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

Horton, G, Christopher Dewdney, Nesteruk, A

Published Jan 1, 2000• 18 citations
OpenAlex

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.

Related papers

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

George Horton, Christopher Dewdney, Alexei Nesteruk

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.

arXiv:quant-ph/0103114•Journal of Physics A Mathematical and General•Oct 5, 2000•28 citations

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

Roderich Tumulka

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.

arXiv:quant-ph/0202140•Journal of Physics A Mathematical and General•Sep 5, 2002•2 citations

A de Broglie-Bohm Like Model for Klein-Gordon Equation

O. Chavoya-Aceves

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.

arXiv:quant-ph/0304133•Il Nuovo Cimento B•Apr 21, 2003•0 citations

de Broglie's pilot wave theory for the Klein-Gordon equation

Horton, G, Christopher Dewdney

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.

arXiv:quant-ph/0109059•Sep 13, 2001•0 citations

The de Broglie-Schr\"odinger Trajectories

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De Broglie's quest for a wave-like approach capable of representing the position of moving particles is satisfied by assuming that they run along the trajectories associated, under assigned starting conditions, with a time-independent Schrodinger (or Klein-Gordon) equation. Just like in usual Dynamics, the starting conditions determine a set of virtual trajectories independent from the number, and even from the very presence, of moving particles.

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From Classical to Wave-Mechanical Dynamics

A. Orefice, Raffaele Giovanelli, Domenico Ditto

The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory.

arXiv:1406.4968•Jun 19, 2014•2 citations
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.
The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - were recently shown to be associated with exact sets of ray-trajectories (coupled by a "Wave Potential" function encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory.