The classical approximation has given useful results, e.g. for the sphaleron rate in 1+1 and 3+1 dimensions. Classical fields at finite temperature T suffer from divergencies, which are made finite in the numerical computations by the finite lattice spacing a, chosen such that aT is of order one. The results then suffer from lattice artefacts. In the sphaleron case these can be mimimized by incorporating continuum hard thermal loop effects. The state of the art is decribed in Guy Moore's contribution.

It of course preferable to stay in the quantum theory where we know how to get rid of regularization effects by renormalization. A relatively simple approximation consists of a gaussian density matrix centered on some mean field (Hartree, large N). However, at finite temperature the equilibrium mean field is zero and a gaussian density matrix is a poor approximation. The idea is now to combine the good aspects of the classical and the gaussian approximation by identifying the mean field with the classical field. The picture is then a classical field describing a semiclassical path through phase space with quantum fluctuations acting back on it. Such fluctations should incorporate hard thermal loop effects, and more. The microcanonical ensemble at finite temperature and/or density may be obtained by averaging over time.

The equations for the mean field plus fluctuation modes can be derived from a classical effective hamiltonian. A crucial test of these ideas is now thermalization: is it classical or quantal? At the Santa Barbara meeting we conjectured quantal equilibration for intermediate times, going over to classical equipartition at large (huge?) times.

The typical classical field is inhomogeneous, which makes the computational effort quite large. For this reason we studied toy models in 1+1 dimensions. In an abelian Higgs model with classical Bose fields and quantal fermions (playing the role of fluctuations) we found the fermions to equilibrate approximately with time dependent termperature and chemical potential (see the contribution of Gert Aarts). In this talk I presented preliminary results for the phi^4 model. There appears to be an intermediate time scale on which the fluctuations thermalize quantally (with a Bose-Einstein distribution). However, at much larger times (of order 1000 times the `plasmon' period) we see indications of classical equipartition.

This work was done in collaboration with Mischa Salle, continuing previous work with Gert Aarts.