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Canadian Journal of Physics

Pilot-wave theory for a relativistic spin zero particle

Dan N. Vollick

Published May 11, 2020• 1 citation
Publisher

Abstract

In the usual approach to the pilot-wave theory for a spin zero particle one starts with the Klein–Gordon equation, which is the relativistic generalization of the Schrodinger equation. This approach encounters several difficulties including superluminal motion and particle trajectories that move backwards in time. In this paper, I start with the relativistic classical Hamilton–Jacobi equation and introduce the quantum potential in a way that avoids the preceding difficulties. Particle trajectories are time-like or null and are future-pointing. The wave equation satisfied by the field is a nonlinear generalization of the Klein–Gordon equation.

DOI
10.1139/cjp-2020-0012

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In the usual approach to the pilot-wave theory for a spin zero particle one starts with the Klein–Gordon equation, which is the relativistic generalization of the Schrodinger equation. This approach encounters several difficulties including superluminal motion and particle trajectories that move backwards in time. In this paper, I start with the relativistic classical Hamilton–Jacobi equation and introduce the quantum potential in a way that avoids the preceding difficulties. Particle trajectories are time-like or null and are future-pointing. The wave equation satisfied by the field is a nonlinear generalization of the Klein–Gordon equation.
We illustrate, using a simple model, that in the usual formulation the time-component of the Klein-Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if one uses a covariant extension of the canonical formulation of relativistic point particle dynamics proposed by Gitman and Tyutin.
A relativistic wave equation describes the motion of a particle that has speed approaching that of light. A characteristic feature of relativistic wave equation is that the spin of the particle is built into the theory from the start and cannot be added afterwards as Pauli added electron spin to Schrödinger non-relativistic equation. In the next section is described a spin zero particle known as Klein Gordon equation. The subsequent section will deal with Dirac equation describing spin 1 / 2 particles. We shall continue to use three dimensional vector notations rather than the the four dimensional notation of special relativity theory. However, the invariance of an equation under Lorentz transformation can usually be inferred from its symmetry between the space coordinate and time.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.
The goal of this paper is an extension of the de Broglie wave mechanics model for a single spinless particle in an electromagnetic field. The analysis indicates that the motion of a particle separates naturally into particle dynamics through the classical Hamilton-Jacobi equation and quantum wave behavior through a pilot or interaction wave equation. The interaction wave equation travels at the classical particle velocity. We study gauge invariance and interpret it in the light of the interaction wave. The Heisenberg uncertainty relations are shown to be implicit in the interaction wave. We also develop a complex quantum-mechanical, relativistic energy-momentum conservation expression using a complex quantum-mechanical four-vector.
This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.