This chapter starts with an introductory survey on the physical background and historical events that lead to the emergence of the density matrix renormalization group (DMRG) and its tensor network generalization. We then briefly overview the major progress on the renormalization group methods of tensor networks and their applications in the past three decades. The tensor network renormalization was initially developed to solve quantum many-body problems, but its application field has grown constantly. It has now become an irreplaceable tool for investigating strongly correlated problems, statistical physics, quantum information, quantum chemistry, and artificial intelligence.

This chapter is devoted to basic aspects of quantum field theory, ranging from the foundations to perturbation theory and renormalization, and is limited to the canonical formalism (functional methods are treated in Chapter 2) and to the traditional workflow (Lagrangian --> Feynman rules --> time-ordered products of fields --> scattering amplitudes) for the calculation of scattering amplitudes (the spinor-helicity formalism and on-shell recursion are considered in Chapter 4).The problems of this chapter deal with questions in scalar field theory and quantum electrodynamics, while non-Abelian gauge theories are discussed in Chapter 3.

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.