This chapter discusses perturbation theory, applied to the λϕ4 model, with a focus on dimensional regularization. It characterizes different types of Feynman diagrams. We explain the meaning of renormalization and discuss the conditions for renormalizability.

This chapter presents the technology of quantum field theory at the one-loop level including general comments and results of two and higher loop calculationsthat we will need later on. The main goal here isthe presentation of the dimensional regularization and the development of the technologyfor one-loop calculations. The next goal isthe presentation of therenormalization and the renormalization group methods. While our presentation is rather general, we will show explicit results only in QCD and QED. The case of the renormalization of electroweak interactions willbe discussed briefly. One of the important results of this chapter will be the collection of integrals necessary not only to performthe renormalization of QED and QCD but also those that will be crucial for one-loop calculations at all stages of our expedition. We collected these integrals in an appendix.

This chapter presents the familiar Schwinger–DeWitt effective action in the ‘in-out’ formalism, suitable for the computation of S-matrix scattering or transition amplitudes. The effective action method is well suited to the treatment of backreaction problems for quantum processes in dynamical background spacetimes, as it yields equations of motion for both the quantum field and the spacetime in a self-consistent way. In the second part, after a quick refresh of basic field theory and quantum fields in curved spacetime, we construct the ‘in-out’ effective action of an interacting quantum field and apply it to the effects of particle creation and interaction in the Friedmann–Lemaitre–Robertson–Walker universe. We illustrate how dimensional regularization is implemented. The third part treats the case where changes in the background spacetime and fields are gradual enough that one can perform a derivative expansion beyond the constant background, introduce momentum space representation for the propagators and obtain a quasi-local effective action in a closed form. The fourth part discusses dimensional regularization and the derivation of renormalization group equations, using the Phi-4 theory as an example.

As a second application of stochastic gravity, we discuss in this chapter the backreaction problem in cosmology when the gravitational field couples to a quantum conformal matter field, and derive the Einstein–Langevin equations describing the metric fluctuations on the cosmological background. Conformal matter may be a reasonable assumption, because matter fields in the standard model of particle physics are expected to become effectively conformally invariant in the very early universe. We consider a weakly perturbed spatially flat Friedman–Lemaitre–Robertson–Walker spacetime and derive the Einstein–Langevin equation for the metric perturbations off this spacetime, using the CTP functional formalism described in previous chapters. With this calculation we also obtain the probability for particle creation. The CTP effective action is also used to derive the renormalized expectation value of the quantum stress-energy tensor and the corresponding semiclassical Einstein equation.