This volume describes the advances in the quantum theory of fields that have led to an understanding of the electroweak and strong interactions of the elementary particles. These interactions have all turned out to be governed by principles of gauge invariance, so we start here in Chapters 15-17 with gauge theories, generalizing the familiar gauge invariance of electrodynamics to non-Abelian Lie groups.

Some of the most dramatic aspects of gauge theories appear at high energy, and are best studied by the methods of the renormalization group. These methods are introduced in Chapter 18, and applied to quantum chromodynamics, the modern non-Abelian gauge theory of strong interactions, and also to critical phenomena in condensed matter physics.

Chapter 19 deals with general spontaneously broken global symmetries, and their application to the broken approximate SU(2) × SU(2) and SU(3) × SU(3) symmetries of quantum chromodynamics. Both the renormalization group method and broken symmetries find some of their most interesting applications in the context of operator product expansions, discussed in Chapter 20.

The key to the understanding of the electroweak interactions is the spontaneous breaking of gauge symmetries, which are explored in Chapter 21 and applied to superconductivity as well as to the electroweak interactions. Quite apart from spontaneous symmetry breaking is the possibility of symmetry breaking by quantum-mechanical effects known as anomalies. Anomalies and various of their physical implications are presented in Chapter 22.

It is often useful to consider quantum field theories in the presence of a classical external field. One reason is that in many physical situations, there really is an external field present, such as a classical electromagnetic or gravitational field, or a scalar field with a non-vanishing vacuum expectation value. (As we shall see in Chapter 19, such scalar fields can play an important role in the spontaneous breakdown of symmetries of the Lagrangian.) But even where there is no actual external field present in a problem, some calculations are greatly facilitated by considering physical amplitudes in the presence of a fictitious external field. This chapter will show that it is possible to take all multiloop effects into account by summing ‘tree’ graphs whose vertices and propagators are taken from a quantum effective action, which is nothing but the one-particle-irreducible connected vacuum-vacuum amplitude in the presence of an external field. It will turn out in the next chapter that this provides an especially handy way both of completing the proof of the renormalizabilty of non-Abelian gauge theories begun in Chapter 15, and of calculating the charge renormalization factors that we need in order to establish the crucial property of asymptotic freedom in quantum chromodynamics.

The Quantum Effective Action

Consider a quantum field theory with action I[ϕ], and suppose we ‘turn on’ a set of classical currents Jr(x) coupled to the fields ϕr(x) of the theory.

The quantum field theories that have proved successful in describing the real world are all non-Abelian gauge theories, theories based on principles of gauge invariance more general than the simple U(1) gauge invariance of quantum electrodynamics. These theories share with electrodynamics the attractive feature, outlined at the end of Section 8.1, that the existence and some of the properties of the gauge fields follow from a principle of invariance under local gauge transformations. In electrodynamics, fields ψn(x) of charge en undergo the gauge transformation ψn(x) → exp(ienΛ(x))ψn(x) with arbitrary Λ(x). Since ∂μψn(x) does not transform like ψn(x), we must introduce a field Aμ(x) with the gauge transformation property Aμ(x) → Aμ(x)+∂μΛ(x), and use it to construct a gauge-covariant derivative ∂μψn(x)–ienAμ(x)ψn(x), which transforms just like ψn(x) and can therefore be used with ψn(x) to construct a gauge-invariant Lagrangian. In a similar way, the existence and some of the properties of the gravitational field gμn(x) in general relativity follow from a symmetry principle, under general coordinate transformations. Given these distinguished precedents, it was natural that local gauge invariance should be extended to invariance under local non-Abelian gauge transformations.

In the original 1954 work of Yang and Mills, the non-Abelian gauge group was taken to be the SU(2) group of isotopic spin rotations, and the vector fields analogous to the photon field were interpreted as the fields of strongly-interacting vector mesons of isotopic spin unity.

