I consider unequal time correlation functions in a hot non-Abelian gauge theory. Here "time" refers to real (Minkowski-) time. Such correlation functions determine the dynamics of a system out of but close to thermal equilibrium.

Suppose the correlation function of interest receives a contribution from soft gauge fields with a typical wavelength of order of the magnetic screening length 1/(g^2 T). At this scale finite temperature perturbation theory breaks down. Even equal time or equilibrium quantities associated with this scale are difficult to deal with. So how difficult will be unequal time?

Getting the complete result would require a real time lattice simulation of the full quantum field theory. So far, nobody has been able to perform such a simulation. Obtaining the result at leading logarithmic order in the gauge coupling turns out to be surprisingly simple . One can use an effective classical theory which is described by a Langevin equation in (3+1) dimensions and which is relatively easy to treat on a lattice.

To arrive at this effective theory one has to integrate out all field modes with momenta larger than g^2 T, which can be done in perturbation theory. First one integrates out the so called hard modes with momenta of order T. At leading order in the gauge coupling one obtains the well known hard thermal loop effective theory (HTLET) . This is an effective theory for momenta of order gT and of order g^2 T.

The next step is to integrate out the field modes with momenta of order g T using the HTLET. The hard (but instructive) way of integrating out the scale g T is to compute loop diagrams within the HTLET for loop momenta of order gT and external momenta small compared to gT. The easy way (for more details click here) is to use the formulation of the HTLET in terms of non-Abelian Vlasov equations (see [Blaizot93]). In this way one takes full advantage of the fact that the field modes which are left in the problem have momenta small compared to the temperature and thus behave classically. One can then integrate out the scale g T by using classical field equations of motion. One obtains a Vlasov-Boltzmann equation for the soft, non-perturbative field modes. In a leading logarithmic approximation this equation can be solved which yields the Langevin equation. A very nice feature of this effective theory is that it is UV finite (see [Arnold98]).

How to apply this effective theory to get a physical result (the electroweak baryon number violation rate) has been explained by Guy Moore in his talk. Closely related are the talks by Peter Arnold, and Christina Manuel. A more detailed but (hopefully) not too technical discussion of this subject can be found in my contributions to the proceedings of the Thermal-Field-Theory workshop TFT-98, Regensburg, Aug 98, and of the SEWM 98 conference, Copenhagen, Dec 98.