General Relativity and Gravitation, Vol. 13, No. 8, 1981
Book Review
Recent Developments in Gravitation-Carg6se 1978. Edited by M. L~vy and S. Deser. Plenum, New York, 1979, 596 pp.
This volume contains the proceedings of the Carg~se Summer Institute on Recent Developments in Gravitation which was held in Carg~se, Corsica in July, 1978. The sixteen articles are divided among three general topics: Classical relativity, quantum gravity, and supergravity, with the latter two categories comprising the bulk of the volume. This reflects the preoccupation of theorists in recent years with the relationship between gravitation and quantum theory. There are three articles devoted to classical relativity. The first, by Bruno Bertotti, reviews the status of experimental gravitation. The various observations which have steadily narrowed the domain of viable theories of gravity are discussed. Bertotti also has a few words to say about theories of theories, and he suggests that schemes such as the PPN formalism which have in the past served to interpret observational data may need to be superseded by something better. The article by Yvonne Choquet-Bruhat begins with an introduction to classical general relativity and then moves on to a discussion of the Cauchy problem for Einstein's equations, including a treatment of existence and uniqueness theorems. The contribution of Brandon Carter, entitled "Underlying Mathematical Structures of Classical Gravitation Theory," focuses upon the theory of Lie group fiber bundles. The stated motivation is to provide a mathematical framework suitable for the treatment of gravitation as a gauge theory. Although most of the article deals with the mathematical aspects of fiber bundles, some discussion of applications to spinors in general relativity is given. The articles appearing under the heading of Quantum Gravity may be divided into two categories: those which deal with quantum field theory in a classical background space-time, and those dealing with the quantization of the gravitational field itself. The former group include the lectures of Leonard Parker, which are devoted to the proper time formalism and the method of zeta function regularization as applied to linear (noninteracting) quantum field theory in curved space-time. A clear and detailed account of these two topics is given, culminating in a derivation of the conformal anomaly for a scalar field. Parker also provides a discussion of the significance of the anomaly for particle creation in Robertson-Walker universes. Related topics are taken up by David Boulware, who discusses regularization and renormalization at the one-loop level of a self813 0001-7701/81/08004)813503.00/0
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interacting scalar field. Various regularization methods are compared, with particular attention to dimensional regularization. The article by S. J. Avis and C. J. Isham is concerned with quantum field theory in topologically nontrivial spacetimes. The first two sections of their article give a lucid description of how nontrivial space-time topology allows the possibility of several physically inequivalent field configurations ("twisted fields"). The remaining three sections delve into a rather abstract discussion of the mathematics required to characterize these field configurations: the classification of fiber bundles. Among the articles concerned with the quantization of gravity is that of Bryce DeWitt, entitled "The Formal Structure of Quantum Gravity." DeWitt emphasizes the formal similarities between gravitation and other gauge theories such as the Yang-Mills theory, the gauge group of the former being the diffeomorphism group. Particular attention is paid to the Feynman path integral approach and the role of the gauge group in that approach. G. 't Hooft considers quantum gravity as perturbations around flat space and discusses the one-loop finiteness of pure gravity. He concludes with some speculations on the possible need to abandon the continuum description in favor of a discrete space-time. Stephen Hawking, in his lectures on "Euclidean Quantum Gravity," takes the viewpoint that the basic difficulty of quantum gravity lies in the use of perturbation theory rather than with general relativity itself. He outlines a program to develop a nonperturbative theory in which physical quantities are calculated by Feynman path integrals defined as sums over Euclidean (signature i I I I) metrics. Although such a theory is yet to be developed, it is a bold and imaginative program and is clearly presented. Another approach to the problem of quantum gravity is that of supergravity, which is the topic of the third part of the volume. The lectures of Bruno Zumino provide a brief but clear introduction to supergravity, followed by a detailed treatment of the superspace formalism. S. Deser treats the Hamiltonian formulation of supergravity and uses this formulation to discuss supergravity as the square root of general relativity and the positiv!ty of energy. The contribution by J. Scherk is devoted to extended shpersymmetry and extended supergravity. That of P. van Nieuwenhuizen treats the tensor calculus of supergravity and the problem of auxiliary fields. The volume concludes with short articles by D. Z. Freedman on irreducible representations of Poincar6 supersymmetry, by S. Ferrara on massive gravitinos arising from spontaneous breakdown of local supersymmetry, and by M. T. Grisaru on the role of anomalies in supersymmettic theories. In summary, the articles in this book provide a good account of current research in gravitation, with emphasis on quantum gravity and supergravity. Although the ratio of mathematical formalism to physical insight is higher than one might consider ideal, this probably reflects the present status of our knowl-
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edge of the relationship of quantum theory and relativity theory. This volume will be of interest to researchers and students of gravitation and related areas of theoretical physics.
L. H. Ford Department of Physics Tufts University Medford, Massachusetts 02155
