Intuitive

Ever since Werner Heisenberg’s 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. […] However, they too are wrong. Quantum mechanics (QM) can be consistently and autonomously formulated in phase space, with c-number position and momentum variables simultaneously placed on an equal footing, in a way that fully respects Heisenberg’s principle. […] The net result is that quantum mechanics works smoothly and consistently in phase space, where position coordinates and momenta blend together closely and symmetrically. Thus, sharing a common arena and language with classical mechanics [14], QMPS connects to its classical limit more naturally and intuitively than in the other two familiar alternate pictures, namely, the standard formulation through operators in Hilbert space, or the path integral formulation. […]

Still, as every physics undergraduate learns early on, classical phase space is built out of “c-number” position coordinates and momenta, x and p, ordinary commuting variables characterizing physical particles; whereas such observables are usually represented in quantum theory by operators that do not commute. How then can the two be reconciled? The ingenious technical solution to this problem was provided by Groenewold in 1946, and consists of a special binary operation, the ⋆-product (see Star Product), which enables x and p to maintain their conventional classical interpretation, but which also permits x and p to combine more subtly than conventional classical variables; in fact to combine in a way that is equivalent to the familiar operator algebra of Hilbert space quantum theory. Nonetheless, expectation values of quantities measured in the lab (observables) are computed in this picture of quantum mechanics by simply taking integrals of conventional functions of x and p with a quasi-probability density in phase space, the Wigner function — essentially the density matrix in this picture. But, unlike a Liouville probability density of classical statistical mechanics, this density can take provocative negative values and, indeed, these can be reconstructed from lab measurements [11]. Quantum Mechanics in Phase space by Curtright and Zachos

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

Time Evolution

The time evolution in the phase space formulation of quantum mechanics is described by the vonNeumann equation

$$ \frac{\partial f}{\partial t} = -\frac{1}{i \hbar} (f \star H - H \star f ). $$

For the Wigner function, this equation reads

$$ \frac{\partial W}{\partial t} = - \{\{ W,H \}\} = \frac{2}{ \hbar} W \left( \frac{\hbar}{2 } ( \overleftarrow \partial_x \overrightarrow \partial_p - \overleftarrow \partial_p \overrightarrow \partial_x)\right) H = - \{ W,H\} + \mathcal{O}(\hbar^2) , $$

where $\{\{ ,\}\}$ denotes the Moyal bracket and $\{ ,\}$ the Poisson bracket.

In the limit, $\hbar \to 0$ the von Neumann equation reduces to the Liouville equation.

The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently, the probability fluid is diffusive and compressible.

Reading Recommendations

• A great introduction is The algebraic way by Hiley
• 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.
• https://en.wikipedia.org/wiki/Phase_space_formulation

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.

The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner–Weyl transform.

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