Dynamical equations for time-ordered Green's functions: from the Keldysh time-loop contour to equilibrium at finite and zero temperature

TL;DR

This paper investigates the dynamical equations governing time-ordered Green's functions across different temperature regimes, revealing that a modified Dyson equation is necessary outside the zero-temperature equilibrium case, with implications for non-equilibrium quantum systems.

Contribution

It derives a generalized dynamical equation for time-ordered Green's functions valid at finite temperature and non-equilibrium, extending the conventional Dyson equation framework.

Findings

At equilibrium and zero temperature, the Dyson equation holds.

At finite temperature or non-equilibrium, a modified Dyson equation applies.

The formalism aligns with Matsubara Green's functions and justifies previous correction terms.

Abstract

We study the dynamical equation of the time-ordered Green's function at finite temperature. We show that the time-ordered Green's function obeys a conventional Dyson equation only at equilibrium and in the limit of zero-temperature. In all other cases, i.e. finite-temperature at equilibrium or non-equilibrium, the time-ordered Green's function obeys instead a modified Dyson equation. The derivation of this result is obtained from the general formalism of the non-equilibrium Green's functions on the Keldysh time-loop contour. At equilibrium, our result is fully consistent with the Matsubara temperature Green's function formalism and also justifies rigorously the correction terms introduced in an ad hoc way with Hedin and Lundqvist. Our results show that one should use the appropriate dynamical equation for the time-ordered Green's function when working beyond the equilibrium zero-temperature limit.