Abstract
We present and amplify some of our previous statements on non-canonical interrelations between the solutions to free Dirac equation (DE) and Klein–Gordon equation (KGE). We demonstrate that all the solutions to the DE (possessing point- or string-like singularities) can be obtained via differentiation of a corresponding pair of the KGE solutions for a doublet of scalar fields. In this way, we obtain a “spinor analogue” of the mesonic Yukawa potential and previously unknown chains of solutions to DE and KGE, as well as an exceptional solution to the KGE and DE with a finite value of the field charge (“localized” de Broglie wave). The pair of scalar “potentials” is defined up to a gauge transformation under which corresponding solution of the DE remains invariant. Under transformations of Lorentz group, canonical spinor transformations form only a subclass of a more general class of transformations of the solutions to DE upon which the generating scalar potentials undergo transformations of internal symmetry intermixing their components. Under continuous turn by one complete revolution the transforming solutions, as a rule, return back to their initial values (“spinor two-valuedness” is absent). With an arbitrary solution of the DE, one can associate, apart from the standard one, a non-canonical set of conserved quantities, positive definite “energy” density among them, and with any KGE solution-positive definite “probability density”, etc. Finally, we discuss a generalization of the proposed procedure to the case when the external electromagnetic field is present.
Highlights
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Not always one can put in correspondence with such a singular solution some δ-like source: for example, this is impossible for the “flat limit” electromagnetic field of the Kerr–Newman solution in GTR with a ring-like singularity, because of the twofold structure of the solution
It has been shown that the free Dirac equation (DE) is, essentially, no more than an identical interdependency between derivatives of a doublet of a Klein–Gordon field
Summary
The close relationship that has been established in the framework of relativistic field theory between the physical Minkowski space-time geometry M and the two observed types of elementary particles, bosons and fermions, is, perhaps, the most remarkable achievement in theoretical physics. In the case of, say, massive fields it is usually postulated that the Dirac equation (DE) and the equation of Klein–Gordon (KGE) are responsible for description of particles of different spins, possess different sets of conserved quantities and transform through different representations of the Lorentz group As for their solutions, everybody knows that each component of the Dirac field identically satisfies the KGE, but not vice versa. The discovered possibility to obtain a general solution to the DE from the solutions for scalar fields poses, in the framework of the considered approach, the question about the origin of the spinor law of transformation of the Dirac field The solution of this problem had been proposed in [6] and is based on the use of the internal symmetry of the KGE system with respect to the transformations of the group SL(4, C), intermixing the components of the quadruple of the Klein–Gordon “potentials”.
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