  We consider quantum phase-space dynamics using Wigner's representation of
quantum mechanics. We stress the usefulness of the integral form for the
description of Wigner's phase-space current~$\bm J$ as an alternative to the
popular Moyal bracket. The integral form brings out the symmetries between
momentum and position representations of quantum mechanics, is numerically
stable, and allows us to perform some calculations using elementary integrals
instead of Groenewold star products. Our central result is an explicit,
elementary proof which shows that only systems up to quadratic in their
potential fulfill Liouville's theorem of volume preservation in quantum
mechanics. Contrary to a recent suggestion, our proof shows that the
non-Liouvillian character of quantum phase-space dynamics cannot be transformed
away.
