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

The aim of this paper is twofold. to probe the statistical mechanical properties of interacting quantum fields, and to provide 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 set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master (n = infinity PI) effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the nPI effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called staving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical Version of which bring the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum held theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.