Abstract:
A family of local eight-parameter transformations that carry the massless Dirac equation into Maxwell's equations, and also connect the symmetry properties of these equations is found. It is shown that such transformations also relate the conserved quantities for the spinor and electromagnetic fields. On the basis of these connections, a new method is proposed for investigating the symmetry properties of Maxwell's equations together with a convenient
method for finding the conserved quantities for the electromagnetic field. The symmetry properties of Maxwell's equations are derived from the symmetries of the massless Dirac equation, and the conservation laws for the electromagnetic field are obtained from those for the Dirac equation by replacing the spinor ψ by a definite combination of components of the electromagnetic field strengths and passage to the limit m→0. A 128-dimensional invariance algebra of the free Maxwell equations in Dirac-like form is established, and 64 electromagnetic conservation laws are obtained.
Citation:
V. M. Simulik, “Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws”, TMF, 87:1 (1991), 76–85; Theoret. and Math. Phys., 87:1 (1991), 386–393
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\by V.~M.~Simulik
\paper Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws
\jour TMF
\yr 1991
\vol 87
\issue 1
\pages 76--85
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\jour Theoret. and Math. Phys.
\yr 1991
\vol 87
\issue 1
\pages 386--393
\crossref{https://doi.org/10.1007/BF01016578}
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Linking options:
https://www.mathnet.ru/eng/tmf5471
https://www.mathnet.ru/eng/tmf/v87/i1/p76
This publication is cited in the following 20 articles:
V M Simulik, “The Dirac equation near centenary: a contemporary introduction to the Dirac equation consideration”, J. Phys. A: Math. Theor., 58:5 (2025), 053001
Arkady L. Kholodenko, “Maxwell-Dirac Isomorphism Revisited: From Foundations of Quantum Mechanics to Geometrodynamics and Cosmology”, Universe, 9:6 (2023), 288
Volodimir Simulik, Igor Gordievich, Taras Zajac, “Slightly generalized Maxwell system and longitudinal components of solution”, J. Phys.: Conf. Ser., 1416:1 (2019), 012033
Michael K.-H. Kiessling, A. Shadi Tahvildar-Zadeh, “On the quantum-mechanics of a single photon”, Journal of Mathematical Physics, 59:11 (2018)
Albert H. Sultanov, Valeriy K. Bagmanov, Vladimir A. Andreev, Vladimir A. Burdin, Oleg G. Morozov, Albert H. Sultanov, Anton V. Bourdine, Optical Technologies in Telecommunications 2017, 2018, 59
Andrea Marini, Fabio Biancalana, Odyssey of Light in Nonlinear Optical Fibers, 2015, 507
““Problems of Theoretical Physics””, Ukr. J. Phys., 58:6 (2013), 511
V.M. Simulik, I.Yu. Krivsky, I.L. Lamer, “Bosonic Symmetries, Solutions, and Conservation Laws for the Dirac Equation with Nonzero Mass”, Ukr. J. Phys., 58:6 (2013), 523
Matteo Conforti, Andrea Marini, Truong X. Tran, Daniele Faccio, Fabio Biancalana, “Interaction between optical fields and their conjugates in nonlinear media”, Opt. Express, 21:25 (2013), 31239
Tullio Rozzi, Davide Mencarelli, Luca Pierantoni, Electromagnetics and Network Theory and their Microwave Technology Applications, 2011, 211
Tullio Rozzi, Davide Mencarelli, Luca Pierantoni, “Towards a Unified Approach to Electromagnetic Fields and Quantum Currents From Dirac Spinors”, IEEE Trans. Microwave Theory Techn., 59:10 (2011), 2587
T. Rozzi, D. Mencarelli, L. Pierantoni, “Deriving Electromagnetic Fields From the Spinor Solution of the Massless Dirac Equation”, IEEE Trans. Microwave Theory Techn., 57:12 (2009), 2907
V. V. VARLAMOV, “MAXWELL FIELD ON THE POINCARÉ GROUP”, Int. J. Mod. Phys. A, 20:17 (2005), 4095
V.M. Simulik, I.Yu. Krivsky, “Relationship between the Maxwell and Dirac equations: Symmetries, quantization, models of atom”, Reports on Mathematical Physics, 50:3 (2002), 315
V. M. Simulik, I. Yu. Krivsky, “Bosonic symmetries of the massless Dirac equation”, AACA, 8:1 (1998), 69
V. M. Simulik, “A theorem on the structure of a complete set of conformal-like series of conserved quantities for massless fields”, Ukr Math J, 49:12 (1997), 1927
A. G. Meshkov, “Conservation laws and Lie-B�cklund symmetry”, Russ Phys J, 38:7 (1995), 657
I.Yu. Krivsky, V.M. Simulik, Z.Z. Torich, “A covariant form for the complete set of first-order electromagnetic conservation laws”, Physics Letters B, 320:1-2 (1994), 96
I. Yu. Krivsky, V. M. Simulik, “Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations”, Theoret. and Math. Phys., 90:3 (1992), 265–276
N. P. Khvorostenko, “Longitudinal electromagnetic waves”, Soviet Physics Journal, 35:3 (1992), 223
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