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arXiv:quant-ph/0103114 · Journal of Physics A Mathematical and General

Time-like flows of energy momentum and particle trajectories for the Klein-Gordon equation

George Horton, Christopher Dewdney, Alexei Nesteruk

Published Oct 5, 2000• 28 citations
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Abstract

The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density, which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.

DOI
10.1088/0305-4470/33/41/306

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The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.

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Comment on Time-like flows of energy momentum and particle trajectories for the Klein Gordon equation

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Horton et al ( 2000 J. Phys. A: Math. Gen. 33 7337 ) have proposed Bohm-type particle trajectories accompanying a Klein–Gordon wavefunction ψ on Minkowski space. From two vector fields on space–time, W + and W − , defined in terms of ψ, they intend to construct a time-like vector field W , the integral curves of which are the possible trajectories, by the following rule: at every space–time point, take either W = W + or W = W − depending on which is time-like. This procedure, however, is ill-defined as soon as both are time-like, or both space-like. Indeed, they cannot both be time-like, but they can well both be space-like, contrary to the central claim of Horton et al . We point out the gap in their proof, we provide a counter example, and we argue that, even for a rather arbitrary wavefunction, the points where both W + and W − are space-like can form a set of positive measure.

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The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density, which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.
Horton et al ( 2000 J. Phys. A: Math. Gen. 33 7337 ) have proposed Bohm-type particle trajectories accompanying a Klein–Gordon wavefunction ψ on Minkowski space. From two vector fields on space–time, W + and W − , defined in terms of ψ, they intend to construct a time-like vector field W , the integral curves of which are the possible trajectories, by the following rule: at every space–time point, take either W = W + or W = W − depending on which is time-like. This procedure, however, is ill-defined as soon as both are time-like, or both space-like. Indeed, they cannot both be time-like, but they can well both be space-like, contrary to the central claim of Horton et al . We point out the gap in their proof, we provide a counter example, and we argue that, even for a rather arbitrary wavefunction, the points where both W + and W − are space-like can form a set of positive measure.
A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
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.
We have discussed the quantum particle model for the case of a free particle solution in the one-dimensional Klein-Gordon nonlinear equations. This model was obtained through the two equations from the conservation laws of the classical physics, namely, the Hamilton-Jacobi equation for the relativistic motion and continuity equations. In this case, the Hamilton-Jacobi equation describes a part of the particle while the continuity equation describes the wave side. The derivation of this equation did not use the two postulates in the quantum mechanics, namely, the Einstein and de Broglie’s postulates regarding the quantization of energy and momentum. According to this derivation, the particle side has almost most of the energy of quantum particle that accumulate at a point while the wave part has only a small portion of the energy of the quantum particle that surrounds the part of the particle. In addition, this paper also shows the form of mathematical functions that represent the particle and wave parts for a free particle solution. This form is obtained through a special solution of the free particle which is a plane-wave solution.