The Global Approach to Quantum Field Theory

Thanks to its impressive success in the second half of the 20th century, both in high-energy physics and in critical phenomena, quantum field theory has enjoyed an abundant literature. We therefore greet yet another book on this subject with caution: what can a monograph on quantum field theory bring now that is new, either conceptually or pedagogically? But when it is written by a physicist such as Bryce DeWitt, who has made his own contribution to the collection of field theory books with The Global Approach to Quantum Field Theory, all suspicion is naturally abandoned. One knows in advance that this book can only lead to a genuine enrichment of the literature. DeWitt has made a formidable contribution to various areas of physics: general relativity, the interpretation of quantum mechanics, and most of all the quantization of non-Abelian gauge theories and quantum gravity. In addition, his pedagogical publications, especially the Les Houches schools of 1963 and 1983 [1, 2], have had a great impact on quantum field theory. All this makes the reader keen to pick up his new work and a deeper reading confirms the reviewer's initial enthusiasm. We must begin by alerting the potential readers of this book that it cannot be compared to any other book in the field (unless of course we are talking about references [1] and [2], of which the book under review is an extension and reworking). This uniqueness applies to both the scientific content and the way the ideas are presented. A quick description of this book and a brief explanation of its title should convince the reader of the book's unique quality. For DeWitt, a central concept of field theory is that of `space of histories'. For a field varphii defined on a given spacetime M, the set of all varphii(x) for all x in all charts of M defines its history. It is the space Phi of all possible histories (dynamically allowed or not) of the fields defined on M which is called the `space of histories' by DeWitt. If only bosonic fields are considered, the space of histories is an infinite-dimensional manifold and if fermionic fields are also present, it must be viewed as an infinite-dimensional supermanifold [3]. The fields can then be regarded as coordinates on these structures, and the geometrical notions of differentiation, metric, connections, measure, as well as the geodesics which can be defined on it, are of fundamental importance in the development of the formalism of quantum field theory. This is the so-called global approach to quantum field theory where time does not play any particular role, and quantization is then naturally realized covariantly using tools such as the Peierls bracket (a covariant generalization of Poisson bracket), the Schwinger variational principle and Feynman sums over histories. However, it should be noted that the boycott of canonical methods by DeWitt is not total: when he judges they genuinely illuminate the physics of a problem, he does not hesitate to descend from the global point of view and to use them. In a few words, we have in fact described the research program initiated by DeWitt forty years ago, which has progressively evolved in order to take into account the latest development of gauge theories. While the Les Houches Lectures of 1963 [1] were mainly concentrated on the formal structure and the quantization of Yang--Mills and gravitational fields, the present book also deals with more general gauge theories including those with open gauge algebras and structure functions, and therefore supergravity theories. More precisely, the book, more than a thousand pages in length, consists of eight parts and is completed by six appendices where certain technical aspects are singled out. An enormous variety of topics is covered, including the invariance transformations of the action functional, the Batalin--Vilkovisky formalism, Green's functions, the Peierls bracket, conservation laws, the theory of measurement, the Everett (or many worlds) interpretation of quantum mechanics, decoherence, the Schwinger variational principle and Feynman functional integrals, the heat kernel, aspects of quantization for linear systems in stationary and non-stationary backgrounds, the S-matrix, the background field method, the effective action and the Vilkovisky--DeWitt formalism, the quantization of gauge theories without ghosts, anomalies, black holes and Hawking radiation, renormalization, and more. It should be noted that DeWitt's book is rather difficult to read because of its great breadth. From the start he is faithful to his own view of field theory by developing a powerful formalism which permits him to discuss broad general features common to all field theories. He demands a considerable effort from the reader to penetrate his formalism, and a reading of Appendix~A which presents the basics of super-analysis is a prerequisite. To keep the reader on course, DeWitt offers a series of exercises on applications of global formalism in Part 8, nearly 200 pages worth. The exercises are to be worked in parallel with reading the text, starting from the beginning. It should be noted that these exercises previously appeared in references [1], [2] and [3], but here they have been worked out in some detail by the author. Before concluding, some criticisms. DeWitt has anticipated some criticism himself in the Preface, where he warns the reader that `this book is in no sense a reference book on quantum field theory and its application to particle physics. The selection of topics is idiosyncratic. ' But the reviewers should add a few more remarks: (1) There are very few references. Of course, this is because the work is largely original. Even where the work of other researchers is presented, it has mostly been transformed by the DeWittian point of view. (2) There are very few diagrams, which sometimes hinders the exposition. In summary, in our opinion, this is one of the best books dealing with quantum field theory existing today. It will be of great interest for graduate and postgraduate students as well as workers in the domains of quantum field theory in flat and in curved spacetime and string theories. But we believe that the reader must have previously studied standard textbooks on quantum field theory and general relativity. Even with this preparation, it is by no means an easy book to read. However, the reward is to be able to share the deep and unique vision of the quantum theory of fields and its formalism by one of its greatest expositors. References [1] DeWitt B S 1965 Dynamical Theory of Groups and Fields (Les Houches Lectures 1963) (New York: Gordon and Breach) [2] DeWitt B S 1984 Relativity, Groups and Topology II (Les Houches Lectures 1983) ed R Stora and B S DeWitt (Amsterdam: North-Holland) [3] DeWitt B S 1994 Supermanifolds (Cambridge: Cambridge University Press)