Semiclassical evolution of correlations between observables
Resumo
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 nonoscillatory
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.
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