Abstract
The quest for a quantum gravity phenomenology has inspired a quantum notion of space-time, which motivates us to study the fate of the relativistic symmetries of a particular model of quantum space-time, as well as its intimate connection with the plausible emergent curved “physical momentum space”. We here focus on the problem of Poincare symmetry of κ-Minkowski type non-commutative (quantum) space-time, where the Poincare algebra, on its own, remains undeformed, but in order to preserve the structure of the space-time non-commutative (NC) algebra, the actions of the algebra generators on the operator-valued space-time manifold must be enveloping algebra valued that lives in entire phase space i.e. the cotangent bundle on the space-time manifold (at classical level). Further, we constructed a model for a spin-less relativistic massive particle enjoying the deformed Poincare symmetry, using the first order form of geometric Lagrangian, that satisfies a new deformed dispersion relation and explored a feasible regime of a future Quantum Gravity theory in which the momentum space becomes curved. In this scenario there is only a mass scale (Planck mass mp), but no length scale. Finally, we relate the deformed mass shell to the geodesic distance in this curved momentum space, where the mass of the particle gets renormalized as a result of noncommutativity. We show, that under some circumstances, the Planck mass provides an upper bound for the observed renormalized mass.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
LIGO Scientific and Virgo collaborations, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965) 57 [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, U.K. (1973) [https://doi.org/10.1017/cbo9780511524646].
S. Doplicher, K. Fredenhagen and J.E. Roberts, Space-time quantization induced by classical gravity, Phys. Lett. B 331 (1994) 39 [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
C. Marletto and V. Vedral, Why we need to quantise everything, including gravity, npj Quantum Inf. 3 (2017) 29.
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S. Bose, A. Mazumdar, M. Schut and M. Toroš, Mechanism for the quantum natured gravitons to entangle masses, Phys. Rev. D 105 (2022) 106028 [arXiv:2201.03583] [INSPIRE].
C. Marletto and V. Vedral, Witness gravity’s quantum side in the lab, Nature 547 (2017) 156.
C. Marletto and V. Vedral, Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity, Phys. Rev. Lett. 119 (2017) 240402 [arXiv:1707.06036] [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 k deformed field theory, Phys. Lett. B 293 (1992) 344 [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].
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 et al., Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach, Phys. Lett. B 777 (2018) 1 [arXiv:1710.09772] [INSPIRE].
G. Amelino-Camelia, Testable scenario for relativity with minimum length, Phys. Lett. B 510 (2001) 255 [hep-th/0012238] [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].
J. Kowalski-Glikman, Introduction to doubly special relativity, Lect. Notes Phys. 669 (2005) 131 [hep-th/0405273] [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].
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Relative locality: a deepening of the relativity principle, Gen. Rel. Grav. 43 (2011) 2547 [arXiv:1106.0313] [INSPIRE].
M. Born, A suggestion for unifying quantum theory and relativity, Proc. Roy. Soc. Lond. A 165 (1938) 291 [INSPIRE].
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].
S. Majid, Meaning of noncommutative geometry and the Planck scale quantum group, Lect. Notes Phys. 541 (2000) 227 [hep-th/0006166] [INSPIRE].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
L. Freidel and S. Speziale, On the relations between gravity and BF theories, SIGMA 8 (2012) 032 [arXiv:1201.4247] [INSPIRE].
J. Kowalski-Glikman, De Sitter space as an arena for doubly special relativity, Phys. Lett. B 547 (2002) 291 [hep-th/0207279] [INSPIRE].
J. Kowalski-Glikman and S. Nowak, Doubly special relativity and de Sitter space, Class. Quant. Grav. 20 (2003) 4799 [hep-th/0304101] [INSPIRE].
F. Koch and E. Tsouchnika, Construction of θ-Poincaré algebras and their invariants on Mθ, Nucl. Phys. B 717 (2005) 387 [hep-th/0409012] [INSPIRE].
J. Kowalski-Glikman and A. Starodubtsev, Effective particle kinematics from quantum gravity, Phys. Rev. D 78 (2008) 084039 [arXiv:0808.2613] [INSPIRE].
S. Kresic-Juric, S. Meljanac and M. Stojic, Covariant realizations of κ-deformed space, Eur. Phys. J. C 51 (2007) 229 [hep-th/0702215] [INSPIRE].
J. Lukierski et al., Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach, Phys. Lett. B 777 (2018) 1 [arXiv:1710.09772] [INSPIRE].
S. Meljanac, A. Samsarov, M. Stojic and K.S. Gupta, κ-Minkowski space-time and the star product realizations, Eur. Phys. J. C 53 (2008) 295 [arXiv:0705.2471] [INSPIRE].
