Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.

This chapter introduces the simplest example of a relativistic field theory, that provided by a scalar (spin zero) field. First we consider the free, noninteracting theory. After making contact with the non-relativistic, particle creation and destruction fields of the previous chapter, the relativistic, free-particle propagation functions are described. The interacting theory is then examined by computing the two-particle scattering cross section in lowest order. Using the general structure of the scattering amplitude, which is illustrated by the simple scalar theory, we derive a formula for the cross section that is easily generalized to any two-particle scattering process. This we do with some rigor by using a wave-packet technique. This discussion is an example of what will often be done in subsequent chapters: The scalar field theory will be employed to motivate and illustrate many features of relativistic quantum field theory that hold in general. We use the scalar field so as to make these features clear and not encumbered with complicated notation. This chapter also contains the calculation of the scalar-scalar scattering cross section to second order. This requires regularization — which we do by the method of dimensional continuation — and renormalization. The renormalization is done with a method that naturally leads to the idea of the running coupling constant which is of great importance in modern elementary particle theory.

Many of the aspects of relativistic quantum field theory have now been described, using the simple scalar theory to bring out the essential features. With the spin ½ theory also explained, we can turn at last to describe the theory of quantum electrodynamics and several of the experimental tests that verify it. Since to low order, the non-Abelian character of the strong interaction QCD theory plays no important role, there are QCD analogs of QED processes, and these will also be discussed. We begin by explaining the quantization of the electromagnetic field. This will be done in a way which is easily generalized to the non-Abelian case. We shall not discuss many of the traditional topics in QED which are well-described in several texts, but we will illustrate the character of the theory with an example of modern interest and with a traditional topic treated with modern methods. Thus we shall not discuss the traditional topics of relativistic Coulomb and Compton scattering, but we shall instead compute the cross section for e+ + e− → µ+ + µ− which is essentially the same as the quark-antiquark production in high-energy electron-positron collisions. The effects of vacuum polarization will be treated with modern methods, and it will be related to the e+e− cross section. The lowest order electron self energy and vertex functions will be calculated and their properties examined — one cannot avoid computing the magnetic moment of the electron.

This is half a book on quantum field theory. Non-Abelian gauge theory including the electro-weak theory and more recent developments are not covered. On the other hand, the topics which are included are treated in a completely modern manner. For example, the functional integral representation is used as the basis for the theory, and it is presented in the first chapter. The nature of spontaneous symmetry breakdown is illustrated early on by the example of superfluid helium. Dimensional regularization is used throughout, and the renormalization group is introduced as soon as renormalization is discussed. The final chapter on quantum electro-dynamics does contain some calculations whose strong-interaction QCD (non-Abelian) counterparts are discussed. In general, the material which is included is thoroughly covered starting from first principles. This has made for a rather long volume even though its range is limited. Perhaps someday I will write a second volume to make a whole book, but the remaining topics are treated in the references which are listed at the end of this volume.

A set of problems appears at the end of each chapter. The role of these problems should be explained. Many are exercises (some simple, some not) which help the reader gain facility with the ideas and techniques introduced in the text. Other problems review material which is assumed to be known in the text such as those which describe determinants at the end of Chapter 1 and special relativity at the end of Chapter 3.

The character of relativistic quantum field theory will continue to be illustrated by the simple example of scalar field theory with the λϕ4 interaction. Previously we have computed a few basic processes by working out the perturbative development provided by the functional integral representation. Based upon this experience, we can now systematize what has been done in the form of Feynman rules that enable one to compute an amplitude in momentum space directly from its graphical representation. With this in hand, we then turn to describe the character of the divergencies which appear in the theory and how these divergencies are removed by renormalization. We shall present only a very rough outline of how the renormalization process works. A detailed description and proof of the renormalizability of the theory is a non-trivial task whose methods are of little use in other contexts, and so this we will not do. We shall, however, describe in some detail the nature of the parameter renormalization in the minimally subtracted, dimensionally regulated scheme, for this leads directly to the renormalization group which is a very useful tool. The behavior that the renormalization group implies for a general field theory — a theory which could have a fixed point of the renormalization group or a theory such as quantum chromodynamics which is asymptotically free — will be described. Composite operators are introduced, and the relationship of the stress-energy tensor to the renormalization group is described. The chapter concludes with a sketch of the operator product expansion.

