A classical limit for the Dirac equation in the context of Magueijo--Smolin model of the doubly special relativity using the Ehrenfest's theorem

Ilyas Haouam

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 4, pp. 435-447.

Résumé

In this article, in the context of the Magueijo\textendash Smolin model and employing Ehrenfest's theorem, we investigate the classical limit of the Dirac equation within doubly special relativity. This leads to obtaining deformed classical equations. Here, we assess the effectiveness of Ehrenfest's theorem in deriving the classical limit in the presence of Magueijo–Smolin model. Besides, we explore the deformed classical equations under the discrete, $\mathcal{CPT}$ and Lorentz symmetries.

Détail

Keywords: doubly special relativity, Magueijo–Smolin model, Ehrenfest's theorem, classical limit.
Mots-clés : Dirac equation

Bibliographie

[1] P.A.Tipler, R.A.Llewellyn, Modern Physics, 5 ed., W. H. Freeman and Company, 2008, 160–161

[2] N.Bohr, “Über die Serienspektra der Elemente”, Z. Physik, 2 (1920), 423 | DOI

[3] P.Ehrenfest, “Bemerkung Über die angendherte Gltigkeit der klassischen Mechanik innerhalb der Quantenmechanik”, Z. Physik, 45 (1927), 455 | DOI

[4] G.Friesecke, M.Koppen, “On the Ehrenfest theorem of quantum mechanics”, J. Math. Phys., 50 (2009), 082102 | DOI | MR | Zbl

[5] H.Krger, “Classical limit of real Dirac theory: Quantization of relativistic central field orbits”, Found. Phys., 23 (1993), 1265 | DOI | MR

[6] W.R.Greenberg, A.Klein, C.T.Li, “Invariant tori and Heisenberg matrix mechanics: a new window on the quantum-classical correspondence”, Phys. Rep., 264:1-5 (1996), 167 | DOI | MR

[7] A.O.Bolivar, “Classical limit of fermions in phase space”, J. Math. Phys., 42:9 (2001), 4020 | DOI | MR | Zbl

[8] A.J.Makowski, “Exact classical limit of quantum mechanics: Central potentials and specific states”, Phys. Rev. A, 65:3 (2002), 032103 | DOI | MR

[9] K.G.Kay, “Exact wave functions from classical orbits. II. The Coulomb, Morse, Rosen-Morse, and Eckart systems”, Phys. Rev. A, 65:3 (2002), 032101 | DOI

[10] R.Alicki, “Search for a border between classical and quantum worlds”, Phys. Rev. A, 65:3 (2002), 034104 | DOI

[11] M.L.Liang, H.B.Wu, “Quantum and classical exact solutions of the time-dependent driven generalized harmonic oscillator”, Phys. Scr., 68:1 (2003), 41 | DOI | MR | Zbl

[12] M.L.Liang, Y.J.Sun, “Quantum-classical correspondence of the relativistic equations”, Ann. Phys., 314:1 (2004) | DOI | MR

[13] M.L.Liang et al., “Quantum-classical correspondence of the Dirac equation with a scalar-like potential”, Pramana. J. Phys., 72 (2009), 777 | DOI

[14] A.A.Hnilo, “Simple Explanation of the Classical Limit”, Found. Phys., 49 (1365), 1365 | DOI | MR

[15] H.Spohn, “Semiclassical limit of the Dirac equation and spin precession”, Ann. Phys., 282:2 (2000), 420 | DOI | MR | Zbl

[16] I.Haouam, L.Chetouani, “The Foldy-Wouthuysen transformation of the Dirac equation in noncommutative phase-space”, J. Mod. Phys., 9 (2018), 2021 | DOI

[17] I.Haouam, “Foldy–Wouthuysen Transformation of Noncommutative Dirac Equation in the Presence of Minimal Uncertainty in Momentum”, Few-Body. Syst., 64:1 (2023), 9 | DOI

[18] G.Amelino-Camelia, “Testable scenario for relativity with minimum length”, Phys. Lett. B, 510:1-4 (2001), 255 | DOI | Zbl

[19] G.Amelino-Camelia, “Special treatment”, Nature, 418 (2002), 34 | DOI

[20] G.Amelino-Camelia, “Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale”, Int. J. Mod. Phys. D, 11:01 (2002), 35 | DOI | MR | Zbl

[21] J.Magueijo, L.Smolin, “Lorentz invariance with an invariant energy scale”, Phys. Rev. Lett., 88 (2002), 190403 | DOI | MR

[22] J.Magueijo, L.Smolin, “Generalized Lorentz invariance with an invariant energy scale”, Phys. Rev. D, 67:4 (2003), 044017 | DOI | MR

[23] G.Amelino-Camelia, “Doubly-Special Relativity: Facts, Myths and Some Key Open Issues”, Symmetry, 2010:2 (2010), 230 | DOI | MR

