Exact classical stochastic representations of the many-body quantum dynamics

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.