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Historically, correspondence rules and quantum quasi-distributions were motivated by classical mechanics as a guide for obtaining quantum operators and quantum corrections to classical results. In this paper, we start with quantum mechanics and show how to derive the infinite number of quantum quasi-distributions and corresponding c-functions. An interesting aspect of our approach is that it shows how the c-numbers of position and momentum arise from the quantum operator.

We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics., 50 pages, to appear in SIAM J. Math. Anal

The Weyl procedure associates a function of two ordinary variables, called the c-function or symbol, with an operator, called the Weyl operator of the symbol. One generally formulates this association by defining the operator corresponding to a given symbol. In this paper we consider the reverse problem: Given the Weyl operator, what is the matching symbol? We give a number of explicit formulas for obtaining the symbol that would generate an arbitrary Weyl operator, and we illustrate each form with an example.

Quantum phase-space distributions offer a royal road into the fascinating quantum–classical interface; the Wigner function being the first and best example. However, the subject is frequent with subtleties and textbook-level misinformation; e.g. “The Wigner distribution can give wrong answers for some operator expectation values” [1]. Since the Wigner distribution is just another representation of the density matrix, it must yield correct answers. To that end, Marlan Scully has asked at several international conferences (the 2015 Prague conference being one of them) the following question: “Starting with the density matrix (not the Moyal characteristic function), could you give me a simple direct derivation of the Wigner distribution?” Section 'A jump start' contains his answer. In Appendix D, we give a related treatment and make contact with other approaches. We hope that as a result of our studies, the Wigner distribution will become more deeply appreciated.