New symmetries, conserved quantities and gauge nature of a free Dirac field

TL;DR

This paper explores the deep interrelations between Dirac and Klein-Gordon solutions, revealing new symmetries, conserved quantities, and gauge structures, including a novel spinor analogue of the Yukawa potential and localized wave solutions.

Contribution

It introduces a new framework linking Dirac and Klein-Gordon solutions via scalar potentials, uncovering hidden symmetries and conserved quantities, and generalizes to include external electromagnetic fields.

Findings

Dirac solutions can be derived from Klein-Gordon solutions through differentiation.

A spinor analogue of the Yukawa potential is constructed.

New localized solutions with finite charge are identified.

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. On 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 continious turn by one complete revolution the transforming solutions, as a rule, return back to their initial values ("spinor two-valuedness" is absent). With arbitrary solution of the DE one can associate, apart of 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