Finiteness of Hot Classical Scalar Field Theory and the Plasmon Damping Rate

We investigate the renormalizability of the classical $\phi^4$ theory at finite temperature. We calculate the time-dependent two point function to two loop order and show that it can be rendered finite by the counterterms of the classical static theory. As an application the classical plasmon damping rate is found to be $\gamma = \lambda^2 T^2/1536 \pi m$. When we use the high temperature expression for $m$ given by dimensional reduction, the rate is found to agree with the quantum mechanical result.