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arXiv:0812.4880

Real-Valued Charged Fields and Interpretation of Quantum Mechanics II

Andrey Akhmeteli

Published Dec 29, 2008• 0 citations
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Abstract

In the first part of this work (http://www.arxiv.org/abs/quant-ph/0509044), it was shown that the Klein-Gordon-Maxwell electrodynamics in the unitary gauge allows natural elimination of the particle wave function and describes independent evolution of the electromagnetic field. Therefore, the electromagnetic field can be regarded as the guiding field in the Bohm interpretation of quantum mechanics. An extension of those results to the Dirac-Maxwell electrodynamics was less general, but represented at least an interesting toy model of quantum theory. Another model based on the Dirac-Maxwell electrodynamics is considered in this work. The model also typically allows elimination of the wave function and describes independent evolution of the electromagnetic field.

DOI
10.48550/arxiv.0812.4880

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In the first part of this work (http://www.arxiv.org/abs/quant-ph/0509044), it was shown that the Klein-Gordon-Maxwell electrodynamics in the unitary gauge allows natural elimination of the particle wave function and describes independent evolution of the electromagnetic field. Therefore, the electromagnetic field can be regarded as the guiding field in the Bohm interpretation of quantum mechanics. An extension of those results to the Dirac-Maxwell electrodynamics was less general, but represented at least an interesting toy model of quantum theory. Another model based on the Dirac-Maxwell electrodynamics is considered in this work. The model also typically allows elimination of the wave function and describes independent evolution of the electromagnetic field.
Schroedinger (Nature, v.169, p.538 (1952)) demonstrated that, contrary to the widespread belief, charged particles may be described by real fields. Therefore the sets of solutions with real-valued charged fields are considered in the present work for some versions of (non-second-quantized) quantum electrodynamics (for Dirac spinors "real-valued" is understood as "satisfying the Majorana condition"). In some of the versions any solution may be obtained from a solution from those sets by a gauge transform. The solutions from those sets have common features suggesting a natural interpretation along the lines of the Bohm interpretation, but no quantum potentials arise, and it is the electromagnetic field, not the wave function, that plays the role of the guiding field.
The article contains a review and new results of some mathematical models relevant to the interpretation of quantum mechanics and emulating well-known quantum gauge theories, such as scalar electrodynamics (Klein–Gordon–Maxwell electrodynamics), spinor electrodynamics (Dirac–Maxwell electrodynamics), etc. In these models, evolution is typically described by modified Maxwell equations. In the case of scalar electrodynamics, the scalar complex wave function can be made real by a gauge transformation, the wave function can be algebraically eliminated from the equations of scalar electrodynamics, and the resulting modified Maxwell equations describe the independent evolution of the electromagnetic field. Similar results were obtained for spinor electrodynamics. Three out of four components of the Dirac spinor can be algebraically eliminated from the Dirac equation, and the remaining component can be made real by a gauge transformation. A similar result was obtained for the Dirac equation in the Yang–Mills field. As quantum gauge theories play a central role in modern physics, the approach of this article may be sufficiently general. One-particle wave functions can be modeled as plasma-like collections of a large number of particles and antiparticles. This seems to enable the simulation of quantum phase-space distribution functions, such as the Wigner distribution function, which are not necessarily non-negative.
The matter field can be naturally eliminated from the equations of the Klein-Gordon-Maxwell electrodynamics in the unitary gauge. The resulting equations describe independent dynamics of the electromagnetic field: if components of the 4-potential of the electromagnetic field and their first derivatives with respect to time are known in the entire space at some time point, the values of their second derivatives with respect to time can be calculated for the same time point, so the Cauchy problem can be posed, and integration yields the 4-potential in the entire space-time. This surprising result both permits mathematical simplification and can be useful for interpretation of quantum theory. For example, in the Bohm interpretation, the electromagnetic field can replace the wave function as the guiding field. Independent of the interpretation, quantum phenomena can be described in terms of electromagnetic field only. For the system of nonlinear partial differential equations of the Klein-Gordon-Maxwell electrodynamics, a generalized Carleman linearization procedure generates a system of linear equations in the Hilbert space, which looks like a second-quantized theory and is equivalent to the original nonlinear system on the set of solutions of the latter. Similar, but less general results are obtained for the Dirac-Maxwell electrodynamics.
Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schrödinger [Nature (London), 169, 538 (1952)], the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schrödinger conclusion that charged fields do not necessarily require complex representation. One of the two resulting real equations for the real function describes current conservation and can be obtained from the Maxwell equations in spinor electrodynamics (the Dirac-Maxwell electrodynamics). As the Dirac equation is one of the most fundamental equations, these results both belong in textbooks and can be used for development of new efficient methods and algorithms of quantum chemistry.
This article builds on recent work (Akhmeteli in Int. J. Quantum Inf. 9(Supp01):17, 2011 ; J. Math. Phys. 52:082303, 2011 ), providing a theory that is based on spinor electrodynamics, is described by a system of partial differential equations in 3+1 dimensions, but reproduces unitary evolution of a quantum field theory in the Fock space. To this end, after introduction of a complex four-potential of electromagnetic field, which generates the same electromagnetic fields as the initial real four-potential, the spinor field is algebraically eliminated from the equations of spinor electrodynamics. It is proven that the resulting equations for electromagnetic field describe independent evolution of the latter and can be embedded into a quantum field theory using a generalized Carleman linearization procedure. The theory provides a simple and at least reasonably realistic model, valuable for interpretation of quantum theory. The issues related to the Bell theorem are discussed.