Lagrangian reformulation of Maxwell electrostatics based on a one-parameter extension of the Heisenberg algebra in a six-dimensional phase space

Document Type : Full length research Paper

Authors

Department of Physics, Faculty of Basic Sciences, Arak University, Arak

Abstract

In a series of papers, Frydryszak and Tkachuk (Czechoslovak Journal of Physics, 53 (2003) 1035-1040; Found. Phys, 46 (2016) 1666-1679) introduced a one-parameter extension of the Heisenberg algebra with a deformation parameter in a six-dimensional phase space. In this paper, after Lagrangian reformulation of Maxwell electrostatics from the viewpoint of the above one-parameter extension of the Heisenberg algebra in a six-dimensional phase space the electrostatic potential and the electric field of a point charge are calculated analytically in this generalized electrostatics (generalized Maxwell electrostatics). We show that in contrast with the conventional Maxwell electrostatics the electrostatic potential and the electric field of a point charge are not singular at the position of the point charge in this generalized electrostatics. We show that in the low-energy limit (large spatial distances), the results of this generalized electrostatics are compatible with the results of the conventional Maxwell electrostatics.

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References

 
[1] A. Rostami, S.K. Moayedi, Exact solution for light propagation through inhomogeneous media, Indian Journal of Physics 75 B 4 (2001) 357-361.
[2] S.K. Moayedi, A. Rostami, PT-invariant Helmholtz optics and its applications to slab waveguides, European Physical Journal B 36 (2003) 359-363. https://doi.org/10.1140/epjb/e2003-00354-5
[3] A. Rostami, S.K. Moayedi, TM mode in inhomogeneous slab waveguide as an exactly solvable oscillator-like Hamiltonian, Journal of Optics A: Pure and Applied Optics 5 (2003) 380-385. https://doi.org/10.1088/1464-4258/5/4/313
[4] S.M. Blinder, Singularity-free electrodynamics for point charges and dipoles: a classical model for electron self-energy and spin, European Journal of Physics 24 (2003) 271-275. https://doi.org/10.1088/0143-0807/24/3/307
[5] R.B. Santos, Plasma-like vacuum in Podolsky regularized classical electrodynamics, Modern Physics Letters A 26 (2011) 1909-1915. https://doi.org/10.1142/S0217732311036395
[6] M. Born, L. Infeld, Foundations of the new field theory, Proceedings of the Royal Society of London A 144 (1934) 425-451. https://doi.org/10.1098/rspa.1934.0059
[7] S.K. Moayedi, M. Shafabakhsh, F. Fathi, Analytical calculation of stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in the framework of Born-Infeld electrostatics, Advances in High Energy Physics 2015 (2015) 180185. https://doi.org/10.1155/2015/180185
[8] F. Bopp, Eine lineare Theorie des Elektrons, Annalen der Physik 430 (1940) 345-384. https://doi.org/10.1002/andp.19404300504
[9] B. Podolsky, A generalized electrodynamics part I-non-quantum, Physical Review 62 (1942) 68-71. https://doi.org/10.1103/PhysRev.62.68
[10] A. Pais, G.E. Uhlenbeck, On field theories with non-localized action, Physical Review 79 (1950) 145-165. https://doi.org/10.1103/PhysRev.79.145
[11] H.S. Snyder, Quantized space-time, Physical Review 71 (1947) 38-41. https://doi.org/10.1103/PhysRev.71.38
[12] S.K. Moayedi, M.R. Setare, H. Moayeri, Formulation of an electrostatic field with a charge density in the presence of a minimal length based on the Kempf algebra, EPL 98 (2012) 50001. https://doi.org/10.1209/0295-5075/98/50001
[13] S.K. Moayedi, M.R. Setare, B. Khosropour, Formulation of electrodynamics with an external source in the presence of a minimal measurable length, Advances in High Energy Physics 2013 (2013) 657870. https://dx.doi.org/10.1155/2013/657870
[14] A.V. Silva, E.M.C. Abreu, M.J. Neves, Quantum electrodynamics and the electron self-energy in a deformed space with a minimal length scale, International Journal of Modern Physics A 31 (2016) 1650096. https://dx.doi.org/10.1142/S0217751X16500962
[15] A. Izadi, S.K. Moayedi, Lagrangian formulation of an infinite derivative real scalar field theory in the framework of the covariant Kempf–Mangano algebra in a (D+1)-dimensional Minkowski space–time, Annals of physics 411 (2019) 167956. https://doi.org/10.1016/j.aop.2019.167956
[16] M. Ranaiy, S.K. Moayedi, The short-distance behavior of an Abelian Proca model based on a one-parameter extension of the covariant Heisenberg algebra, Modern Physics Letters A 35 (2020) 2050038. https://dx.doi.org/10.1142/S0217732320500388
[17] A.M. Frydryszak, V.M. Tkachuk, Aspects of pre-quantum description of deformed theories, Czechoslovak Journal of Physics 53 (2003) 1035-1040. https://doi.org/10.1023/B:CJOP.0000010529.32268.03
[18] V.M. Tkachuk, Galilean and Lorentz transformations in a space with generalized uncertainty principle, Foundations of Physics 46 (2016) 1666–1679. https://doi.org/10.1007/s10701-016-0036-5
[19] W.S. Chung, H. Hassanabadi, New generalized uncertainty principle from the doubly special relativity, Physics Letters B 785 (2018) 127-131. https://doi.org/10.1016/j.physletb.2018.07.064
[20] K. Nozari, M.A. Gorji, V. Hosseinzadeh, B. Vakili, Natural cutoffs via compact symplectic manifolds, Classical and Quantum Gravity 33 (2015) 025009. https://dx.doi.org/10.1088/0264-9381/33/2/025009
[21] K. Nozari, A. Etemadi, Minimal length, maximal momentum, and Hilbert space representation of quantum mechanics, Physical Review D 85 (2012) 104029. https://dx.doi.org/10.1103/PhysRevD.85.104029
[22] N. Moeller, B. Zwiebach, Dynamics with infinitely many time derivatives and rolling tachyons, JHEP 10 (2002) 034. https://doi.org/10.1088/1126-6708/2002/10/034

[23] M.R. Spiegel, S. Lipschutz, J. Liu, Schaum's Outline of Mathematical Handbook of Formulas and Tables, Fifth Edition, McGraw-Hill, (2018).

[24] A. Kempf, G. Mangano, R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Physical Review D 52 (1995) 1108-1118. https://doi.org/10.1103/PhysRevD.52.1108
[25] M.M. Stetsko, V.M. Tkachuk, Scattering problem in deformed space with minimal length, Physical Review A 76 (2007) 012707. https://dx.doi.org/10.1103/PhysRevA.76.012707
Journal of Research on Many-body Systems
Volume 12, Issue 4 - Serial Number 35
winter 2023
March 2023
Pages 51-69

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History

  • Receive Date: 27 June 2021
  • Revise Date: 25 December 2022
  • Accept Date: 06 February 2023

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