Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action

TL;DR

This paper reveals a symmetry in the Schwinger-Keldysh action that characterizes quantum systems in thermal equilibrium, linking it to Gibbs states and fluctuation-dissipation relations, thus providing a new theoretical tool for identifying equilibrium.

Contribution

It introduces a symmetry transformation of the Schwinger-Keldysh action that uniquely characterizes quantum systems in thermal equilibrium, connecting it to established thermodynamic principles.

Findings

Symmetry transformation of the Schwinger-Keldysh action characterizes equilibrium.

This symmetry leads to the emergence of Gibbs thermal states.

Fluctuation-dissipation relations are derived as Ward-Takahashi identities.

Abstract

The time evolution of an extended quantum system can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently encodes the information about the dynamics. We show here that the action of quantum systems evolving in thermal equilibrium is invariant under a symmetry transformation which distinguishes them from generic open systems. A unitary or dissipative dynamics having this symmetry naturally leads to the emergence of a Gibbs thermal stationary state. Moreover, the fluctuation-dissipation relations characterizing the linear response of an equilibrium system to external perturbations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient check for the onset of thermodynamic equilibrium and it makes testing the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry renders the one which is known to characterize equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.