Published online by Cambridge University Press: 20 January 2020
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.