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+1 different complete commuting sets of observables (e.g.
vecq,
vecp and
n−1 other sets). In general the particle velocities do not coincide with the de Broglie-Bohm velocities.
In two previous papers a causal interpretation of the quantum theory was developed which involved the hypothesis that a quantum-mechanical system contains a precisely defined particle variable x but that, at present, we are restricted to calculating the probability density
P(x,t) that the particle is at the position x. It was shown that the assumption that $P(\mathrm{x}, t)={|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$ is consistent, in the sense that if it holds initially, the equations of motion of the particles will cause this relation to be maintained for all time. In this paper, we extend the theory by showing that as a result of random collisions, an arbitrary probability density will ultimately decay into one with a density of ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$. Since all quantum-mechanical experiments to date have been concerned with statistical ensembles of systems that have been colliding with other systems for a very long time, it is therefore inevitable that as we draw samples from such ensembles, the probability density of systems with particles at the point x will be equal to ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$.In the previous papers we also pointed out that, within the conceptual framework of the causal interpretation, it was possible to suggest mathematical theories more general than are permitted by the usual interpretation and that these more general theories might be needed in the domain of ${10}^{\ensuremath{-}13}$ cm, where present theories seem to fail. However, if these more general theories should apply at the level of ${10}^{\ensuremath{-}13}$ cm, then there would be a tendency to create discrepancies between
P and ${|\ensuremath{\psi}|}^{2}$, a tendency whose cumulative effects should be felt even at the atomic level, where the more general theory ought to approach the usual theory. However, because those discrepancies have been shown to die out as a result of collisions, we can expect that under normal conditions the difference between
P and ${|\ensuremath{\psi}|}^{2}$ would be negligible. Conditions are suggested, however, in which this difference might be appreciable, and experiments are indicated which might be able to test for the existence of such discrepancies.
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.