Dynamics of the wigner distribution function: conservation of positivity in time

The necessary and sufficient conditions are found under which the Wigner distribution function conserves (i) positivity, and (ii) factorization. The proof is based on the study of the continuous representation of three-dimensional linear-canonical-transformed minimum-uncertainty-states. The validity of (i) and (ii) for all initial conditions leads to quadratic Hamiltonians; when the condition is relaxed to the validity for only some initial conditions, non-quadratic systems are obtained. In all the above cases, the Wigner function is a true probability and follows the classical trajectory in phase space. The calculational advantage of the method here proposed is shown in the study of parabolic tunneling in a uniform magnetic field.