- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

κ-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra, and dispersion relations

D. Meljanac1,*, S. Meljanac2,†, S. Mignemi3,4,‡, and R. Štrajn5,§

  • 1Division of Materials Physics, Ruđer Bošković Institute, Bijenička cesta 54, 10002 Zagreb, Croatia
  • 2Division of Theoretical Physics, Ruđer Bošković Institute, Bijenička cesta 54, 10002 Zagreb, Croatia
  • 3Dipartimento di Matematica e Informatica, Università di Cagliari, viale Merello 92, 09123 Cagliari, Italy
  • 4INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy
  • 5Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20000 Dubrovnik, Croatia

  • *Daniel.Meljanac@irb.hr
  • †meljanac@irb.hr
  • ‡smignemi@unica.it
  • §rina.strajn@unidu.hr
PDF

Phys. Rev. D 99, 126012 – Published 19 June, 2019

DOI: https://doi.org/10.1103/PhysRevD.99.126012

Abstract

We consider κ-deformed relativistic quantum phase space and possible implementations of the Lorentz algebra. There are two ways of performing such implementations. One is a simple extension where the Poincaré algebra is unaltered, while the other is a general extension where the Poincaré algebra is deformed. As an example we fix the Jordanian twist and the corresponding realization of noncommutative coordinates, coproduct of momenta, and addition of momenta. An extension with a one-parameter family of realizations of the Lorentz generators, dilatation and momenta closing the Poincaré-Weyl algebra is considered. The corresponding physical interpretation depends on the way the Lorentz algebra is implemented in phase space. We show how the spectrum of the relativistic hydrogen atom depends on the realization of the generators of the Poincaré-Weyl algebra.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. H. S. Snyder, Quantized space-time, Phys. Rev. 71, 38 (1947).
  2. J. E. Moyal, Quantum Mechanics as a Statistical Theory, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 45 (Cambridge University Press, Cambridge, England, 1949); V. P. Nair and A. P. Polychronakos, Quantum mechanics on the noncommutative plane and sphere, Phys. Lett. B 505, 267 (2001).
  3. A. Connes, Noncommutative Geometry (Academic Press, New York, 1994).
  4. S. Doplicher, K. Fredenhagen, and J. E. Roberts, Spacetime quantization induced by classical gravity, Phys. Lett. B 331, 39 (1994); The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys. 172, 187 (1995).
  5. S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, Cambridge, England, 1995).
  6. M. Chaichian and A. Demichev, Introduction to Quantum Groups (World Scientific, Singapore, 1996); J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, England, 1999); V. G. Drinfeld (Quantum groups), Proceedings of the ICM, Rhode Island, USA, 1987 (Berkeley, 1986).
  7. P. Aschieri, M. Dimitrijević, P. Kulish, F. Lizzi, and J. Wess, Noncommutative Spacetimes: Symmetries in Noncommutative Geometry and Field Theory, Lecture Notes in Physics (Springer, New York, 2009); P. Aschieri, F. Lizzi, and P. Vitale, Twisting all the way: From classical mechanics to quantum fields, Phys. Rev. D 77, 025037 (2008); V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32, 254 (1985); Quasi Hopf algebras, Alg. Anal. 1, 114 (1989); Leningrad Math. J. 1, 1419 (1990).
  8. J. Lukierski and H. Ruegg, Quantum κ Poincaré in any dimension, Phys. Lett. B 329, 189 (1994); J. Lukierski, A. Nowicki, H. Ruegg, and V. N. Tolstoy, Q-deformation of Poincaré algebra, 264, 331 (1991); J. Lukierski, A. Nowicki, and H. Ruegg, New quantum Poincaré algebra and κ-deformed field theory, 293, 344 (1992).
