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

Abstract

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 = ∞P I) 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.

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

“…This issue was solved by Hu and one of us in ref. [49]. Similar issues appear at every order in the Schwinger-Dyson hierarchy.…”

“…Finally, let us recover the noise self-correlation given in [49]. The situation is similar to the one in the 1PI theory.…”

“…This approach works for any effective action, but seems to be subject to a sign ambiguity. In this Section we shall develop this second approach in a systematic way, dispelling the apparent paradox afflicting the discussion in [49]. Let us reformulate the 1PI problem in terms of the fluctuation -dissipation inspired approach.…”

“…In [49] the same authors follow a different strategy, which may be regarded as a nonequilibrium generalization of the fluctuation -dissipation theorem. We shall consider here only the case of a field theory with no background mean fields.…”

Noise kernel in stochastic gravity and stress energy bitensor of quantum fields in curved spacetimes

“…We begin our studies here with the lowest order term, i.e., the 2 point function of the energy momentum tensor which contains the 4th order correlation of the quantum field (or gravitons when they are considered as matter source). 4 Progress is made now on how to characterize the higher order correlation functions of an interacting field systematically from the Schwinger-Dyson equations in terms of 'correlation noise' [19,20], after the BBGKY hierarchy. This may prove useful for a correlation dynamics /stochastic semiclassical approach to quantum gravity [6].…”