Working in phase-space with Wigner and Weyl

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 . 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 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.