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Gauge field theories. (English) Zbl 0637.53103
Cambridge Monographs on Mathematical Physics. Cambridge etc.: Cambridge University Press. XIII, 394 p.; £50.00; $ 89.50 (1987).

According to the preface this book is addressed to graduate students and research workers in theoretical physics “who have some knowledge of quantum field theory in its canonical formulation”. This warning, taken in conjunction with a subsequent statement to the effect that “the book is intended to be a relatively concise reference to some of the field theoretical tools used in contemporary research”, should indeed be taken quite literally. From the outset it is assumed that the reader is so thoroughly familiar with the notations and concomitant conventions that are prevalent in the modern theory of fundamental interactions that little if any attempt is made to identify many of the symbols that are used or to give a precise indication of their background (as elements of some well defined class or space).

The volume covers a wide range of fundamentally important topics, including some of which there have not as yet appeared comprehensive expositions and that are the objects of intense current research activity. Consequently the book is potentially a most useful addition to the literature; however, its value would have been enhanced considerably, and its appeal extended to a much wider audience, had the author been just slightly more sensitive to the needs of readers (physicists or mathematicians) who are not entirely familiar with the prevalent terminology of elementary particle theory.

After some brief introductory remarks concerning gauge invariance, U(1) gauge symmetry, and non-abelian gauge symmetry, a fairly detailed account of the path integral formulation of quantum field theory is given. This is followed by an introduction to the theory of renormalization, together with chapters on quantum electrodynamics and the renormalization group. Particularly useful features of the book are (i) an account of quantum chromodynamics, which is the theory of interactions of quarks and gluons, and (ii) an account of chiral symmetry, spontaneous and explicit global symmetry breaking, spontaneous breaking of gauge symmetry and the Higgs mechanism for the case of the U(1) gauge theory with the gauge field coupled to a complex scalar field. Chiral anomalies and the so-called effective Lagrangians are described towards the end of the book, which is concluded by a very terse introduction to supersymmetry.

Each chapter is augmented by a set of problems, some of which are difficult and contribute substantially to the contents of the text. Although the book cannot be recommended unreservedly as a source for independent study, it will no doubt prove to be useful to experts in the field.

Reviewer: H.Rund

MSC:
53C80Applications of global differential geometry to physics
53C05Connections, general theory
81-02Research monographs (quantum theory)
81T08Constructive quantum field theory
81T17Renormalization group methods (quantum theory)
81V10Electromagnetic interaction; quantum electrodynamics
81Q05Closed and approximate solutions to quantum-mechanical equations
81V15Weak interaction