The topic of this chapter is the wave function – what it is, how it is to be interpreted and how information can be extracted from it. To this end, the notion of operators in quantum physics is introduced. And the statistical interpretation called the Born interpretation is discussed. This discussion also involves terms such as expectation values and standard deviations. The first part, however, is dedicated to a brief outline of how quantum theory came about – who were the key people involved, and how the theory grew out of a need for understanding certain natural phenomena. Parallels are drawn to the historical development of our understanding of light. At a time when it was generally understood that light is to be explained in terms of travelling waves, an additional understanding of light consisting of small quanta turned out to be required. It was in this context that Louis de Broglie introduced the idea that matter, which finally was known to consist of particles – atoms – must be perceived as waves as well. Finally, formal aspects such as Dirac notation and inner products are briefly addressed. And units are introduced which allow for convenient implementations in the following chapters.

In this chapter we present the Schwinger–Keldysh effective action in the so-called ‘in-in’, or ‘closed-time-path’ (CTP) formalism necessary for the derivation of the dynamics of expectation values. The real and causal equation of motion derived therefrom ameliorates the deficiency of the ‘in-out’ effective action which produces an acausal equation of motion for an effective geometry that is complex, thus marring the physical meaning of effects related to backreaction, such as dissipation. We construct the in-in effective action for quantum fields in curved spacetime, show that the regularization required is the same as in the in-out formulation, and show how it can be used to treat problems in nonequilibrium quantum processes by considering initial states described by a density matrix. We then show two applications: (1) the damping of anisotropy in a Bianchi Type I universe from the semiclassical Einstein equation for conformal fields derived from the CTP effective action; and (2) higher-loop calculations, renormalization of the in-in effective action, and the calculation of vacuum expectation values of the stress-energy tensor for a Phi-4 field. The last part describes thermal field theory in its CTP formulation.