The preceding chapter showed that the coupling constant must be renormalized. In the present chapter we shall describe, to lowest order in the coupling, the other basic or elementary processes that give divergent results. These divergencies are compensated by renormalizing other parameters of the theory, the mass and field strength. To render the theory completely well defined, another renormalization is needed — the addition of an (infinite) constant Λ0 to the Lagrange function. This constant corresponds to an infinite renormalization of the “zero-point energy” — in rough terms, to zeroth order in the coupling, the field theory corresponds to an infinite collection of harmonic oscillators with the a-th oscillator having a zero-point or ground state energy given by ½ωa, and the sum Σa½ωa diverges. Higher order terms in the coupling give additional zero-point divergencies. This divergence was previously hidden in a re-definition of the measure [dϕ]. However, the divergence depends upon the scalar field's mass, and it is best to define the functional integral, and thereby the theory, in a mass-independent fashion. This requires that the zero-point divergence be displayed explicitly. Moreover, when gravitational couplings are introduced into theory, this additional constant in the Lagrange function appears as a contribution to the “cosmological constant” Λ0. The exceedingly small value of this constant on the scale set by elementary particles is an outstanding problem in contemporary physics, and this puzzle further motivates our discussion of Λ0.

Non-relativistic quantum states of any given angular momentum S can be built out of the direct product of spin ½ states. Suitable linear combinations of these product states can be formed to produce the irreducible representation of the rotation group which describes angular momentum S. In a similar fashion, relativistic quantum fields describing particles with spin S can be built out of spin ½ fields. We shall, therefore, dwell on the nature of spin ½ fields at some length. Moreover, it is the basic spin ½ field, the Weyl spinor, which is used in the modern electro-weak theory of weak interactions, and so these spinors will be discussed extensively with the more commonly known Dirac field related to them. Since the behavior of integer-spin fields, such as the spin-one photon field, is described by tensor fields that should be familiar, not much attention will be paid to them. Throughout this chapter, we shall work in our observable world of four dimensions. Higher dimensional spaces, however, are needed in the regularization procedure produced by dimensional continuation which we employ throughout this book. Moreover, there has been considerable theoretical speculation which involves higher-dimension space times. Much of the machinery for the extension to higher dimensions appears in the problems at the end of this chapter. The problems also deal with some of the peculiarities of a world with two space-time dimensions.

Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.

The purpose of the effective lagrangian method is to represent in a simple way the dynamical content of a theory in the low energy limit, where effects of heavy particles can be incorporated into a few constants. The basic plan of attack is to write out the most general set of lagrangians consistent with the symmetries of the theory. At sufficiently low energies only one, or perhaps a few, of the lagrangians are relevant, and it is straightforward to read off the predictions of the theory.

Effective lagrangians are used in all aspects of the Standard Model and beyond, from QED to superstrings. Perhaps the best setting for learning about them is that of chiral symmetry. Besides being historically important in the development of effective lagrangian techniques, chiral symmetry is a rather subtle theory which can be used to illustrate all aspects of the method, viz. the low energy expansion, non-leading behavior, symmetry breaking and loops. In addition, the results can be tested directly by experiment since effective lagrangians provide a framework for understanding the very low energy limit of QCD.

Nonlinear lagrangians and the sigma model

The linear sigma model, introduced in Sects. I–3, I–5, provides a ‘user friendly’ introduction to effective lagrangians because all the relevant manipulations can be explicitly demonstrated. The Goldstone boson fields, the pions, are present at all stages of the calculation.