"After reviewing some basic principles of quantum field theory in Chapter 1, we now turn to a series of problems exploring various aspects of functional methods. Although it lacks a robust mathematical foundation in the case of interacting theories (but the situation in this respect is no better within the canonical formalism), the formulation of QFT in terms of path integrals considerably simplifies many manipulations that would otherwise be extremely tedious because of the need to keep track of the ordering of operators.

Besides the conventional representation of expectation values of time-ordered products of field operators and their generating functionals in terms of path integrals, we also briefly discuss the worldline representation for propagators and for one-loop effective actions, which provides an alternative point of view on quantization. Moreover, these ideas are not limited to ordering products of field operators, and can be useful in managing products of other types of non-commuting objects."

After having explored various basic aspects of quantum field theory in Chapter 1 and functional methods in Chapter 2, we now turn to non-Abelian gauge theories (mostly with an SU(N) gauge group). The problems in this chapter explore various mathematical questions relevant for non-Abelian gauge theories (in particular, manipulations of su(N) generators), some questions related to their perturbative expansion, as well as some non-perturbative ones, in particular in connection with the rich structure of ground states in Yang–Mills theory.

This chapter is devoted tonovel methods (spinor helicity formalism, on-shell recursion, generalized unitarity), developed in the past 20 years for computing scattering amplitudes, that bypass the traditional workflow (Lagrangian ? Feynman rules ? diagrams ? amplitudes via LSZ formulas). The recurring theme of this approach is to avoid, as much as possible, direct references to the underlying Lagrangian, whose gauge invariance is a source of unnecessary redundancies. This chapter covers both the case of tree-level amplitudes (with some extensions to treat a few examples with scalar particles or fermions) and a few problems about one-loop amplitudes (this is limited to some rather simple examples, since calculating loop amplitudes by hand remains a challenging task, even with modern tools).

In this chapter, we explore aspects of quantum field theory related to the topics of lattice field theory, field theory at finite temperature and strong fields. The connection between the last two subjects is that they depart from quantum field theory in the vacuum (e.g., scattering amplitudes for high-energy physics) by considering situations with a large density of particles, in which the quantum field theory formalism needs to be supplemented with aspects of many-body physics. The reason for the inclusion of lattice field theory in this chapter is that, being a Euclidean formulation of QFT, it has a natural finite-temperature interpretation if one views the inverse of the extent of the time direction as a temperature.