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Foundations of Physics

Trajectories and causal phase-space approach to relativistic quantum mechanics

Peter Holland, Anastasios Kyprianidis, Jean Pierre Vigier

Published May 1, 1987• 2 citations
Publisher
DOI
10.1007/bf01559700

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We show how the causal interpretation provides a physical basis for the Wigner phase-space formulation of quantum mechanics by analyzing the case of a relativistic spin-zero particle in an external electromagnetic field. The Wigner function can be interpreted as both a generalized transformation and a field which defines a quantum potential in phase space. We also propose a Lorentz-vector distribution function which always associates positive probability densities with the real motions of particles and antiparticles in phase space.
The constraints imposed by relativistic covariance on the law of motion of fermions in the trajectory interpretation of quantum mechanics are examined. It is found that the quantum current satisfies a uniqueness property that fixes the guidance law. This in turn fixes the nonrelativistic equation which generally differs from that originally given by de Broglie and Bohm for electrons. These results extend to Bohm's noncovariant many-body theory. The demonstration does not exclude other guidance equations and this is shown by the explicit construction of an alternative Lorentz covariant many-body theory possessing future-causal space-time paths.