  The trace of an arbitrary product of quantum operators with the density
operator is rendered as a multiple phase space integral of the product of their
Weyl symbols with the Wigner function. Interspersing the factors with various
evolution operators, one obtains an evolving correlation. The kernel for the
matching multiple integral that evolves within the Weyl representation is
identified with the trace of a single compound unitary operator. Its evaluation
within a semiclassical approximation then becomes a sum over the periodic
trajectories of the corresponding classical compound canonical transformation.
  The search for periodic trajectories can be bypassed by an exactly equivalent
initial value scheme, which involves a change of integration variable and a
reduced compound unitary operator. Restriction of all the operators to
observables with smooth non-oscillatory Weyl symbols reduces the evolving
correlation to a single phase space integral. If each observable undergoes
independent Heisenberg evolution, the overall correlation evolves classically.
Otherwise, the kernel acquires a nonclassical phase factor, though it still
depends on a purely classical compound trajectory: e.g. the fase for a double
return of the quantum Loschmidt echo does not coincide with twice the phase for
a single echo.
