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Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws. (English. Russian original) Zbl 1189.78007
Theor. Math. Phys. 87, No. 1, 386-393 (1991); translation from Teor. Mat. Fiz. 87, No. 1, 76-85 (1991).
Summary: 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 $$\psi$$ by a definite combination of components of the electromagnetic field strengths and passage to the limit $$m\to 0$$. A 128-dimensional invariance algebra of the free Maxwell equations in Dirac-like form is established, and 64 electromagnetic conservation laws are obtained.

##### MSC:
 78A25 Electromagnetic theory (general) 81V45 Atomic physics 78A02 Foundations in optics and electromagnetic theory 81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
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##### References:
 [1] O. Laporte and G. E. Uhlenbeck, Phys. Rev.,37, 1380 (1931). · Zbl 0002.09001 [2] J. R. Oppenheimer, Phys. Rev.,38, 725 (1931). · Zbl 0002.30601 [3] R. Mignani, E. Recami, and M. Baldo, Lett. Nuovo Cimento,11, 568 (1974). [4] A. A. Borgardt, Zh. Eksp. Teor. Fiz.,34, 334 (1958). [5] H. E. Moses, Nuovo Cimento Suppl.,7, 1 (1958). [6] J. S. Lomont, Phys. Rev.,111, 1710 (1958). · Zbl 0086.22401 [7] V. I. Fushchich and A. G. Nikitin, Symmetry of Maxwell’s Equations [in Russian], Naukova Dumka, Kiev (1983). · Zbl 0531.35003 [8] V. M. Simulik, ?Lagrangian and algebra-theoretical analysis of the Dirac-like form of Maxwell’s equations,? in: Group-Theoretical Investigations of the Equations of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainain SSR Academy of Sciences, Kiev (1985), pp. 130-133. [9] E. Giannetto, Lett. Nuovo Cimento,44, 140 (1985). [10] H. Sallhofer, Z. Naturforsch. Teil A,41, 463 (1986). [11] V. M. Simulik, ?A new method of finding conserved quantities for the electromagnetic field,? in: Quantum Field Theory and High-Energy Physics [in Russian], Moscow State University, Moscow (1989) pp. 185-190. [12] E. Bessel-Hagen, Math. Ann.,84, 258 (1921). · JFM 48.0877.02 [13] I. Yu. Krivskii and V. M. Simulik, Ukr. Fiz. Zh.,30, 1457 (1985). [14] I. Yu. Krivskii and V. M. Simulik, Teor. Mat. Fiz.,80, 274 (1989). [15] I. Yu. Krivskii and V. M. Simulik, Teor. Mat. Fiz.,80, 340 (1989). [16] W. I. Fushchich, I. Yu. Krivsky, and V. M. Simulik, Nuovo Cimento B,103, 423 (1989). [17] N. Kh. Ibragimov, Teor. Mat. Fiz.,1, 350 (1969). [18] I. Yu. Krivskii and V. M. Simulik, ?Quasirelativistic Hamiltonian for the interaction of charged particles with the electromagnetic field in terms of field strengths,? in: Ninth All-Union Conference on the Theory of Atoms and Atomic Spectra. Abstracts of Papers, Uzhgorod State University, Uzhgorod (1985), p. 157. [19] I. Yu. Krivskii and V. M. Simulik, in: Problems of Atomic Science and Technologies, Ser. General and Nuclear Physics, No. 1(34) [in Russian] (1986), p. 29. [20] A. Sudbery, J. Phys. A,19, L33 (1986). [21] V. I. Fushchich, I. Yu. Krivskii, and V. M. Simulik, ?On vector Lagrangians for the electromagnetic and spinor fields,? Preprint 87.54 [in Russian], Institute of Mathematics, Ukrainian SSR Academy of Sciences, Kiev (1987). [22] V. I. Fushchich, I. Yu. Krivskii, and V. M. Simulik, ?Non-Lie symmetries and Noether analysis of conservation laws for the Dirac equation,? Preprint 89.49 [in Russian], Institute of Mathematics, Ukrainian SSR Academy of Sciences, Kiev (1989). [23] I. Yu. Krivskii and V. M. Simulik, ?On a Lagrangian approach for the electromagnetic field in terms of the field strengths and conservation laws,? Preprint 85-13 [in Russian], Institute of Nuclear Physics, Ukrainian SSR Academy of Sciences, Kiev (1985). [24] I. Yu. Krivskii and V. M. Simulik, ?Noether’s theorem for transformations of three types,? Preprint 85-12 [in Russian], Institute of Nuclear Physics, Ukrainian SSR Academy of Sciences, Kiev (1985). [25] D. M. Lipkin, J. Math. Phys.,5, 696 (1964). · Zbl 0117.22505 [26] D. M. Fradkin, J. Math. Phys.,6, 879 (1965). [27] D. B. Fairlie, Nuovo Dimento,37, 897 (1965). · Zbl 0131.44004 [28] V. S. Vladimirov and I. V. Volovich, Usp. Mat. Nauk,40, 17 (1985). [29] V. S. Vladimirov and I. V. Volovich, Dokl. Akad. Nauk SSSR,280, 328 (1985). [30] V. I. Fushchich, V. M. Shtelen’, and I. I. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989). · Zbl 0727.58002
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