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arXiv:quant-ph/0004109 · International Journal of Theoretical Physics

Hidden Variables or Positive Probabilities?

Tony Rothman, E. C. G. Sudarshan

Published Aug 1, 2001• 20 citations
arXivHTMLPDF
DOI
10.1023/a:1017565730083

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It is proved that in non-relativistic quantum mechanics (without spin) the transition probability may be described in terms of particle paths, every path having a (positive) probability. This leads to a stochastic hidden variables theory providing an intuitive picture of particle motion. The change of velocity at every time has a probability that depends on the potential over a large region, at a difference with the local action of classical dynamics. Thus the hidden variables theory is non-local like Bohm's, but not deterministic.
A simple relativistic quantum hidden-variable theory of particle trajectories, similar to the Bohm theory but without nonlocal forces between the particles, is proposed. To provide compatibility with statistical predictions of quantum mechanics one needs to assume the initial probability density |psi|^2 of particle positions in spacetime, which is the only source of nonlocality in the theory. This demonstrates that the usual Bohm hidden-variable theory contains much more nonlocality than required by the Bell theorem.
The paper argues that far from challenging-or even refuting-Bohm's quantum theory, the no-hidden-variables theorems in fact support the Bohmian ontology for quantum mechanics. The reason is that (i) all measurements come down to position measurements; and (ii) Bohm's theory provides a clear and coherent explanation of the measurement outcome statistics based on an ontology of particle positions, a law for their evolution and a probability measure linked with that law. What the no-hidden-variables theorems teach us is that (i) one cannot infer the properties that the physical systems possess from observables; and that (ii) measurements, being an interaction like other interactions, change the state of the measured system.