Stochastic dynamics of correlations in quantum field theory: From Schwinger-Dyson to Boltzmann-Langevin equation

The aim of this paper is two-fold: in probing the statistical mechanical properties of interacting quantum fields, and in providing a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the Schwinger - Dyson equations for the correlation functions, which we describe by a closed-time-path master ($n = \infty PI$) effective action. When the hierarchy is truncated, one obtains the ordinary closed-system of correlation functions up to a certain order, and from the nPI effective action, a set of time-reversal invariant equations of motion. But when the effect of the higher order correlation functions is included (through e.g., causal factorization-- molecular chaos -- conditions, which we call 'slaving'), in the form of a correlation noise, the dynamics of the lower order correlations shows dissipative features, as familiar in the field-theory version of Boltzmann equation. We show that fluctuation-dissipation relations exist for such effectively open systems, and use them to show that such a stochastic term, which explicitly introduces quantum fluctuations on the lower order correlation functions, necessarily accompanies the dissipative term, thus leading to a Boltzmann-Langevin equation which depicts both the dissipative and stochastic dynamics of correlation functions in quantum field theory.