  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.
