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 h 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 < v+n parallel to alpha<^>(t)parallel to v ><->alpha(n)e(iota n omega t)parallel to(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.