  We consider quantum corrections to classical real time correlation functions
at finite temperature. We derive a semi-classical expansion in powers of
$\hbar$ with coefficients including all orders in the coupling constant. We
give explicit expressions up to order $\hbar^2$. We restrict ourselves to a
scalar theory. This method, if extended to gauge theories, might be used to
compute quantum corrections to the high temperature baryon number violation
rate in the Standard Model.
