Intuitive

Classical statistical mechanics is a ' crypto-deterministic' theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distributions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as radioactive decay or spontaneous emission (cf. Whittaker (2)). Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibility of 'diffusion' of the probability 'fluid', so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics.

Quantum Mechanics as a statistical theory by J. E. Moyal

Reading Recommendations

• Quantum Mechanics in Phase space by Curtright and Zachos

Concrete

Reading Recommendations

• See also the corresponding chapter in Ballentine's Quantum Mechanics book, and also
• Canonical transformation in quantum mechanics Y. S. Kim and E. Wigner
• A Concise Treatise on Quantum Mechanics in Phase Space by Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos,  World Scientific, 2014.

Important Papers

• Quantum Mechanics as a statistical theory by J. E. Moyal

Abstract

In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as deformation quantization.

Why is it interesting?

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space".[5] This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (cf. classical limit). https://en.wikipedia.org/wiki/Phase_space_formulation