  9. S. Majid and H. Ruegg, Bicrossproduct structure of κ-Poincaré group and non-commutative geometry, Phys. Lett. B 334, 348 (1994); J. Lukierski, H. Ruegg, and W. J. Zakrzewski, Classical and quantum mechanics of free κ-relativistic systems, Ann. Phys. (N.Y.) 243, 90 (1995).
  10. S. Zakrzewski, Quantum Poincare group related to κ Poincare algebra, J. Phys. A 27, 2075 (1994).
  11. P. Kosinski and P. Maslanka, The duality between κ Poincare algebra and κ Poincare group, arXiv:hep-th/9411033.
  12. J. Lukierski and A. Nowicki, Heisenberg double description of κ-Poincare algebra and κ-deformed phase space, in Proc. XXI Int. Coll. Group. Theor. Methods in Physics, edited by V. K. Dobrev and H. D. Doebner (Heron Press, Sofia, 1997), p. 186.
  13. G. Amelino-Camelia, J. Lukierski, and A. Nowicki, κ-deformed covariant phase space and quantum-gravity uncertainty relations, Phys. At. Nucl. 61, 1811 (1998).
  14. J. Lukierski, Z. Škoda, and M. Woronowicz, κ-deformed covariant quantum phase spaces as Hopf algebroids, Phys. Lett. B 750, 401 (2015).
  15. G. Amelino-Camelia, Doubly-special relativity: First results and key open problems, Int. J. Mod. Phys. D 11, 35 (2002); Testable scenario for relativity with minimum-length, Phys. Lett. B 510, 255 (2001).
  16. J. Magueijo and L. Smolin, Lorentz Invariance with an Invariant Energy Scale, Phys. Rev. Lett. 88, 190403 (2002); Generalized Lorentz invariance with an invariant energy scale, Phys. Rev. D 67, 044017 (2003).
  17. G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, and S. Sarkar, Potential sensitivity of gamma-ray burster observations to wave dispersion in vacuo, Nature (London) 393, 763 (1998).
  18. C. J. Hogan, Interferometers as probes of Planckian quantum geometry, Phys. Rev. D 85, 064007 (2012).
  19. I. R. Berchera, I. P. Degiovanni, S. Olivares, and M. Genovese, Testing Quantum Gravity by Quantum Light, Phys. Rev. Lett. 110, 213601 (2013).
  20. S. Dey, A. Bhat, D. Momeni, M. Faizal, A. F. Ali, T. K. Dey, and A. Rehman, Probing noncommutative theories with quantum optical experiments, Nucl. Phys. B924, 578 (2017).
  21. S. Mignemi and R. Štrajn, Snyder dynamics in a Schwarzschild spacetime, Phys. Rev. D 90, 044019 (2014); S. Mignemi, Classical dynamics on Snyder spacetime, Int. J. Mod. Phys. D 24, 1550043 (2015); S. Mignemi and N. Uras, Noncommutative geometry of the quantum clock, Phys. Lett. A 383, 585 (2019); V. Todorinov, P. Bosso, and S. Das, Relativistic generalized uncertainty principle, arXiv:1810.11761.
  22. S. Meljanac, D. Meljanac, F. Mercati, and D. Pikutić, Noncommutative spaces and poincaré symmetry, Phys. Lett. B 766, 181 (2017).
  23. S. Meljanac, D. Meljanac, A. Pachoł, and D. Pikutić, Remarks on simple interpolation between Jordanian twists, J. Phys. A 50, 265201 (2017).
  24. P. Aschieri, A. Borowiec, and A. Pachoł, Observables and dispersion relations in κ-Minkowski spacetime, J. High Energy Phys. 10 (2017) 152.
  25. S. Meljanac and M. Stojić, New realizations of Lie algebra κ-deformed Euclidean space, Eur. Phys. J. C 47, 531 (2006).
  26. S. Krešić-Jurić, S. Meljanac, and M. Stojić, Covariant realizations of κ-deformed space, Eur. Phys. J. C 51, 229 (2007).
  27. S. Meljanac and S. Krešić-Jurić, Differential structure on κ-Minkowski space, and κ-Poincaré algebra, Int. J. Mod. Phys. A 26, 3385 (2011).
  28. S. Meljanac, D. Meljanac, A. Samsarov, and M. Stojić, κ-deformed Snyder space, Mod. Phys. Lett. A 25, 579 (2010).
  29. D. Kovačević and S. Meljanac, κ-Minkowski spacetime, κ-Poincaré Hopf algebra and realizations, J. Phys. A 45, 135208 (2012).
