Introduction to the Keldysh Formalism

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Abstract

In this chapter we give an introduction to the Keldysh formalism, which is an extremely useful tool for first-principles studies of nonequilibrium manyparticle systems. Of particular interest for TDDFT is the relation to nonequilibrium Green functions (NEGF), which allows us to construct exchangecorrelation potentials with memory by using diagrammatic techniques. For many problems, such as quantum transport or atoms in intense laser pulses, one needs exchange-correlation functionals with memory, and Green function techniques offer a systematic method for developing these. The Keldysh formalism is also necessary for defining response functions in TDDFT and for defining an action functional needed for deriving TDDFT from a variational principle. In this chapter, we give an introduction to the nonequilibrium Green function formalism, intended to illustrate the usefulness of the theory. The formalism does not differ much from ordinary equilibrum theory, the main difference being that all time-dependent functions are defined for time-arguments on a contour, known as the Keldysh contour.