Since K. Wilson's seminal work on lattice gauge theory in 1974 the regularization of quantum field theories by a space–time lattice has become one of the basic methods for non-perturbative studies in field theory. It has been applied to Quantum Chromodynamics, to the electroweak theory, and to various other model field theories. Many techniques have been developed to deal with quantum fields on a lattice, and with their help many insights into non-perturbative phenomena have been gained.

During a workshop in Cargèse in 1989 Peter Landshoff explained to us his impression that the development of this subject in recent years justified a new book after the one by Michael Creutz on lattice gauge theory, which was published by Cambridge University Press in 1983, and he invited us to undertake the task of writing a book about lattice field theory. As usual the writing took more time than anticipated by the authors and was only finished three years later.

The resulting book is by no means homogeneous, which corresponds to our intentions and also reflects the continuous but irregular development of the field. Some parts are of a more introductory nature and may be useful for graduate students who would like to enter the field. Other parts explain specific techniques in some detail and may serve as a reference for active workers on the subject. Still other parts summarize some of the results obtained about particular questions, or can be used as a guide to the literature.

The zero-mass limit is intimately related to high-energy behavior in field theory. As we have seen in Section 12.4, cross sections develop collinear, as well as infrared, divergences as masses vanish. These divergences, however, cancel in certain perturbative quantities, jet cross sections in e+e− annihilation among them. Proofs of finiteness for these and other ‘infrared safe’ quantities require a formalism to treat infrared and collinear divergences to all orders in perturbation theory. Such a formalism can be developed on the basis of the analytic structure of Feynman diagrams, which we shall supplement by a power-counting method for estimating the strength of singularities in massless perturbation theory. We shall use these tools to prove a number of important results, including the infrared finiteness of Wick-rotated Green functions and of the e+e− total cross section, as well as of e+e− jet cross sections. The finiteness of the latter, in turn, is a variant of the famous KLN theorem, which states that suitably averaged transition probabilities are finite in the zero-mass limit for any unitary theory.

Analytic structure of Feynman diagrams

The calculations of Section 12.5 are suggestive and encouraging, but we must still determine whether jet cross sections are really mass-independent at higher orders in perturbation theory.

Loops are the characteristic feature of higher orders in the perturbative expansion of Green functions. In principle, loop diagrams are completely defined by the same Feynman rules that give tree diagrams; the only difference is that there are integrals left to do. In practice, the evaluation of these integrals is not completely straightforward. First, their integrands diverge at propagator poles. The integrals remain well defined, however, because of the infinitesimal term ‘i∊’ in each propagator. We shall show below how this works, and how the ‘i∊-prescription’ defines Green functions as analytic functions of their external momenta. Second, in most field theories, some integrands simply do not fall off fast enough at infinity for the corresponding integrals to converge. Such integrals are said to be ultraviolet divergent. The consideration of this problem will lead us, in due course, to the concepts of regularization and renormalization.

We begin our discussion with a simple example, which will illustrate both of these problems. Later in the chapter, we develop the method of time-ordered perturbation theory, and use it to verify that the perturbation expansion generates a unitary S-matrix order-by-order in perturbation theory. Unitarity is a fundamental property, necessary for a theory to make physical sense.

There is a natural solution to an interacting-field theory whenever its Lagrangian is the sum of a free (quadratic) Lagrangian and interaction terms. We associate with each interaction a multiplicative factor gi, the coupling constant for that term. We then expand S-matrix elements of the interacting-field theory around those of the free-field theory, as a power series in the gi. In this approach, the interaction is conceived as a small perturbation on the free theory; the systematic method for expanding in the coupling constant(s) is perturbation theory.

The development of field-theoretic perturbation theory can be carried out in a number of ways. We shall use the path integral method, initiated by Feynman (1949). The other commonly discussed approach, based on the interaction picture, is sketched in Appendix A.

The Feynman path integral is a reformulation of quantum mechanics in terms of classical quantities. Extensive discussions of the path integral in quantum mechanics may be found, for instance, in Feynman & Hibbs (r1965) and Schulman (r1981). Technical reviews include Keller & McLaughlin (r1975) and Marinov (r1980). For issues of mathematical rigor, which by and large are avoided below, see Glimm & Jaffe (r1981) and DeWitt-Morette, Maheshwari & Nelson (r1979).

The path integral

We first consider a quantum mechanical system with only one coordinate q and time-independent Hamiltonian H(p, q).