  30. T. R. Govindarajan, K. S. Gupta, E. Harikumar, S. Meljanac, and D. Meljanac, Twisted statistics in κ-Minkowski spacetime, Phys. Rev. D 77, 105010 (2008).
  31. T. Juric, S. Meljanac, and R. Štrajn, Twists, realizations and Hopf algebroid structure of κ-deformed phase space, Int. J. Mod. Phys. A 29, 1450022 (2014).
  32. T. Jurić, S. Meljanac, D. Pikutić, and R. Štrajn, Toward the classification of differential calculi on κ-Minkowski space and related field theories, J. High Energy Phys. 07 (2015) 055.
  33. T. Jurić, S. Meljanac, and D. Pikutić, Realizations of κ-Minkowski space, Drinfeld twists and related symmetry algebras, Eur. Phys. J. C 75, 528 (2015).
  34. D. Meljanac, S. Meljanac, and D. Pikutić, Families of vector-like deformed relativistic quantum phase spaces, twists and symmetries, Eur. Phys. J. C 77, 830 (2017).
  35. T. Jurić, S. Meljanac, and R. Štrajn, κ-Poincaré-Hopf algebra and Hopf algebroid structure of phase space from twist, Phys. Lett. A 377, 2472 (2013).
  36. T. Jurić, D. Kovačević, and S. Meljanac, κ-deformed phase space, Hopf algebroid and twisting, SIGMA 10, 106 (2014).
  37. S. Meljanac, A. Samsarov, and R. Štrajn, κ-deformation of phase space; generalized Poincaré algebras and R-matrix, J. High Energy Phys. 08 (2012) 127.
  38. S. Meljanac and Z. Škoda, Hopf algebroid twists for deformation quantization of linear Poisson structures, SIGMA 14, 026 (2018).
  39. A. Borowiec and A. Pachoł, κ-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79, 045012 (2009).
  40. J.-G. Bu, J. H. Yee, and H.-C. Kim, Differential structure on κ-Minkowski spacetime realized as module of twisted Weyl algebra, Phys. Lett. B 679, 486 (2009).
  41. D. Kovačević, S. Meljanac, A. Pachoł, and R. Štrajn, Generalized Poincare algebras, Hopf algebras and κ-Minkowski spacetime, Phys. Lett. B 711, 122 (2012).
  42. N. Loret, S. Meljanac, F. Mercati, and D. Pikutić, Vector-like deformations of relativistic quantum phase-space and relativistic kinematics, Int. J. Mod. Phys. D 26, 1750123 (2017).
  43. S. Meljanac, Z. Škoda, and M. Stojić, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107, 475 (2017).
  44. A. Granik, Maguejo-Smolin transformation as a consequence of a specific definition of mass, velocity, and the upper limit on energy, arXiv:hep-th/0207113.
  45. S. Mignemi, Transformations of coordinates and Hamiltonian formalism in deformed special relativity, Phys. Rev. D 68, 065029 (2003).
  46. D. Kimberly, J. Magueijo, and J. Medeiros, Nonlinear relativity in position space, Phys. Rev. D 70, 084007 (2004).
  47. S. Ghosh and P. Pal, Deformed special relativity and deformed symmetries in a canonical framework, Phys. Rev. D 75, 105021 (2007).
  48. P. Kosiński and P. Maślanka, On the definition of velocity in doubly special relativity theories, Phys. Rev. D 68, 067702 (2003); S. Mignemi, On the definition of velocity in theories with two observer independent scales, Phys. Lett. A 316, 173 (2003); M. Daszkiewicz, K. Imilkowska, and J. Kowalski-Glikman, Velocity of particles in doubly special relativity, 323, 345 (2004).
  49. 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, 150 (2011).
  50. M. Coraddu and S. Mignemi, The nonrelativistic limit of the Magueijo-Smolin model of deformed special relativity, Europhys. Lett. 91, 51002 (2010).
  51. H. Kleinert, Particles and Quantum Fields (World Scientific, Singapore, 2016).

Outline

Information

Phys. Rev. D 99, 126012– Published 19 June, 2019
  • Received 19 April 2019
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation