  We study time dependent correlation functions in hot quantum and classical
field theory for the $\lambda\phi^4$ case. We set up the classical analogue of
thermal field theory and make a direct comparison between the quantum and
classical diagrams. A restriction to time independent correlation functions
gives the connection with conventional dimensional reduction. If the parameters
in the classical theory are chosen according to the dimensional reduction
matching relations, the classical expressions are cutoff independent and they
approximate the quantum expressions, provided that the external momenta and
frequencies are small with respect to the temperature.
