Introduction to the Keldysh formalism and applications to time-dependent density-functional theory

摘要

This paper gives an introduction to the Keldysh formalism, with emphasis onits usefulness in time-dependent density functional theory. In the first partwe introduce the Keldysh contour and the one-particle Green function defined onthis contour. We then discuss how to combine and manipulate functions withtime-arguments on the contour. The effects of electron-electron interaction canbe taken systematically into account, as we illustrate by propagating theKadanoff-Baym equations for the second-order self-energy approximation. It isimportant to use conserving approximations such that the evolution of theelectron density, momentum, and total energy agrees with the macroscopicconservation laws. One of the main topics in this paper is the non-interactingGreen function, which is the relevant quantity for time-dependent densityfunctional theory. We discuss this Green function in detail, and show how theKeldysh contour in a simple way allows us to derive the time-dependentKohn-Sham potential from an action functional. The formalism in a similar wayleads to response functions that obey the causality principle. To illustratethese points, we discuss the time-dependent optimized effective potentialequations.