Up to now we have been living in a world of scalar particles. Of course this is not sufficient when dealing with the real world, where spin one-half and spin one particles play a prominent role, and we have to learn how to quantize Dirac and gauge fields at finite temperature. The quantization of Dirac fields is a rather straightforward generalization of what we have already learned. As in the zero-temperature case, the quantization of gauge fields is more subtle, and we shall study in some detail the gauge field propagator and the role of the unphysical degrees of freedom. In the final section of this chapter we shall generalize the results of section 4.4, by deriving the rate for photon and lepton pair production from a quark–gluon plasma.

The Dirac field at finite temperature

Coherent fermion states and path integrals

As a preliminary step, we generalize to the case of fermions the path integral formalism which was set up in chapters 2 and 3. We first consider an elementary case, which is the transposition to fermions of the simple harmonic oscillator of chapter 2.

The real Klein-Gordon field

We considered in Chapter 2 the simplest relativistic equation, the Klein- Gordon equation, as a single-particle equation, and found the following difficulties with it (i) the occurrence of negative energy solutions, (ii) the current jμ does not give a positive definite probability density ρ, as the Schrodinger equation does. For these reasons we must abandon the interpretation of the Klein-Gordon equation as a single-particle equation. (Historically, this was the motive which led Dirac to his equation.) Can any sense then be made out of the Klein-Gordon equation? After all, spin 0 particles do exist (π, K, η, etc.) so, surely, there must be some interpretation of the equation which makes sense.

What we shall do first is to consider the Klein-Gordon equation as describing a field φ(x). Since the equation has no classical analogue, φ(x) is a strictly quantum field, but nevertheless we shall begin by treating it as a classical field, as we did in the last chapter, and shall find that the negative energy problem does not then exist. We shall then take seriously the fact that φ(x) is a quantum field by recognising that it should be treated as an operator, which is subject to various commutation relations analogous to those in ordinary quantum mechanics.

We have seen in previous chapters that integration over internal loops in Feynman diagrams gives divergent results. Since our approach to field theory is based on perturbation theory, however, it is imperative that we make sense of the perturbation series – and that series is one in which higher order terms involve more and more internal integrations, and therefore the possibility of increasing degrees of divergence. It is obvious that, in order for a field theory to be at all sensible or believable, the problems raised by the divergences must be satisfactorily resolved. In this chapter we show how this is done for φ4 theory, electrodynamics (QED) and Yang-Mills theories (QCD). Our general approach is to proceed order by order in perturbation theory (actually in the loop expansion – see below), and show that at each order the quantities of physical interest (masses, coupling constants, Green's functions) can be renormalised to finite values. Then (for QED and QCD) we show that this is, in principle, possible to all orders', these theories are therefore renormalisable. (So is φ4 theory, but we do not prove that.) We begin with φ4 theory.

Divergences in φ4 theory

We saw in Chapter 6 that Δ(x – x) = Δ(0) is a divergent quantity, which modifies the free particle propagator and contributes to the self-energy.

The most important change that has been made for this edition is the addition of a chapter on supersymmetry. It was approximately twenty years ago that supersymmetry burst on the scene of high energy physics. Despite the fact that there is still almost no experimental evidence for this symmetry, its mathematical formulation continues to have appeal to many theoretical physicists justifying, I think, the inclusion of a chapter on supersymmetry in an introductory text. Beyond this, I have rewritten a few sections of the book and incorporated a large number of corrections. I am particularly grateful to Messrs Chris Chambers, Halvard Fausk, Stephen Lyle, Michael Ody, John Smith and Gerhard Soff for pointing out errors and misconceptions in the first edition. The impetus to prepare this second edition owes a lot to the encouragement and friendly advice of Rufus Neal of Cambridge University Press, to whom I should like to express my thanks. Finally, I should like to express my gratitude to Mrs Janet Pitcher for so expertly typing the new material for this edition.

This book is designed for those students of elementary particle physics who have no previous knowledge of quantum field theory. It assumes a knowledge of quantum mechanics and special relativity, and so could be read by beginning graduate students, and even advanced third year undergraduates in theoretical physics.

I have tried to keep the treatment as simple as the subject allows, showing most calculations in explicit detail. Reflecting current trends and beliefs, functional methods are used almost throughout the book (though there is a chapter on canonical quantisation), and several chapters are devoted to the study of gauge theories, which at present play such a crucial role in our understanding of elementary particles. While I felt it important to make contact with particle physics, I have avoided straying into particle physics proper. The book is pedagogic rather than encyclopaedic, and many topics are not treated; for example current algebra and PC AC, discrete symmetries, and supersymmetry. Important as these topics are, I felt their omission to be justifiable in an introductory text.

