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Introduction to supersymmetry in particle and nuclear physics. (Proceedings of an International School of Supersymmetry, held December 14-18, 1981, in Mexico City, Mexico). (English) Zbl 0572.17001

New York - London: Plenum Press. VIII, 187 p. $ 29.50 (1984).
Supersymmetry is a new type of physical symmetry first discovered in quantum field theory. One of its most prominent features is that it allows to relate fields with different spins and statistics. Moreover, it has opened a new approach to an old problem in particle physics, the unification of all known interactions including gravity. However, while supersymmetric field theories are the objects of intensive research, there is at present no hint that they might have something to do with real nature. In the meantime, supersymmetry has also been applied to different branches of physics such as nuclear and condensed matter physics. A survey can be found in the proceedings of a conference on ”Supersymmetry in Physics”, held at Los Alamos in December 1983, published in Physica 15D (1985).
The most obvious way to describe supersymmetry is in terms of Lie superalgebras. These are \({\mathbb{Z}}_ 2\)-graded algebras of a generalized Lie type whose multiplication can be viewed as a commutator or anti- commutator depending on the degree of the elements under consideration [see V. G. Kac, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)]. An immediate question to ask is then whether some associated global objects also do exist. This has initiated the development of a new branch of mathematics called supermathematics which essentially may be characterized as an attempt to generalize algebra, analysis, and differential geometry by using a Grassmann algebra as the fundamental domain of scalars in place of the fields of real or complex numbers. In particular, this leads to structures like supermanifolds and Lie supergroups. The interested reader may want to look into the monograph by B. DeWitt on ”Supermanifolds”, Cambridge University Press (1984; Zbl 0551.53002), where these topics are presented from a theoretical physicist’s point of view and where several references to the pertinent literature are given. Some very recent publications containing further important references are those by J. M. Rabin and L. Crane [Commun. Math. Phys. 102, 123-137 (1985)] and R. F. Picken and K. Sundermeyer [ibid. 102, 585-604 (1986)].
The present work contains the proceedings of a School which was to provide both students and researchers with an introduction to supersymmetry as well as an overview of some topics of current research. In the first two lectures by D. Z. Freedman ”Introduction to supersymmetry” (pp. 1-27), and M. T. Grisaru ”Superfields” (pp. 29- 52), respectively, the basic methods for constructing a supersymmetric field theory are described. Freedman uses the more conventional ”component approach”, while Grisaru introduces the ”superspace formulation”, i.e., he makes use of superanalysis and super differential geometry. These ideas are then applied to particle physics.
E. Witten ”Grand unification with and without supersymmetry” (pp. 53-76) discusses the construction of ”grand unified theories” (unifying the so-called gauge theories of strong, weak, and electromagnetic interactions) with and without supersymmetry.
S. Ferrara ”Yang-Mills theories with local and global supersymmetry - Higgs and superhiggs effect in unified field theories” (pp. 77-105) first deals with the construction of supersymmetric field theoretical models, then he describes the ”spontaneous symmetry breaking” of supersymmetry as well as some attempts to build supersymmetric models of electroweak and strong interactions, and finally he proceeds to construct supersymmetric theories involving gauge and gravitational interactions.
The final lecture by I. Bars ”Supergroups and their representations” (pp. 107-184) aims at a different subject, it deals with the representation theory of the special superunitary groups SU(N/M). These Lie supergroups are analogous to the classical special unitary groups and are basic to the applications of supersymmetry to nuclear physics. Bars’ approach is useful in practical applications and is frequently employed in the physical literature, however, I doubt that a mathematician will be satisfied with it.
Reviewer: M.Scheunert

MSC:

17A70 Superalgebras
00B25 Proceedings of conferences of miscellaneous specific interest
17-06 Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras
81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E70 Applications of Lie groups to the sciences; explicit representations
81T08 Constructive quantum field theory
83C45 Quantization of the gravitational field
83E99 Unified, higher-dimensional and super field theories