[24] G.Gibbons, S.Gielen, “Deformed general relativity and torsion”, Class. Quantum Grav., 26 (2009), 135005 | DOI | MR | Zbl

[25] I.Haouam, “On the noncommutative geometry in quantum mechanics”, J. Phys. Stud., 24:2 (2020) | DOI

[26] J.Madore, “An introduction to noncommutative geometry”, Geometry and Quantum Physics, Lecture Notes in Physics, 543, eds. H. Gausterer, L. Pittner, H. Grosse, Springer, Berlin–Heidelberg, 2000 | DOI | MR

[27] I.Haouam, “On the Fisk-Tait equation for spin-3/2 fermions interacting with an external magnetic field in noncommutative space-time”, J. Phys. Stud., 24:1 (2020), 1801 | DOI

[28] I.Haouam, “Analytical solution of (2+1) dimensional Dirac equation in time-dependent noncommutative phase-space”, Acta. Polytech., 60:2 (2020), 111 | DOI

[29] N.Seiberg, E.Witten, “String theory and noncommutative geometry”, J. High. Energy. Phys., 1999:09 (1999), 032 | DOI | MR

[30] I.Haouam, “Dirac oscillator in dynamical noncommutative space”, Acta. Polytech., 61:6 (2021), 689 | DOI

[31] I.Haouam, “Two-dimensional Pauli equation in noncommutative phase-space”, Ukr. J. Phys., 66:9 (2021), 771 | DOI

[32] P.Martinetti, “Beyond the standard model with noncommutative geometry, strolling towards quantum gravity”, Journal of Physics: Conference Series, 634 (2015), 012001 | DOI

[33] I.Haouam, “On the three-dimensional Pauli equation in noncommutative phase-space”, Acta Polytech., 61:1 (2021), 230 | DOI

[34] I.Haouam, H.Hassanabadi, “Exact solution of (2+1)-dimensional noncommutative Pauli equation in a time-dependent background”, Int. J. Theor. Phys., 61:8 (2022), 215 | DOI | MR | Zbl

[35] D.M.Gingrich, “Noncommutative geometry inspired black holes in higher dimensions at the LHC”, J. High. Energy. Phys., 2010 (2010), 22 | DOI | MR

[36] I.Haouam, S.A.Alavi, “Dynamical noncommutative graphene”, Int. J. Mod. Phys. A, 37:10 (2022), 2250054 | DOI | MR

[37] I.Haouam, “Solutions of noncommutative two-dimensional position-dependent mass dirac equation in the presence of rashba spin-orbit interaction by using the Nikiforov-Uvarov Method”, Int. J. Theor. Phys., 62:6 (2023), 111 | DOI | MR

[38] B.Hamil et al., “Relativistic oscillators in the context of MS noncommutative model”, EPL, 131:1 (2020), 10003 | DOI

[39] S.Mignemi, A.Samsarov, “Vacuum energy from noncommutative models”, Phys. Lett. B, 779 (2018), 244 | DOI | Zbl

[40] M.Coraddu, S.Mignemi, “The nonrelativistic limit of the Magueijo-Smolin model of deformed special relativity”, EPL, 91:5 (2010), 51002 | DOI

[41] I.Haouam, “Classical limit and Ehrenfest's theorem versus non-relativistic limit of noncommutative Dirac equation in the presence of minimal uncertainty in momentum”, Int. J. Theor. Phys., 62:8 (2023), 189 | DOI | MR | Zbl

[42] I.Haouam, “Ehrenfest's theorem for the Dirac equation in noncommutative Phase-Space”, Math. Comput. Sci., 4:4 (2024), 53 | DOI

[43] H.S.Snyder, “Quantized space-time”, Phys. Rev., 71:1 (1947), 38 | DOI | MR | Zbl

[44] S.Majid, H.Ruegg, “Bicrossproduct structure of $\kappa$-Poincari group and noncommutative geometry”, Phys. Lett. B, 334:3-4 (1994), 348–354 | DOI | MR | Zbl

[45] J.Lukierski et al., “Quantum deformations of nonsemisimple algebras: the example of D=4 in homogeneous rotations”, J. Math. Phys., 35:5 (1994), 2607 | DOI | MR | Zbl

[46] D.Benedetti, “Fractal properties of quantum spacetime”, Phys. Rev. Lett., 102:11 (2009), 111303 | DOI | MR

[47] M.Arzano, T.Trześniewski, “Diffusion on $\kappa$-Minkowski space”, Phys. Rev. D, 89 (2014), 124024 | DOI

[48] A.Granik, Maguejo-Smolin transformation as a consequence of a specific definition of mass, velocity, and the upper limit on energy, 2002, arXiv: hep-th/0207113