The finite temperature real time \hbar^2 corrections in quantum mechanics

We study non-perturbative real time correlation functions at finite temperature. In order to see whether the classical term gives a good approximation in the high temperature limit T >> \hbar\omega, we consider the first \hbar^2 quantum corrections. We find that for the simplest non-trivial case, the quantum mechanical anharmonic oscillator, the classical result is reliable only for moderately large times: after some time t_* the classical approximation breaks down even at high temperatures. Moreover, the result for the first quantum corrections cannot, in general, be reproduced by modifying the parameters of the classical theory.