Much of the physics of this century has been built on principles of symmetry: first the space time symmetries of Einstein's 1905 special theory of relativity, and then internal symmetries, such as the approximate SU(2) isospin symmetry of the 1930s. It was therefore exciting when in the 1960s it was discovered that there are more internal symmetries than could be guessed by inspection of the spectrum of elementary particles. There are exact or approximate symmetries of the underlying theory that are ‘spontaneously broken,’ in the sense that they are not realized as symmetry transformations of the physical states of the theory, and in particular do not leave the vacuum state invariant. The breakthrough was the discovery of a broken approximate global SU(2) × SU(2) symmetry of the strong interactions, which will be discussed in detail in Section 19.3. This was soon followed by the discovery of an exact but spontaneously broken local SU(2) × U(1) symmetry of the weak and electromagnetic interactions, which will be taken up along with more general broken local symmetries in Chapter 21. In this chapter we shall begin with a general discussion of broken global symmetries, and then move on to physical examples.

Degenerate Vacua

We do not have to look far for examples of spontaneous symmetry breaking. Consider a chair. The equations governing the atoms of the chair are rotationally symmetric, but a solution of these equations, the actual chair, has a definite orientation in space.

The method of the renormalization group was originally introduced by Gell-Mann and Low as a means of dealing with the failure of perturbation theory at very high energies in quantum electrodynamics. An n-loop contribution to an amplitude involving momenta of order q, such as the vacuum polarization Πμν(q), is found to contain up to n factors of In() as well as a factor αn, so perturbation theory will break down when is large, even though the fine structure constant a is small. Even in a massless theory like a non-Abelian gauge theory we must introduce some scale μ to specify a renormalization point at which the renormalized coupling constants are to be defined, and in this case we encounter logarithms In(E/μ), so that perturbation theory may break down if E ≫ μ or E ≪ μ, even if the coupling constant is small.

Fortunately, there is a modified version of perturbation theory that can often be used in such cases. The key idea of this approach consists in the introduction of coupling constants gμ defined at a sliding renormalization scale μ — that is, a scale that is not related to particle masses in any fixed way. By then choosing μ to be of the same order of magnitude as the energy E that is typical of the process in question, the factors In(E/μ) are rendered harmless. We can then do perturbation theory as long as gμ remains small.

There are subtleties in the implications of symmetries in quantum field theory that have no counterpart in classical theories. Even in renormalizable theories, the infinities in quantum field theory require that some sort of regulator or cut-off be used in actual calculations. The regulator may violate symmetries of the theory, and even when this regulator is removed at the end of the calculation it may leave traces of this symmetry violation. This problem first emerged in trying to understand the decay rate of the neutral pion, in the form of an anomaly that violates a global symmetry of the strong interactions. Anomalies can also violate gauge symmetries, but in this case the theory becomes inconsistent, so that the condition of anomaly cancellation may be used as a constraint on physical gauge theories. The importance of anomalies will become even more apparent in the next chapter, where we shall study the non-perturbative effects of anomalies in the presence of topologically non-trivial field configurations.

The π° Decay Problem

By the mid-1960s the picture of the pion as a Goldstone boson associated with a spontaneously broken SU(2)⊗ SU(2) symmetry of the strong interactions had scored a number of successes, outlined here in Chapter 19. However, this picture also had a few outstanding failures. The most disturbing had to do with the rate of the dominant decay mode of the neutral pion, π0→ 2γ.

Most of this book has been devoted to applications of quantum field theory that can at least be described in perturbation theory, whether or not the perturbation series actually works well numerically. In using perturbation theory, we expand the action around the usual spacetime-independent vacuum values of the fields, keeping the leading quadratic term in the exponential exp(iI), and treating all terms of higher order in the fields as small corrections. Starting in the mid-1970s, there has been a growing interest in effects that arise because there are extended spacetime-dependent field configurations, such as those known as instantons, that are also stationary ‘points’ of the action. In principle, we must include these configurations in path integrals and sum over fluctuations around them. (In Section 20.7 we have already seen an example of an instanton configuration, applied in a different context.) Although such non-perturbative contributions are often highly suppressed, they are large in quantum chromodynamics, and produce interesting exotic effects in the standard electroweak theory.

There are also extended field configurations that occur, not only as correction terms in path integrals for processes involving ordinary particles, but also as possible components of actual physical states. These configurations include some that are particle-like, such as magnetic monopoles and skyrmions, which are concentrated around a point in space or, equivalently, around a world line in spacetime. There are also string-like configurations, similar to the vortex lines in superconductors discussed in Section 21.6, which are concentrated around a line in space or, equivalently, around a world sheet in spacetime.