Fourth-order full quantum correlations from a Langevin-Schwinger-Dyson equation

It is well known that some quantum and statistical fluctuations of a quantum field may be recovered by adding suitable stochastic sources to the mean field equations derived from the Schwinger-Keldysh (closed-time path) effective action. In this paper we show that this method can be extended to higher correlations and higher (n-particle irreducible) effective actions. As an example, we investigate third- and fourth-order correlations by adding stochastic sources to the Schwinger-Dyson equations derived from the 2-particle irreducible effective action. This method is a simple way to investigate the nonlinear dynamics of quantum fluctuations.