During the past 25 years or so a standard model of particle physics has emerged in which the fundamental constituents of matter, the quarks and leptons (such as the electron), interact by exchanging quanta of the force fields (such as the photon in the case of electromagnetic interactions and the gluons for the strong nuclear force). The mathematical framework that describes these interactions is quantum field theory.

Since Kenneth Wilson's pioneering work in the 1970s, lattice quantum field theory (LQFT) has been a rapidly developing area of research in theoretical elementary particle physics. This book provides a thorough review of the foundations of LQFT, as well as a discussion of the main areas of application to particle physics. It will serve as an excellent technical introduction for research students, supplementing and updating Quarks, Gluons and Lattices by M. Creutz, which has played this role for ten years now, and as a useful reference for active researchers in the field. Istvan Montvay and Gernot Munster have written this book for the practitioner, assuming that the reader is already motivated to study LQFT and, regrettably, providing essentially no explanation of the physical significance of the subject before beginning to present the technical material.

A major chapter of this book is devoted to the lattice formulation of quantum chromodynamics (QCD) and its applications. The theoretical background material is presented in depth; it is pleasing in particular to see a detailed discussion of the breaking and restoration of chiral symmetry in lattice formulations of QCD. Two important applications of lattice QCD are discussed particularly carefully, so the reader should get a clear understanding of how the calculations are performed, and of the theoretical background. The first of these is the evaluation of the masses of hadrons (strongly interacting particles such as the proton) from first principles. The second application is to the study of the phase transition between the low-temperature (and/or density) phase for which the quarks are confined inside hadrons, and the high-temperature (and/or density) phase in which the quarks and gluons form a plasma. Such a phase transition (from the plasma to the confined phase) is believed to have occurred as the universe cooled during the early stages of its evolution, and a considerable effort is being made to produce the quark-gluon plasma in heavy ion collisions.

But there is a serious omission in the chapter on lattice QCD, namely the absence of a detailed discussion of the computation of "weak matrix elements", which contain the strong interaction effects in processes mediated by the weak nuclear force. The study of weak processes is one of the main areas of investigation in particle physics, and one of the principle applications of lattice QCD, and yet this topic is mentioned only very briefly.

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The feature of the standard model of particle physics that is least well understood is the generation of the masses of the intermediate vector bosons and of the quarks and leptons. In the standard model this occurs through the "Higgs" mechanism that is based on the spontaneous breaking of the electro-weak gauge symmetry. It is not known whether the generation of mass in nature occurs via the Higgs mechanism or not, and if it does how many Higgs bosons there are and what their masses are. Lattice techniques provide the opportunity of studying the Higgs sector of the standard model nonperturbatively, and have contributed significantly to a qualitative understanding of the mechanism of mass generation. There is a detailed discussion of the treatment of the Higgs sector in lattice field theory that illustrates the conceptual difficulties involved. It contains a review of the important studies dealing with the upper bound on the Higgs mass obtained using scalar field theories, and a detailed description of Higgs and Yukawa models that are designed to incorporate the effects of gauge fields and fermions, in particular heavy fermions such as the top quark. Despite not having the same predictive power as lattice QCD, analytic numerical studies of Higgs and Yukawa models are able to test the consistency of the field theoretical formulation.

The early chapters of this book contain a detailed exposition of the foundations of lattice quantum field theory, describing how one studies the propagation and interactions of particles of spin 0 (such as the Higgs bosons), spin 1/2 (for example, the quarks) and spin 1 (in particular the force carriers such as the photon or the gluons) on a discrete lattice. This material, which is mostly well established, has been skilfully collected and will be a useful teaching and reference source for many years. Whereas most of the book deals with theoretical questions concerning the formulation of quantum field theory on a discrete lattice in Euclidean space, the final chapter contains a review of the algorithms used in numerical simulations of quantum field theories. This is a necessary and valuable addition to the material required by the practising lattice field theorist.

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Montvay and Munster are to be congratulated on producing a book that will be valued by most researchers using the lattice formulation of quantum field theory.

C. T. Sachrajda is professor of theoretical physics, University of Southampton.