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Journal of Physics A Mathematical and General

Reply to Comment on Timelike flows of energy momentum and particle trajectories for the Klein Gordon equation

George Horton, Christopher Dewdney

Published Sep 5, 2002• 3 citations
Publisher

Abstract

We demonstrate that a straightforward limiting process can be used to continue trajectories in spacetime through exceptional regions in the velocity field (identified by Tumulka ( 2000 J. Phys. A: Math. Gen. 35 7691 )) using the formalism we previously proposed (Horton G, Dewdney C and Nesteruk A 2000 J. Phys. A: Math. Gen. 33 7337 ).
DOI
10.1088/0305-4470/35/37/402

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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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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.
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.