: derivatives, in Proceedings BW2003 workshop, Vrnjacka Banja, Serbia and Montenegro, G. Djordjevic, L. Nesic and J. Wess eds., World Scientific Singapore, (2005) [ISBN:9789812561305 (Print), 9789814481137 (Online)] [hep-th/0408080] [INSPIRE].
J.-G. Bu, H.-C. Kim, Y. Lee, C.H. Vac and J.H. Yee, κ-deformed spacetime from twist, Phys. Lett. B 665 (2008) 95 [hep-th/0611175] [INSPIRE].
T.R. Govindarajan, K.S. Gupta, E. Harikumar, S. Meljanac and D. Meljanac, Twisted statistics in κ-Minkowski spacetime, Phys. Rev. D 77 (2008) 105010 [arXiv:0802.1576] [INSPIRE].
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].
G. Amelino-Camelia, N. Loret and G. Rosati, Speed of particles and a relativity of locality in κ-Minkowski quantum spacetime, Phys. Lett. B 700 (2011) 150 [arXiv:1102.4637] [INSPIRE].
A. Borowiec, K.S. Gupta, S. Meljanac and A. Pachol, Constarints on the quantum gravity scale from κ-Minkowski spacetime, EPL 92 (2010) 20006 [arXiv:0912.3299] [INSPIRE].
P. Aschieri, A. Borowiec and A. Pachol, Dispersion relations in noncommutative cosmology, to appear.
S.R. Coleman and E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].
C. Englert, J. Jaeckel, V.V. Khoze and M. Spannowsky, Emergence of the electroweak scale through the Higgs portal, JHEP 04 (2013) 060 [arXiv:1301.4224] [INSPIRE].
K.A. Meissner and H. Nicolai, Conformal symmetry and the Standard Model, Phys. Lett. B 648 (2007) 312 [hep-th/0612165] [INSPIRE].
K.A. Meissner and H. Nicolai, Effective action, conformal anomaly and the issue of quadratic divergences, Phys. Lett. B 660 (2008) 260 [arXiv:0710.2840] [INSPIRE].
A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano and R.A. Tacchi, Generalizing the Noether theorem for Hopf-algebra spacetime symmetries, Mod. Phys. Lett. A 22 (2007) 1779 [hep-th/0607221] [INSPIRE].
P. Aschieri, L. Castellani and M. Dimitrijević, Dynamical noncommutativity and Noether theorem in twisted \( \phi \) ∗4 theory, Lett. Math. Phys. 85 (2008) 39 [arXiv:0803.4325] [INSPIRE].
P. Aschieri, F. Lizzi and P. Vitale, Twisting all the way: from classical mechanics to quantum fields, Phys. Rev. D 77 (2008) 025037 [arXiv:0708.3002] [INSPIRE].
P. Aschieri, M. Dimitrijević, P. Kulish, F. Lizzi and J. Wess, Noncommutative spacetimes: symmetries in noncommutative geometry and field theory, Lect. Notes Phys. 774 (2009) 1 [INSPIRE].
S.L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1989) 125 [INSPIRE].
P. Aschieri and P. Schupp, Vector fields on quantum groups, Int. J. Mod. Phys. A 11 (1996) 1077 [q-alg/9505023] [INSPIRE].
P. Aschieri, On the geometry of inhomogeneous quantum groups, Edizioni della Scuola Normale, collana Tesi, Springer-Birkhäuser, (1999) [math/9805119] [INSPIRE].
P. Aschieri, M. Dimitrijević, F. Meyer and J. Wess, Noncommutative geometry and gravity, Class. Quant. Grav. 23 (2006) 1883 [hep-th/0510059] [INSPIRE].
P. Aschieri and L. Castellani, An introduction to noncommutative differential geometry on quantum groups, Int. J. Mod. Phys. A 8 (1993) 1667 [hep-th/9207084] [INSPIRE].
P. Kulish and A. Mudrov, Twisting adjoint module algebras, Lett. Math. Phys. 95 (2011) 233 [arXiv:1011.4758] [INSPIRE].
P. Aschieri and A. Schenkel, Noncommutative connections on bimodules and Drinfeld twist deformation, Adv. Theor. Math. Phys. 18 (2014) 513 [arXiv:1210.0241] [INSPIRE].
P. Aschieri, Twisting all the way: from algebras to morphisms and connections, Int. J. Mod. Phys. Conf. Ser. 13 (2012) 1 [arXiv:1210.1143] [INSPIRE].
M. Nakahara, Geometry, topology and physics, second edition, Graduate Student Series in Physics, Taylor & Francis, (2003) [ISBN-13:978-0750306065].
P. Aschieri and L. Castellani, Bicovariant calculus on twisted ISO(N), quantum Poincaré group and quantum Minkowski space, Int. J. Mod. Phys. A 11 (1996) 4513 [q-alg/9601006] [INSPIRE].
J. Lukierski and A. Nowicki, Doubly special relativity versus κ-deformation of relativistic kinematics, Int. J. Mod. Phys. A 18 (2003) 7 [hep-th/0203065] [INSPIRE].
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, Q deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
S. Majid and H. Ruegg, Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B 334 (1994) 348 [hep-th/9405107] [INSPIRE].
T. Jurić, S. Meljanac, D. Pikutić and R. Štrajn, Toward the classification of differential calculi on κ-Minkowski space and related field theories, JHEP 07 (2015) 055 [arXiv:1502.02972] [INSPIRE].
S. Judes and M. Visser, Conservation laws in ‘doubly special relativity’, Phys. Rev. D 68 (2003) 045001 [gr-qc/0205067] [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].
A. Borowiec and A. Pachol, κ-Minkowski spacetimes and DSR algebras: fresh look and old problems, SIGMA 6 (2010) 086 [arXiv:1005.4429] [INSPIRE].
A. Borowiec and A. Pachol, κ-deformations and extended κ-Minkowski spacetimes, SIGMA 10 (2014) 107 [arXiv:1404.2916] [INSPIRE].
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The principle of relative locality, Phys. Rev. D 84 (2011) 084010 [arXiv:1101.0931] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1703.08726
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Aschieri, P., Borowiec, A. & Pachoł, A. Observables and dispersion relations in κ-Minkowski spacetime. J. High Energ. Phys. 2017, 152 (2017). https://doi.org/10.1007/JHEP10(2017)152
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)152