Non-Equilibrium Field Theory for Dynamics Starting from Arbitrary Athermal Initial Conditions

TL;DR

This paper extends the Schwinger Keldysh field theory to describe non-equilibrium quantum dynamics starting from arbitrary initial states, providing exact solutions for non-interacting systems and insights into initial correlations.

Contribution

It introduces a formalism to handle arbitrary initial density matrices in non-equilibrium quantum field theory, applicable to both Bosons and Fermions, and for open and closed systems.

Findings

Exact expressions for correlation functions in non-interacting systems with arbitrary initial states.

Method to quantify Wick's theorem violation due to initial correlations.

Application to density and current profiles in open quantum systems.

Abstract

Schwinger Keldysh field theory is a widely used paradigm to study non-equilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe non-equilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both Bosons and Fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions w.r.t. initial sources to obtain physical observables. The set of derivatives depend on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for non-interacting systems. However our formalism constructs intermediate ``n-particle Green's functions'' which obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many body correlation functions of a non-interacting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem, and relate it to presence of connected multi-particle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many body Fermionic and Bosonic open quantum systems initialized to non-trivial density matrices. We have also shown how this formalism can be extended to interacting many body systems.