Quantum-classical correspondence in response theory

The correspondence principle between the quantum commutator [A, B] and the classical Poisson brackets iota h{A, B} is examined in the context of response theory. The classical response function is obtained as the leading term of the expansion of the phase space representation of the response function in terms of Weyl-Wigner transformations and is shown to increase without bound at long times as a result of ignoring divergent higher-order contributions. Systematical inclusion of higher-order contributions improves the accuracy of the h expansion at finite times. Resummation of all the higher-order terms establishes the classical-quantum correspondence <--> alpha n e iota n omega t|Jv + nh/2. The time interval of the validity of the simple classical limit [A(t), B(0)] --> iota h{A(t), B(0)} is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity.