Yang Model Revisited
DOI:
https://doi.org/10.15407/ujpe69.7.492Keywords:
noncommutative geometry, de Sitter spacetime, Yang modelAbstract
A long time ago, C.N. Yang proposed a generalization of the Snyder model to the case of a curved background spacetime, based on an algebra isomorphic to o(1, 5) which includes, as subalgebras both the Snyder and the de Sitter algebras. His proposal can, therefore, be interpreted as a model of noncommutative curved spacetime, and could be useful for relating physics on very small and very large scales. We review this model and some recent progress concerning its generalizations and its interpretation in the framework of Hopf algebras. We also report some possibilities to relate it to more phenomenological aspects.
References
L.J. Garay. Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145 (1995).
https://doi.org/10.1142/S0217751X95000085
S. Hossenfelder. Minimal length scale scenarios for quantum gravity. Liv. Rev. Rel. 16, 2 (2013).
https://doi.org/10.12942/lrr-2013-2
S. Doplicher, K. Fredenhagen, J.E. Roberts. The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187 (1995).
https://doi.org/10.1007/BF02104515
S. Majid. Algebraic approach to Quantum Gravity II: noncommutative spacetime. In: Approaches to Quantum Gravity. Edited by D. Oriti (Cambridge Univ. Press, 2009), p. 466.
https://doi.org/10.1017/CBO9780511575549.029
J. Madore. An Introduction to Noncommutative Geomtry and Its Physical Applications (Cambridge Univ. Press, 1995).
M. Arzano M., J. Kowalski-Glikman. Deformation of Spacetime Symmetries - Gravity, Group-Valued Momenta, and Noncommutative Fields (Springer-Verlag, 2021).
https://doi.org/10.1007/978-3-662-63097-6
G. Rosati, G. Amelino-Camelia, A. Marciano, M. Matassa. Planck-scale-modified dispersion relations in FRW spacetime. Phys. Rev. D 92, 124042 (2015).
https://doi.org/10.1103/PhysRevD.92.124042
C.N. Yang. On Quantized space-time. Phys. Rev. 72, 874 (1947).
https://doi.org/10.1103/PhysRev.72.874
H.S. Snyder. Quantized space-time. Phys. Rev. 71, 38 (1947).
https://doi.org/10.1103/PhysRev.71.38
S. Meljanac, S. Mignemi. in preparation.
M. Born. Reciprocity theory of elementary particles. Rev. Mod. Phys. 21 463 (1949).
https://doi.org/10.1103/RevModPhys.21.463
H.G. Guo, C.G. Huang, H.T. Wu. Yang's model as triply special relativity and the Snyder's model-de Sitter special relativity duality. Phys. Lett. B 663 270 (2008).
https://doi.org/10.1016/j.physletb.2008.04.012
J. Kowalski-Glikman, L. Smolin. Triply special relativity. Phys. Lett. D 70, 065020 (2004).
https://doi.org/10.1103/PhysRevD.70.065020
C. Chryssomakolos, E. Okon. Linear form of 3-scale special relativity algebra and the relevance of stability. Int. J. Mod. Phys. D 13, 1817 (2004).
https://doi.org/10.1142/S0218271804005225
A. Das, O.C.W. Kong. Physics of quantum relativity through a linear realization. Phys. Rev. D 73, 124029 (2006).
https://doi.org/10.1103/PhysRevD.73.124029
S. Mignemi. The Snyder model and quantum field theory. Class. Quantum Grav. 26, 245020 (2009).
https://doi.org/10.1088/0264-9381/26/24/245020
R. Banerjee, K. Kumar, D. Roychowdhury. Symmetries of Snyder-de Sitter space and relativistic particle dynamics. J. High Energ. Phys. 1103, 060 (2011).
https://doi.org/10.1007/JHEP03(2011)060
S. Meljanac, R. ˇStrajn. Deformed quantum phase spaces, realizations, star products and twists. Symmetry, Integrability and Geometry: Methods and Applications 18, 022 (2022).
J. Lukierski, S. Meljanac, S. Mignemi, A. Pachol. Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking. Phys. Lett. B 847, 138261 (2023).
https://doi.org/10.1016/j.physletb.2023.138261
T. Martini'c-Bila'c, S. Meljanac, S. Mignemi. Hermitian realizations of the Yang model. J. Math. Phys. 64, 122302 (2023).
https://doi.org/10.1063/5.0157268
S. Meljanac, T. Martini'c-Bila'c, S. Kreˇsi'c-Juri'c. Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz, and Poincar'e algebras and their dual extensions. J. Math. Phys. 61, 051705 (2020).
https://doi.org/10.1063/5.0006184
S. Meljanac, S. Mignemi. Generalizations of Snyder model to curved spaces. Phys. Lett. B 833, 137289 (2022).
https://doi.org/10.1016/j.physletb.2022.137289
S. Meljanac, S. Mignemi. Noncommutative Yang model and its generalizations. J. Math. Phys. 64, 023505 (2023).
https://doi.org/10.1063/5.0135492
S. Meljanac, S. Mignemi. Realizations of the Yang-Poisson model on canonical phase space. Int. J. Mod. Phys. A 38, 2350182 (2023).
https://doi.org/10.1142/S0217751X23501828
T. Martini'c-Bila'c, S. Meljanac, S. Mignemi. Generalized Yang-Poisson models on canonical phase space. Symmetry, Integrability and Geometry: Methods and Applications 20, 049 (2024).
T. Martini'c-Bila'c, S. Meljanac, S. Mignemi. Realizations and star-product of doubly κ-deformed Yang models. arXiv:2404.01792.
V.V. Khruschev, A.N. Leznov. The relativistic invariant Lie algebra for the kinematical observables in quantum space-time. Grav. Cosmol. 9, 159 (2003).
J. Lukierski, M. Woronowicz. Spinorial Snyder and Yang models from superalgebras and noncommutative quantum superspaces. Phys. Lett. B 824, 136783 (2021).
https://doi.org/10.1016/j.physletb.2021.136783
J. Lukierski, S. Meljanac, S. Mignemi, A. Pachol. From Snyder space-times to doubly κ-dependent Yang quantum phase spaces and their generalizations. Phys. Lett. B 854, 138729 (2024).
https://doi.org/10.1016/j.physletb.2024.138729
S. Mignemi. Classical and quantum mechanics of the nonrelativistic Snyder model in curved space. Class. Quantum Grav. 29, 215019 (2012).
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