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.

This chapter introduces the Hamiltonian operator in the language of second quantization, which is associated with the many-particle system to be considered throughout, including its time-dependent part. An expression is derived for the corresponding time-evolution operator, which depends only on the Hamiltonian and not on the initial preparation of the system before the time-dependent part begins to act. The connection between the Schrödinger and Heisenberg representations is discussed.

Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.

This chapter converts the Dyson equation for the contour single-particle Green’s function to real-time variables. The corresponding equations for the Matsubara, right Keldysh, left Keldysh, lesser, greater, retarded, advanced, and Keldysh components are explicitly considered. A connection is also established with the original Kadanoff–Baym equations.

This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.

This chapter applies the Wick’s theorem to the contour single-particle Green’s function. The corresponding average is represented in the interaction picture, with no need to specify at the outset the kind of contour that is used. This procedure is summarized in a set of Feynman diagrammatic rules, which are reported schematically. Here, only the normal phase is considered, while, in Part II, the Feynman rules are extended to superfluid Fermi systems.

This chapter considers an open quantum system, exemplified by a junction made up of a central region of finite size and of (at least two) connected terminals, with a time-dependent bias superposed on the terminals. For simplicity, fermions in the terminals are assumed to be noninteracting, while those in the central region are interacting. In particular, the time-dependent current flowing through the system is calculated using the Schwinger–Keldysh formalism developed in Part I for the normal phase. To this end, the present problem is framed in a more general context by adapting the Zwanzig P-Q projector operators technique. In this way, “memory” effects arise due to the transfer of information from P to Q subspaces (and vice versa).

This chapter considers the initial preparation of the many-particle system, whose control is achieved before the reference time t₀ when the time-dependent perturbation begins to act on the system. After t₀, the system is let to evolve in time according to the full time-dependent Hamiltonian. The initial control can be either full or partial. Full control signifies that at t₀ the system is prepared in a definite “pure” quantum state (like the ground state), while partial control signifies that initially the system is only known to be in a “mixture” of states with given probabilities, such that the information on the phases of the superposition is lost. These two cases are here treated separately.

The effects of the coupling to the environment can also manifest itself in a superfluid Fermi system. This chapter explicitly considers this case, by addressing the time-dependent behavior of the gap parameter following a sharp quench of the coupling parameter of the contact interaction. In this case, coupling the system to the environment is important for reaching equilibrium eventually. Several simplifying assumptions are adopted along the way for treating the problem in an as simple as possible way.

This chapter considers some of the items discussed in the previous chapters and cast them in a more formal way so as to adapt them for future developments. In this way, the integro-differential form of the Dyson equation for the contour single-particle Green’s function (as well as its integral counterpart) is obtained, which play an important role in the following chapters for capturing the dynamical evolution of the physical system.