Wigner functions and Weyl transforms of operators offer a formulation of quantum mechanics that is equivalent to the standard approach given by the Schrödinger equation. We give a short introduction and emphasize features that give insight into the nature of quantum mechanics and its relation to classical physics. A careful discussion of the classical limit and its difficulties is also given. The discussion is self-contained and includes complete derivations of the results presented.
The careful reader may be concerned about the dropping of the terms at infinity that occur during the integration by parts. This neglect can be justified by replacing in the definition, Eq. (3), with , where is a small positive constant. The calculation can now be carried out and allowed to go to zero afterward giving the same result as was found previously.