By Vilgelm Ilich Fushchich, A. G. Nikitin, W. I. Fushchich

This publication is dedicated to the research of outdated (classical) and new (non-Lie) symmetries of the basic equations of quantum mechanics and classical box conception, and to the class and algebraic-theoretical deduction of equations of movement of arbitrary spin debris in either Poincaré invariant method. The authors current targeted information regarding the representations of the Galilei and Poincaré teams and their attainable generalizations, and expound a brand new method for investigating symmetries of partial differential equations; this results in discovering formerly unknown algebras and teams of invariance of the Dirac, Maxwell and different equations.

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1. Introduction Maxwell’s equations are one of the main foundations of modern physics. Describing a very extensive branch of physical phenomena, these equations are distinguished by their extremely simple and elegant form. But the source of this simplicity and elegance lies in the remarkably rich symmetry of Maxwell’s equations. The investigation of the symmetry of Maxwell’s equations has a long and glorious history. In 1893, having written these equations in the vector notations, Heaviside [219] pointed out that they are invariant under the change E→H, H→ E, 28 Chapter 1.

G. for the construction of nonlinear generalizations of Maxwell’s equations, being invariant under the conformal group. 6. Symmetry Under the P-, T- and C-Transformations Invariance under the transformations considered above does not exhaust all symmetry properties of Maxwell’s equations. We will see later that these equations are invariant also with respect to nonlocal transformations not connected with geometrical space-time symmetries. But there exist discrete symmetry transformations of dependent and independent variables which we not considered in the previous subsection.

In conclusion we notice that the algebra AC(1,3) is isomorphic to the Lie algebra of the group O(2,4) (the group of pseudoorthogonal matrices conserving the vector length in the (2+4) Minkowski space. 49) g44 g55 0, gmn 0, m≠ n. The existence of this isomorphism means that the problem of the description of the representations of the algebra AC(1,3) reduces to the description of the representations of the algebra AO(2,4). 8. 22) inasmuch as such representations are used for description of actual physical fields.