The Global Approach to Quantum Field Theory Bryce DeWitt Oxford U. Press, New York, 2003. Vols. 1 and 2. $230.00 set (1042 pp. set). ISBN 0-19-851093-4

Quantum field theory is a (very big) piece of 20th-century physics that is still falling into the 21st—at least if one judges by the number of recent monographs dedicated to passing down that legacy. Bryce DeWitt’s half-century devotion to QFT has produced a two-volume treatise of some 1040 pages, elegantly if expensively produced by Oxford University Press. The set is very much a self-contained exposition, with extremely few references, of his personal development of QFT. DeWitt’s stated goal is to expound the covariant rather than the “3 + 1” description, and he does so in terms of his notoriously idiosyncratic notation in which almost everything is suppressed.

Much of the work is based on DeWitt’s earlier expositions, including his 1992 book on supermanifolds and his two famous sets—spanning two decades—of Les Houches lectures. The range of topics in DeWitt’s latest offering is astonishing in both breadth and treatment. For example, Yang–Mills and gravitational fields are— together—given a chapter with 40 or so pages and three physics references; quantum electrodynamics rates 23 pages and three references. On the other hand, DeWitt devotes considerable space to the foundations of quantum mechanics, with detailed treatments of measurement theory, decoherence, and the many-worlds interpretation.

A short review can only indicate the general plan of the presentation, which is organized into eight roughly equal divisions plus assorted appendixes. DeWitt introduces his covariant approach, notation, and technology—including the Peierls bracket—via classical field theory. That introduction is an essential prerequisite for understanding the rest of the work. Quantization is ushered in through measurement theory and implemented mostly through Schwinger’s variational principle and “its corollary, the Feynman functional integral.” Standard applications, approximations, and methodology follow, notably including the heatkernel expansion. With considerable formal refinement, DeWitt discusses the quantization of free fields in curved backgrounds. His exposition of interactions, particularly of gauge theories, the notions of renormalizability, and the guiding idea of effective actions, emphasizes the background-field method and the role of ghosts. Heatkernel and regularization methods are applied to anomalies and other vacuum phenomena. DeWitt gives special attention to black-hole radiation, and to the effects of background geometries on matter in general.

Special topics ranging from Euclideanization to discrete symmetries, to quantum electrodynamics, to Yang–Mills and gravitational fields are dealt with in some 25 separate “exercises.” Notable among them are DeWitt’s discussions of free particles; Bose and Fermi oscillators; a higher derivative model; an entire catalog of free massive and massless fields, including forms and spin-$32$ fields; and quantization of fields on closed manifolds. Notable too is the relegation of the re normalization group and spontaneous symmetry breaking to a short exercise.

Supersymmetry is confined to a highly formal and lengthy appendix, essentially a sort of supermanifold dictionary. Unfortunately, no realization of actual supersymmetric systems, let alone supergravity, is to be found. The remaining appendixes are mostly technical, meant to provide some help for particular points in the text. Even there, however, one finds the unpredictable—for example, an elementary exposition of the criteria for conformal flatness in various dimensions.

Over the past five decades, DeWitt has worked on the gamut of formal ideas in QFT. *The Global Approach to Quantum Field Theory* crystallizes a pioneer’s view of the subject’s development during those years. It is in no sense a text, given its dearth of references and elision of many major topics including, for example, any aspect of the standard model. Nevertheless, we in physics would be the poorer without it: Despite the notation, experts will long consult—and beginners will ever marvel at—this tour de force.

**Stanley Deser**, *like Bryce DeWitt, had Julian Schwinger as his thesis adviser. He is the Enid and Nate Ancell Professor of Physics at Brandeis University in Waltham, Massachusetts. Deser’s research involves quantum fields, gravitational theories, and their generalizations*.