This chapter considers the closed-time-path Green’s functions approach when specified to equilibrium situations and shows that it offers an alternative to the more standard Matsubara plus analytic continuation procedure for obtaining physical quantities directly in real frequency. In this case, the number of independent components of the single-particle Green’s function (as well as of the related self-energy) reduces considerably, thereby making it easier to solve the Kadanoff–Baym equations. The fluctuation–dissipation theorem and the single-particle spectral function (with its related sum rule) are also considered.

Similar to Chapter 20 of Part I and Chapter 31 of Part II, 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 addresses a schematic derivation of the Lindblad Master equation (aimed at helping the reader in retracing and better identifying the essential steps made and approximations adopted in the more general derivation presented in Chapter 35), as well as the physical assumptions underlying the original Kadanoff–Baym ansatz.

This chapter discusses how a classical universe arises out of the quantum wave function of the universe. The process of decoherence is described, first in general and then applied to cosmology. The classicalization of the background spacetime (and the associated reduction in interference between saddle points) as well as the classicalization of long-wavelength perturbation modes is discussed, by studying an example of interactions between background and fluctuations, as well as interaction between perturbation modes of different wavelengths. Comments on the interpretation of the wave function are included.

This chapter starts with a description of quantum tunneling as a process taking place in imaginary or even complex time. This physical picture can be extended to include gravity, which leads to a description of Coleman–DeLuccia instantons and the nucleation of bubble universes. The mathematical analysis is complemented by a derivation of negative modes, which puts the tunneling process on a firm theoretical footing. Very similar methods can also describe the decay of spacetime via bubbles of nothing. A semiclassical view of spacetime may also lead to the existence of wormholes, of both the Lorentzian and the Euclidean variety. Their properties as well as associated puzzles are discussed in detail.

Provides a review of how the standard model of cosmology is built up, emphasizing the interplay between theory and observations. The Robertson–Walker line element is derived and used to find the Friedmann equations. Elementary solutions are discussed. In this way the hot big bang model emerges. Its implications are discussed, especially the thermal history of the universe and the existence of the cosmic microwave background radiation. The chapter concludes with a discussion of the main puzzles of the hot big bang model.

How can one describe the appearance of space and time? This chapter reviews the no-boundary proposal, which allows for a concrete calculation of the nucleation of space and time from nothing. After providing heuristic motivations for this idea, concrete examples are presented, and the stability of solutions as well as the numerical methods required to find generic solutions are discussed. A general prescription for characterizing no-boundary instantons is developed, before examining explicit minisuperspace models. A special emphasis is put on the appropriate boundary conditions, both in the path integral formalism and in the Wheeler–DeWitt equation. The robustness of solutions upon the inclusion of expected quantum gravity corrections is discussed, as well as the question of which kinds of complex metrics should be allowed. This leads to a discussion of both postdictions and predictions of the proposal.

A variational principle for gravity, based on the Einstein–Hilbert action, is presented and augmented with a discussion of surface terms and boundary conditions. The ADM or Hamiltonian formalism is introduced, and gravity is rewritten in a (1+3)-dimensional decomposition. The theory is canonically quantized, which leads to the Wheeler–DeWitt equation. The properties of this equation are discussed, as well as those of JWKB semiclassical solutions. In this way it is shown how time is recovered in a semiclassical setting.

A link between horizons, imaginary time, and temperature is developed at the heuristic level first, before being made precise in the following sections with the use of Bogolyubov transformations. This leads to the derivation of the Unruh effect, which shows that an accelerated observer experiences a temperature. Analogous methods allow one to derive the phenomenon of Hawking radiation by which black holes can evaporate, and an explicit calculation of the closely related Hawking–Page transition is provided via path integral methods in which the background spacetime is also quantized. It is further shown that due to the existence of a horizon, one may in the same way associate a temperature with de Sitter spacetime. An explicit discussion of de Sitter mode functions is included, because it relates directly to the quantization of inflationary fluctuations.