Thermodynamic equilibrium as a symmetry of the Schwinger-Keldysh action

Extended quantum systems can be theoretically described in terms of the Schwinger-Keldysh functional integral formalism, whose action conveniently describes both dynamical and static properties. We show here that in thermal equilibrium, defined by the validity of fluctuation-dissipation relations, the action of a quantum system is invariant under a certain symmetry transformation, and thus it is distinguished from generic systems. In turn, the fluctuation-dissipation relations can be derived as the Ward-Takahashi identities associated with this symmetry. Accordingly, the latter provides an efficient test for the onset of thermodynamic equilibrium and it makes checking the validity of fluctuation-dissipation relations unnecessary. In the classical limit, this symmetry reduces to the well-known one that characterizes equilibrium in the stochastic dynamics of classical systems coupled to thermal baths, described by Langevin equations.