J. Lukierski et al., Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach, Phys. Lett. B 777 (2018) 1 [arXiv:1710.09772] [INSPIRE].
M. Dimitrijevic et al., Field theory on κ-spacetime, Czech. J. Phys. 54 (2004) 1243 [hep-th/0407187] [INSPIRE].
J. Kowalski-Glikman, Living in curved momentum space, Int. J. Mod. Phys. A 28 (2013) 1330014 [arXiv:1303.0195] [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. Meljanac, A. Samsarov, J. Trampetić and M. Wohlgenannt, Scalar field propagation in the phi4 κ-Minkowski model, JHEP 12 (2011) 010 [arXiv:1111.5553] [INSPIRE].
M. Dimitrijevic et al., Deformed field theory on κ space-time, Eur. Phys. J. C 31 (2003) 129 [hep-th/0307149] [INSPIRE].
S. Meljanac, A. Pachol, A. Samsarov and K.S. Gupta, Different realizations of κ-momentum space, Phys. Rev. D 87 (2013) 125009 [arXiv:1210.6814] [INSPIRE].
N.A. Lemos, Short proof of Jacobi’s identity for Poisson brackets, Am. J. Phys. 68 (2000) 88 [physics/0210074] [INSPIRE].
D. Kovacevic and S. Meljanac, κ-Minkowski spacetime, κ-Poincaré Hopf algebra and realizations, J. Phys. A 45 (2012) 135208 [arXiv:1110.0944] [INSPIRE].
R. Banerjee, S. Kulkarni and S. Samanta, Deformed symmetry in Snyder space and relativistic particle dynamics, JHEP 05 (2006) 077 [hep-th/0602151] [INSPIRE].
R. Banerjee and S. Samanta, Gauge symmetries on θ-deformed spaces, JHEP 02 (2007) 046 [hep-th/0611249] [INSPIRE].
A.J. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian systems, Accademia Nazionale dei Lincei, Italy (1976).
J.M. Carmona, J.L. Cortés and J.J. Relancio, Relativistic deformed kinematics from momentum space geometry, Phys. Rev. D 100 (2019) 104031 [arXiv:1907.12298] [INSPIRE].
M. Arzano, G. Gubitosi and J.J. Relancio, Deformed relativistic symmetry principles, arXiv:2211.11684 [INSPIRE].
S.A. Franchino-Viñas and J.J. Relancio, Geometrizing the Klein-Gordon and Dirac equations in doubly special relativity, Class. Quant. Grav. 40 (2023) 054001 [arXiv:2203.12286] [INSPIRE].
T. Padmanabhan, Gravitation: foundations and frontiers, Cambridge University Press, Cambridge, U.K. (2010) [https://doi.org/10.1017/cbo9780511807787].
M. Arzano and J. Kowalski-Glikman, Quantum particles in noncommutative spacetime: an identity crisis, Phys. Rev. D 107 (2023) 065001 [arXiv:2212.03703] [INSPIRE].
F.G. Scholtz, B. Chakraborty, S. Gangopadhyay and A.G. Hazra, Dual families of non-commutative quantum systems, Phys. Rev. D 71 (2005) 085005 [hep-th/0502143] [INSPIRE].
L. Lu and A. Stern, Snyder space revisited, Nucl. Phys. B 854 (2012) 894 [arXiv:1108.1832] [INSPIRE].
S.K. Pal and P. Nandi, Effect of dynamical noncommutativity on the limiting mass of white dwarfs, Phys. Lett. B 797 (2019) 134859 [arXiv:1908.11206] [INSPIRE].
Z. Shen, Lectures on Finsler geometry, World Scientific, Singapore (2001) [https://doi.org/10.1142/4619].
M.J. Strassler, Field theory without Feynman diagrams: one loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
Acknowledgments
PN and SKP, would like to extend their gratitude to S.N. Bose National Centre for Basic Sciences, Kolkata for visiting fellowships during the initial stages of the work. One of the authors, PN, also would like to express his gratitude to Stellenbosch University for providing postdoctoral funds during the last stage of the work. AC and BC thank Prof. A.P. Balachandran and Prof. Kumar S. Gupta for their critical comments and useful discussion. The authors would also like to thank the referee for his/her useful and constructive comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2303.02728
Rights and permissions
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.
About this article
Cite this article
Nandi, P., Chakraborty, A., Pal, S.K. et al. Symmetries of κ-Minkowski space-time: a possibility of exotic momentum space geometry?. J. High Energ. Phys. 2023, 142 (2023). https://doi.org/10.1007/JHEP07(2023)142
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2023)142