Semiclassical evolution of correlations between observables

Abstract

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.