Stochastic interpretation of Kadanoff-Baym equations and their relation to Langevin processes

In this more pedagogical study we want to elucidate on stochastic aspects inherent to the (non-)equilibrium real time Green's function description (or `closed time path Green's function' -- CTPGF) of transport equations, the so called `Kadanoff-Baym equations'. As a toy model we couple a free scalar boson quantum field to an exemplaric heat bath with some given temperature T. It will be shown in detail that the emerging transport equations have to be understood as the ensemble average over stochastic equations of Langevin type. This corresponds to the equivalence of the influence functional approach by Feynman and Vernon and the CTP technique. The former, however, gives a more intuitive physical picture. In particular the physical role of (quantum) noise and the connection of its correlation kernel to the Kadanoff-Baym equations will be discussed. The inherent presence of noise and dissipation related by the fluctuation-dissipation theorem guarantees that the modes or particles become thermally populated on average in the long-time limit. For long wavelength modes with momenta much less than the temperature the emerging wave equation do behave nearly as classical. On the other hand, a kinetic transport description can be obtained in the semi-classical particle regime. Including fluctuations, its form resembles that of a phenomenological Boltzmann-Langevin description. However, we will point out some severe discrepancies in comparison to the Boltzmann- Langevin scheme. As a further byproduct we also note how the occurrence of so called pinch singularities is circumvented by a clear physical necessity of damping within the one-particle propagator.