This chapter considers the analog of the time-dependent Hartree–Fock (mean-field) decoupling treated in Part I and extends it to the broken-symmetry phase for superfluid fermions. Two coupled equations for the “normal” and “anomalous” time-dependent single-particle Green’s functions are obtained, which extend to nonequilibrium situations the equations originally obtained at equilibrium by Gor’kov, soon after the BCS original article on the theory of superconductivity. Accordingly, the time-dependent gap (order) parameter is also introduced.

This chapter derives from first principles the time-dependent Gross–Pitaevskii equation, which describes the time-dependent behavior of the condensate wave function associated with the composite bosons that form on the BEC side of the BCS–BEC crossover at sufficiently low temperature. The derivation relies on the Green’s functions method for nonequilibrium problems developed before and explores the assumption that the fermionic chemical potential, associated with the initial preparation of the system at thermodynamic equilibrium, is the largest energy scale in the problem. The relation between the scattering length for composite bosons and the scattering length for the constituent fermions is also discussed.

This chapter gives a concise overview about a number of specific physical problems, which are of recent, current, and possibly future interest, problems that can be ideally dealt with in terms of the nonequilibrium Schwinger–Keldysh Green’s functions technique developed at a formal level in Parts I and II. Accordingly, this chapter aims at providing a synthetic demonstration of the versatility of the Schwinger–Keldysh technique, especially in the view of possible future applications to scientific problems as well as to technological issues. In particular, it considers the main features associated with closed and driven open quantum systems, spectroscopic problems related to pump and probe photoemission, metastable photo-induced superconductivity, dynamics induced by quenches and rumps in “closed” quantum systems with emphasis on thermalization, and driven “open” quantum systems with emphasis on dissipation. A more detailed treatment of these topics is deferred to the following chapters.

This chapter considers the extension of the t-matrix approximation to the superfluid phase, for which it is convenient to restrict from the outset to a contact-type interparticle interaction. This is because, when addressing nonequilibrium (time-dependent) situations, the extension of the fermionic t-matrix approach from the normal to the superfluid phase requires a careful account for the Nambu indices in the two-particle channels, owing to the presence of the “anomalous” single-particle Green’s functions. The ladder approximation for the many-particle T-matrix is specifically considered.

This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t₀ to the measuring time t and a backward evolution operator from t back to t₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.

The general expressions for the number density and current are first considered under the action of an external time-dependent perturbing potential of arbitrary strength and cast in terms of the single-particle lesser Green’s function. The expansion of the number density up to linear order in the perturbing potential is then considered, yielding the density–density correlation function of linear-response theory. A connection is also considered with the temperature correlation function of the Matsubara formalism via an analytic continuation in frequency space.

This chapter reconsiders the original derivation of the Kadanoff and Baym equations, which relies on a procedure of analytic continuation from imaginary to real time in terms of an “extended” Matsubara approach. The procedure of analytic continuation proves useful for formal developments, like those to be considered in Chapters 28 and 29. The case when the system Hamiltonian does not depend on time is first treated, and the procedure is then extended under appropriate assumptions to the case when the Hamiltonian depends on time.

This chapter considers the procedure originally due to Schwinger, which sets up a number of exact coupled integral equations satisfied by the Green’s functions, avoiding in this way expansions in powers of the coupling constant. This procedure relies on the source field method, where a functional differentiation with respect to a source field is suitably exploited. Specifically, this procedure is here considered for the time-dependent (nonequilibrium) case.

This chapter utilizes the Nambu pseudo-spinor field operators for the superfluid phase, to reformulate in terms of them the closed-time-path Green’s functions, the ensuing nonequilibrium Dyson equations, the conversion of contour-time to real-time arguments, and the Langreth rules. In this way, the results obtained previously within a mean-field decoupling are framed in a more general context, which will later make it possible to include beyond-mean-field effects for the superfluid phase.

This chapter considers the treatment of a few topics, which are relevant to the general purposes of the book, but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Two topics are explicitly considered, which are relevant to a useful partition of the Dyson equation and to the Keldysh formalism.

This chapter gives a general introduction to the book. The book aims to provide the readers with a practical working knowledge on how to use the tools of the contour many-body Green’s functions for time-dependent problems. Its scope is to highlight the universality and versatility of the contour Schwinger–Keldysh formalism to treat a wide class of physical phenomena. A self-contained introduction to the topic is provided together with a considerable amount of detailed derivations, which make the text accessible to graduate students with minimal training in Green’s functions methods. The book also possesses a distinct degree of originality and contains material not commonly found in other books or review articles on the subject.

This chapter considers the boundary conditions on the time variables z₁ and z₂ for the contour single-particle Green’s function, which run over a generic contour in the complex-time z-plane. Different contours of interest are then specified. The boundary conditions for the integral form of the Dyson equation are also considered.