Classical fields in curved spacetime

This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.

Field models

This chapter provides constructions of Lagrangians for various field models and discusses the basic properties of these models. Concrete examples of field models are constructed, including real and complex scalar field models, the sigma model, spinor field models and models of massless and massive free vector fields. In addition, the chapter discusses various interactions between fields, including the interactions of scalars and spinors with the electromagnetic field. A detailed discussion of the Yang-Mills field is given as well.

The scattering matrix and the Green functions

In this chapter, the book begins to develop a perturbative formalism to describe the interactions of quantized fields and, in particular, the interactions of particles in terms of their quantum fields. Quantum scattering requires the description of particle interactions and asymptotic states, which are introduced in detail. The n-point Green functions are defined. An essential part of the chapter is devoted to deriving the reduction of the S-matrix in terms of the Green functions. The compact description of these notions is achieved by introducing the generating functionals of Green functions and the S-matrix. The same constructions are also introduced for the spinor fields.

Canonical quantization of free fields

This chapter discusses canonical quantization in field theory and shows how the notion of a particle arises within the framework of the concept of a field. Canonical quantization is the process of constructing a quantum theory on the basis of a classical theory. The chapter briefly considers the main elements of this procedure, starting from its simplest version in classical mechanics. It first describes the general principles of canonical quantization and then provides concrete examples. The examples include the canonical quantization of free real scalar fields, free complex scalar fields, free spinor fields and free electromagnetic fields.

The renormalization group in perturbative quantum gravity

This is a short chapter summarizing the main results concerning the renormalization group in models of pure quantum gravity, without matter fields. The chapter starts with a critical analysis of non-perturbative renormalization group approaches, such as the asymptotic safety hypothesis. After that, it presents solid one-loop results based on the minimal subtraction scheme in the one-loop approximation. The polynomial models that are briefly reviewed include the on-shell renormalization group in quantum general relativity, and renormalization group equations in fourth-derivative quantum gravity and superrenormalizable models. Special attention is paid to the gauge-fixing dependence of the renormalization group trajectories.

Lagrange formalism in field theory

This chapter is devoted to a general discussion of classical field theory. It presents the minimum information required about classical fields for the subsequent treatment of quantum theory in the rest of the book. The Lagrange formalism for the fields is introduced, based on the least action principle. Global symmetries are described, and the proof of Noether's theorem given. In addition, the energy-momentum tensor for a field system is constructed as an example.

Relativistic symmetry

This chapter discusses relativistic symmetry, starting from the Lorentz transformations. Basic notions of group theory are introduced before a more detailed discussion of the Lorentz and Poincaré groups is given. Tensor representations and spinor representations of the Lorentz group are described, although full proofs of the theorems are not given. The chapter ends with the irreducible representations of the Poincaré group. This chapter provides all of the necessary notions for group theory, although it is not intended to replace a textbook on the subject.

Quantum gauge theories

This chapter, which is the last chapter in Part I, is devoted to an extensive discussion of quantum gauge theories, which is based on functional integrals and Lagrangian quantization. After introducing the notion of a Yang-Mills gauge theory, the Faddeev-Popov method (also known as the DeWitt-Faddeev-Popov procedure) is explained. Starting from this point, the BRST symmetry is formulated, and the corresponding Ward identities (called Slavnov-Taylor identities in some cases) established. More specialized subjects, such as the gauge dependence of effective action and the background field method, are dealt with in detail. In addition, Yang-Mills theory is analyzed as a primary example of general theorems concerning the renormalization of gauge theories.

The renormalization group in curved space

As the main purpose of renormalization is not to remove divergences but to get essential information about the finite part of effective action, this chapter discusses some of the existing methods of solving this problem; such methods can be denoted the renormalization group. First, the minimal subtraction renormalization group in curved space is formulated. Next, the chapter shows how the overall μ‎-independence of the effective action enables one to interpret μ‎-dependence in some situations. As an example, the effective potential is restored from the renormalization group and compared with the expression calculated directly in chapter 13. In addition, the global conformal (scaling) anomaly is derived from the renormalization group.

One-loop renormalization in quantum gravity

This chapter demonstrates the basic methods of one-loop calculations in quantum gravity. Basing its discussion on the general results obtained in chapter 10, it first presents a detailed analysis of the gauge-fixing dependence of one-loop divergences in quantum general relativity and higher-derivative models of quantum gravity. After that, a detailed derivation of divergences in quantum general relativity is given, with the simplest parametrization of the quantum metric and minimal gauge fixing. One-loop divergences in the general (non-conformal) fourth-derivative quantum gravity are then derived in less detail. For a similar calculation in the superrenormalizable polynomial model (superrenormalizable gravity), the chapter just presents and discusses the final result.