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.

This chapter derives the time-dependent Ginzburg–Landau equation from first principles, by relying on the same formal approach that has led to the time-dependent Gross–Pitaevskii equation of Chapter 28. Specifically, the time-dependent Ginzburg–Landau equation holds close to the critical temperature of the initial equilibrium preparation and in the (extreme) BCS limit of the BCS–BEC crossover, when the Cooper pairs are largely overlapping with each other. Care has to be exerted when dealing with the analytic properties in the wave-vector and frequency space of the normal and anomalous particle–particle bubbles.

This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two (t and t′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.

This chapter introduces the Nambu representation for the pseudo-spinor fields and expresses the system Hamiltonian in terms of them. In this way, the anomalous single-particle Green’s function is made to match the form of the single-particle Green’s functions treated in Part I, where an even number of creation and destruction operators appear. On physical grounds, this approach exploits the fact that opposite-spin fermions are coupled in pairs. The special role played by the Hartree–Fock self-energy for a superfluid Fermi system is duly emphasized.

The nonequilibrium diagrammatics and the Dyson equations contain integrations over the time variables that run over a generic contour. In both cases, the time variables run first forward and then backward along the ordinary time axis. For computational purposes, it is then required to convert these time integrals into ordinary time integrals. To this end, it is first necessary to single out all possible combinations of the pair of time variables in the contour single-particle Green’s function. This is what is done in the present chapter.

This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.

Similar to Chapter 20 of Part I, 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. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.

This chapter provides a concise account of the salient features of the BCS–BEC crossover. After a brief historical review of the topic, the key features of the BCS–BEC crossover are recalled. In particular, the BCS wave function for the ground state is shown to contain the BEC state of composite bosons as a limiting situation, and the special role played in this context by the chemical potential is pointed out. The need for pairing fluctuations beyond mean field is also emphasized, together with the occurrence of two (coupling and temperature dependent) lengths, which measure the correlation either within a pair of fermions with opposite spins or among different pairs. The limiting physical situations corresponding to the Ginzburg–Landau and Gross–Pitaevskii equations are finally considered.