I acknowledge my indebtedness to many people. Professors P.W. Higgs, FRS, and J.C. Taylor, FRS, offered me much valuable advice on early drafts of some chapters, and I have benefited (though doubtless insufficiently) from their deep understanding of field theory. I was lucky to have the opportunity of attending Professor J. Wess's lectures on field theory in 1974, and I thank him and the Deutscher Akademischer Austauschdienst for making that visit to Karlsruhe possible.

The foregoing chapters have dealt with field theories, including gauge theories, and their quantisation. The stage is now almost set for applying this knowledge to particle physics. One crucial bit of scenery, however, is still missing – the idea of ‘spontaneous breaking of symmetry’. About 1960 Nambu and Goldstone realised the significance of this notion in condensed matter physics, and Nambu in particular speculated on its application to particle physics. In 1964 Higgs pointed out that the consequences of spontaneous symmetry breaking in gauge theories are very different from those in non-gauge theories. Weinberg and Salam, building on earlier work of Glashow, then applied Higgs’ ideas to an SU(2) × U(1) gauge theory, which they claimed described satisfactorily the weak and electromagnetic interactions together, in other words, in a unified way. Serious interest was shown in this theory when 't Hooft proved, in 1971, that is was renormalisable. It has met with notable experimental successes. These matters are the concern of this chapter (with the exception of renormalisation, which we deal with in the next chapter). We begin by explaining spontaneous symmetry breaking, which, when applied to field theory, is a concept that refines our notion of the vacuum.

What is the vacuum?

We begin by considering two simple physical examples. First, consider the situation illustrated in Fig. 8.1. Place a thin rod of circular cross section vertically on a table, and push down on it along its length, with a force F.

Quantum field theory

Quantum field theory has traditionally been a pursuit of particle physicists. In recent years, some condensed matter physicists have also succumbed to its charms, but the rationale adopted in this book is the traditional one: that the reason for studying field theory lies in the hope that it will shed light on the fundamental particles of matter and their interactions. Surely (the argument goes), a structure that incorporates quantum theory – which was so amazingly successful in resolving the many problems of atomic physics in the early part of this century – and field theory – the language in which was cast the equally amazing picture of reality uncovered by Faraday, Maxwell and Hertz – surely, a structure built on these twin foundations should provide some insight into the fundamental nature of matter.

And indeed it has done. Quantum electrodynamics, the first child of this marriage, predicted (to name only one of its successes) the anomalous magnetic moment of the electron correctly to six decimal places; what more could one want of a physical theory? Quantum electrodynamics was formulated in about 1950, many years after quantum mechanics. Planck's original quantum hypothesis (1901), however, was indeed that the electromagnetic field be quantised; the quanta we call photons. In the years leading up to 1925, the quantum idea was applied to the mechanics of atomic motion, and this resulted in particle-wave duality and the Schrodinger wave equation for electrons.

No-one can deny the success which quantum field theory, in the perturbative approximation, has enjoyed over the last half century. One need only mention the interpretation of quantised fields as particles, the description of scattering processes, the precise numerical agreements in quantum electrodynamics, the successful prediction of the W particle, and the beginnings of an understanding of the strong interaction through quantum chromodynamics. Yet despite these successes, the question of how to describe the basic matter fields of nature has remained unanswered – except, of course, through the introduction of quantum numbers and symmetry groups. As far as field theory goes, the matter fields are treated as point objects. Even in classical field theory these present us with unpleasant problems, in the shape of the infinite self-energy of a point charge. In the quantum theory, these divergences do not disappear; on the contrary, they appear to get worse, and despite the comparative success of renormalisation theory the feeling remains that there ought to be a more satisfactory way of doing things.

Now it turns out that non-linear classical field theories possess extended solutions, commonly known as solitons, which represent stable configurations with a well-defined energy which is nowhere singular.

Since its very beginning, physics has pondered the question of what are the fundamental building blocks of matter. In the last century all matter was shown to be composed of atoms. In this century, science has taken a further step forward, revealing the internal structure of the atom. Today, the fact that the atomic nucleus consists of protons and neutrons, and that electrons move around it, is common knowledge. But the story does not end there. Over the past 40 or 50 years it has been found that in collisions between sub-atomic particles additional particles are formed, whose properties and behaviour can shed light on the basic laws of physics. All these particles were called — somewhat unjustifiably — elementary particles.

The study of the sub-atomic structure has led to the development of two branches of modern physics: nuclear physics, which is concerned with furthering the understanding of the atomic nucleus as a whole and the processes going on inside it, and elementary particle physics, the new frontier, which deals with the properties and structure of the various particles themselves and the interactions between them. Up until the end of the 1940s, elementary particle physics had been regarded as a part of nuclear research, but in the 1950s it achieved the status of an independent branch of physics.