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Journal of Physics A Mathematical and General

Probability current versus charge current of a relativistic particle

B. Rosenstein, Lawrence Paul Horwitz

Published Aug 1, 1985• 28 citations
Publisher

Abstract

The probability density of detecting a relativistic particle has been found by Newton and Wigner (1949) for a point in space. This probability density must, however, be the zero component of the same vector field. This (unique) vector field-the probability current-is constructed and discussed.
DOI
10.1088/0305-4470/18/11/032

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Here the probability density of relativistic particles coordinates, satisfying the formal conditions of the quantum mechanics and the special relativity, is determined (under textbooks view, such density does not exist). It is specified for the boson, neutral and charged, scalar and vector, including the photon, the electron. Uniqueness of this design has been proved. This is a relativistic invariant, describing the probability of the particle observation in points of space-time. The time is deprived of his role of a dynamic parameter of distribution and formally equalized with other coordinates. Also the meaning of the energy-momentum distribution is clarified. Conditions of such distributions observation are described. As applied to the quantum field these structures are transformed into new characteristics of particles distribution in space-time in addition to those for momentum states. Their operators for free fields are obtained. Under such model the Heisenberg's uncertainty relations are grounded for the relativistic particle; both particles and antiparticles turn out to be different states of a single particle; vacuum state of boson quantum field has zero energy.
The quantum mechanics status of the probability vector current density has long seemed to be marginal. On one hand no systematic prescription for its construction is provided, and the special examples of it that are obtained for particular types of Hamiltonian operator could conceivably be attributed to happenstance. On the other hand this concept's key physical interpretation as local average particle flux, which flows from the equation of continuity that it is supposed to satisfy in conjunction with the probability scalar density, has been claimed to breach the uncertainty principle. Given the dispiriting impact of that claim, we straightaway point out that the subtle directional nature of the uncertainty principle makes it consistent with the measurement of local average particle flux. We next focus on the fact that the unique closed-form linear-superposition quantization of any classical Hamiltonian function yields in tandem the corresponding unique linear-superposition closed-form divergence of the probability vector current density. Because the probability vector current density is linked to the quantum physics only through the occurrence of its divergence in the equation of continuity, it is theoretically most appropriate to construct this vector field exclusively from its divergence -- analysis of the best-known "textbook" special example of a probability vector current density shows that it is thus constructed. That special example in fact leads to the physically interesting "Ehrenfest subclass" of probability vector current densities, which are closely related to their classical peers.
We show that the antisymmetric spinor tensor representation of spin-0 relativistic quantum mechanics provides a conserved current with positive definite timelike component, interpretable as probability density. The construction runs in complete analogy to the spin-1/2 case, and provides an analogously natural one-particle Hilbert space description for spin 0. Except for the free particle, the obtained formulation proves to be inequivalent to the one based on the Klein--Gordon equation. The second quantized version may lead to new field theoretical interaction terms for zero-spin particles.
The conditions for observation of the particle coordinates, required by logic of the Special Relativity and filtering the quantum field effects, are described. A general relation between the corresponding density of probability and the wave function is found. It is a relativistic invariant describing probability of the particle emergences in space -time. This density is concretized for bosons, both scalar and vector (including photon), charged and neutral, also electron. The Heisenberg's uncertainty relations have been approved in regards to relativistic particle. As applied to the quantum field, this new construction is transformed to new characteristic of the particles distribution in space-time, which complete distribution throughout impulses. The operators of these distributions and the invariant relativistic description for free quantum fields have been obtained. These new properties of the particles and fields are proposed for experimental investigations.
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.