In this work we analyze the relation between the stochastic representation of nonrelativistic quantum mechanics and the Newtonian stochastic mechanics. In order to understand how the nodes of a wave function are accounted for in stochastic quantum mechanics, we solve the Fokker-Planck equation associated with the quantum-mechanical density ${\ensuremath{\rho}}_{0}$=
x2 and current
j0=0, and find that the stochastically perturbed particle does not cross the nodal surface x=0 of the density ${\ensuremath{\rho}}_{0}$. Further we consider the classical motion of a particle subject to a stochastic perturbation, in the absence of conventional external forces and friction, and moreover we consider the classical stochastic motion of a harmonically bound particle in the presence of a friction force proportional and opposite to the velocity of the particle, to show that although the quantum-mechanical equations for the probability density and current can be put in a form resembling Newton's second law, the actual significance of those equations is not related to Newtonian stochastic mechanics. From the fact that, in the stochastic representation of quantum mechanics, the particles have continuous trajectories, we infer that the interactions would possess, in this theory, a global character. We finally show that the property of reversibility in stochastic quantum mechanics depends on the existence of suitably chosen compensating drifts in the mean forward velocity or acceleration of the particle under consideration.
In this work we investigate the exact classical stochastic representations of many-body quantum dynamics.We focus on the representations in which the quantum states and the observables are linearly mapped onto classical quasiprobability distributions and functions in a certain (abstract) phase space.We demonstrate that when such representations have regular mathematical properties, they are reduced to the expansions of the density operator over a certain overcomplete operator basis.Our conclusions are supported by the fact that all the stochastic representations currently known in the literature (quantum mechanics in generalized phase space and, as it recently has been shown by us, the stochastic wave-function methods) have the mathematical structure of the above-mentioned type.We illustrate our considerations by presenting the recently derived operator mappings for the stochastic wave-function method.