  The Heisenberg evolution of a given unitary operator corresponds classically
to a fixed canonical transformation that is viewed through a moving coordinate
system. The operators that form the bases of the Weyl representation and its
Fourier transform, the chord representation, are, respectively, unitary
reflection and translation operators. Thus, the general semiclassical study of
unitary operators allows us to propagate arbitrary operators, including density
operators, i.e. the Wigner function. The various propagation kernels are
different representations of the superoperators which act on the space of
operators of a closed quantum system. We here present the mixed semiclassical
propagator, that takes translation chords to reflection centres, or vice versa.
In contrast to the centre-centre propagator that directly evolves Wigner
functions, it is guaranteed to be caustic free, having a simple WKB-like
universal form for a finite time, whatever the number of degrees of freedom.
Special attention is given to the near-classical region of small chords, since
this dominates the averages of observables evaluated through the Wigner
function.
