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arXiv:hep-ph/9410378 · Modern Physics Letters A

CAUSAL QUANTUM MECHANICS TREATING POSITION AND MOMENTUM SYMMETRICALLY

Shasanka Mohan Roy, Virendra Singh

Published Mar 14, 1995• 17 citations
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Abstract

De Broglie and Bohm formulated a causal quantum mechanics with a phase space density whose integral over momentum reproduces the position probability density of the usual statistical quantum theory. We propose a causal quantum theory with a joint probability distribution such that the separate probability distributions for position and momentum agree with the usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive-definite and obeys the Liouville condition.

DOI
10.1142/s0217732395000752

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De Broglie and Bohm formulated a causal quantum mechanics with a phase space density whose integral over momentum reproduces the position probability density of the usual statistical quantum theory. We propose a causal quantum theory with a joint probability distribution such that the separate probability distributions for position and momentum agree with the usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive-definite and obeys the Liouville condition.
Present quantum theory, which is statistical in nature, does not predict joint probability distribution of position and momentum because they are noncommuting. We propose a deterministic quantum theory which predicts a joint probability distribution such that the separate probability distributions for position and momentum agree with usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive definite. The theory predicts a correlation between position and momentum in individual events.
We recently constructed a causal quantum mechanics in 2 dim. phase space which is more realistic than the de Broglie-Bohm mechanics as it reproduces not just the position but also the momentum probability density of ordinary quantum theory. Here we present an even more ambitious construction in 2n dim. phase space. We conjecture that the causal Hamiltonian quantum mechanics presented here is `maximally realistic'. The positive definite phase space density reproduces as marginals the correct quantum probability densities of n+1n+1n+1 different complete commuting sets of observables (e.g. vecq\\vec qvecq, vecp\\vec pvecp and n−1n-1n−1 other sets). In general the particle velocities do not coincide with the de Broglie-Bohm velocities.
The non-relativistic quantum theory has been interpreted causally by de Broglie, David Bohm, and others, where a quantum entity is viewed as a particle with a definite position and momentum. This interpretation contrasts with the Copenhagen orthodox interpretation, which conveys uncertainty through the commutator relationship between position and momentum operators. This is further exacerbated by the de Broglie-Bohm interpretation, which uses no operators corresponding to these observables. We reconcile these opposing viewpoints by introducing mathematical nonlinear transforms in Bohmian mechanics for the momentum observable. While these nonlinear operator-like transforms cannot be easily represented as matrices in the Hilbert space, we show that they can be utilized in the de Broglie-Bohm interpretations of quantum theory.
It is shown that phase-space distribution functions that characterize a single-particle quantum system obey a certain "generalized non-negativity condition," which reflects the fact that the density operator is a positive operator. A corresponding criterion is obtained for the associated characteristic functions and is found to resemble, to some extent, Bochner's theorem of classical probability theory. Necessary and sufficient conditions on a phase-space representation of quantum mechanics are also derived, which ensure that all the possible distribution functions in that representation are non-negative; but it is also shown that such distribution functions are not joint probabilities for position and momentum. In fact, our results readily provide a new proof of theorems of Wigner, and of Cohen and Margenau, which imply that quantum mechanics cannot be formulated as a stochastic theory in phase space.