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arXiv:1901.02738

On the probability interpretation of wave functions in the Dirac theory

Yu. M. Poluéktov

Published Jan 8, 2019• 1 citation
arXivHTMLPDF

Abstract

It is suggested symmetric relative to particles and antiparticles formulation of the Dirac theory, in which the states with negative energy are excluded. The fields of particles and antiparticles are associated with the wave functions, for which there is valid the Born interpretation of being the probability amplitudes. In doing so, one eliminates different kinds of "paradoxes" in the theory, which existence is caused by an incorrect account of the states with negative energy.

DOI
10.48550/arxiv.1901.02738

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It is suggested symmetric relative to particles and antiparticles formulation of the Dirac theory, in which the states with negative energy are excluded. The fields of particles and antiparticles are associated with the wave functions, for which there is valid the Born interpretation of being the probability amplitudes. In doing so, one eliminates different kinds of "paradoxes" in the theory, which existence is caused by an incorrect account of the states with negative energy.
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As currently understood since its discovery, the bare Klein-Gordon theory consists of negative quantum probabilities which are considered to be physically meaningless if not outright obsolete. Despite this annoying setback, these negative probabilities are what led the great Paul Dirac in 1928 to the esoteric discovery of the Dirac equation. The Dirac equation led to one of the greatest advances in our understanding of the physical World. In this reading, we ask the seemingly senseless question, �Do negative probabilities exist in quantum mechanics?��. In an effort to answer this question, we arrive at the conclusion that depending on the choice one makes of the quantum probability current, one will obtain negative probabilities. We thus propose a new quantum probability current of the Klein-Gordon theory. This quantum probability current leads directly to positive definite quantum probabilities. Because these negative probabilities are in the bare Klein-Gordon theory, intrinsically a result of negative energies, the fact that we-here arrive at a theory with positive probabilities, it means that negative energy particles are not to be considered problematic as is the case in the bare Klein-Gordon theory. From an abstract-objective stand-point; in comparison with positive energy particles, the corollary is that negative energy particles should have equal chances to exist. As to why these negative energy particles do not exist, this is redolent to asking why is it that Dirac�s antimatter does not exist in equal proportions with matter. This problem of why negative energy particles not exist in equal proportions with positive energy particles is a problem that needs to be solved by a future theory.
As currently understood since its discovery, the bare Klein-Gordon theory consists of negative quantum probabilities which are considered to be physically meaningless if not outright obsolete. Despite this annoying setback, these negative probabilities are what led the great Paul Dirac in 1928 to the esoteric discovery of the Dirac Equation. The Dirac Equation led to one of the greatest advances in our understanding of the physical world. In this reading, we ask the seemingly senseless question, “Do negative probabilities exist in quantum mechanics?” In an effort to answer this question, we arrive at the conclusion that depending on the choice one makes of the quantum probability current, one will obtain negative probabilities. We thus propose a new quantum probability current of the Klein-Gordon theory. This quantum probability current leads directly to positive definite quantum probabilities. Because these negative probabilities are in the bare Klein-Gordon theory, intrinsically a result of negative energies, the fact that we here arrive at a theory with positive probabilities, means that negative energy particles are not to be considered problematic as is the case in the bare Klein-Gordon theory. From an abstract—objective stand-point; in comparison with positive energy particles, the corollary is that negative energy particles should have equal chances to exist. As to why these negative energy particles do not exist, this is analogous to asking why is it that Dirac’s antimatter does not exist in equal proportions with matter. This problem of why negative energy particles do not exist in equal proportions with positive energy particles is a problem that needs to be solved by a future theory.
The Dirac equation consistent with the principles of quantum mechanics and the special theory of relativity, introduces a set of matrices combined with the wave function of a particle in motion to give rise to the relativistic energy-momentum relation. In this paper a new hypothesis, the wave function of a particle in motion is associated with a pair of complementary waves is proposed. This hypothesis gives rise to the same relativistic energy-momentum relation and achieves results identical to those of Dirac. Additionally, both the energy-time and momentum-position uncertainty relations are derived from the complementary wave interpretation. How the complementary wave interpretation of the Dirac equation is related to the time-arrow and the four-vectors are also presented.
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Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.