The variety of approaches to the problem of the derivation of Dirac equation


Purpose. The Dirac equation is one of the fundamental equations of modern theoretical physics. It is in service more than 90 years (1928-2018). The application today is much wider than the areas of quantum mechanics, quantum field theory, atomic and nuclear physics, solid state physics. The successful derivation of some equation of mathematical physics is the first step to successful application. In such process the essence of the corresponding model of nature, the mathematical principles and the physical foundations are visualized. Here we deal with the different approaches to the problem of the Dirac equation derivation.
Methods. The quantum-mechanical, quantum field-theoretical, group-theoretical, algebraic, symmetrical, statistical, stochastical and numerical approaches are used.
Results. The 26 different ways of the Dirac equation derivation are presented. The various physical principles and mathematical formalisms are used. Three original approaches of the authors to the problem are given. They are (i) the generalization of H. Sallhofer derivation, (ii) the obtaining of the massless Dirac equation from the Maxwell equations in maximally symmetrical form, (iii) the derivation of the Dirac equation with nonzero mass from the relativistic quantum mechanics of the fermion-antifermion spin s = 1/2 doublet. In some sense the role of the Dirac equation today is demonstrated.
Conclusions. The original investigation of the problem of the Dirac equation derivation is presented. The different approaches, which are based on the various mathematical and physical principles, are considered (26 methods). The role of the scientists from Uzhhorod is shown. The importance of place of the Dirac equation in modern theoretical physics is discussed

Keywords: the Dirac equation, spinor field, quantum mechanics, field theory