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TWMS Journal of Applied and Engineering Mathematics

THE DIRAC EQUATION AS THE CONSEQUENCE OF THE QUANTUM-MECHANICAL SPIN 1/2 DOUBLET MODEL

Zajac Taras

Published Jan 1, 2013• 6 citations
OpenAlex

Abstract

The detailed consideration of the relativistic canonical quantum-mechanical model of an arbitrary ?s -multiplet is given. The group-theoretical analysis of the algebra of experimentally observable physical quantities for the s = 1/2 doublet is presented. It is shown that both the Foldy-Wouthuysen equation for the fermionic spin s = 1/2 doublet and the Dirac equation in its local representation are the consequences of the relativistic canonical quantum mechanics of the corresponding doublet. The mathematically welldefined consideration on the level of modern axiomatic approaches to the field theory is provided.

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The detailed consideration of the relativistic canonical quantum-mechanical model of an arbitrary ?s -multiplet is given. The group-theoretical analysis of the algebra of experimentally observable physical quantities for the s = 1/2 doublet is presented. It is shown that both the Foldy-Wouthuysen equation for the fermionic spin s = 1/2 doublet and the Dirac equation in its local representation are the consequences of the relativistic canonical quantum mechanics of the corresponding doublet. The mathematically welldefined consideration on the level of modern axiomatic approaches to the field theory is provided.
The detailed consideration of the relativistic canonical quantum-mechanical model of an arbitrary spin-multiplet is given. The group-theoretical analysis of the algebra of experimentally observables physical quantities for the spin 1/2 doublet is presented. It is shown that both the Foldy-Wouthuysen equation for the fermionic spin 1/2 doublet and the Dirac equation in its local representation are the consequences of the relativistic canonical quantum mechanics of the corresponding doublet. The mathematically well-defined consideration on the level of modern axiomatic approaches to the field theory is provided. The hypothesis of the antiparticle negative mass is discussed briefly in the section conclusions.
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). Therole of the scientists from Uzhhorod is shown. The importance of place of the Dirac equation in modern theoretical physics is discussed.
We give a direct link between covariant description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s = 1/2 corresponds to the Foldy-Wouthuysen mean-spin operator. We hope that this formalism will be useful in the theory of relativistic quantum information.
We give a direct link between description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss in detail the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s=1/2s=1/2s=1/2 corresponds to the Foldy-Woutheysen mean-spin operator.