MATHEMATICAL REVIEWS
January 198888a:22037 22E70 81C40.
Kim, Y. S. (1MDR); Noz,Marilyn E. (1NY)
Theory and applications of the Poincaré.
Fundamental Theories of Physics.
D.Reidel Publishing Co., Dordrecht  Boston, Mass, 1986.
xvi+331 pp. $73.00. ISBN 9027721416
This book deals mainly with the applications of the Poincaré group in high energy physics. But the authors also give an introduction to group theory, making use of the physical applications to illustrate and show how the theory works. The material contained in this book is divided into twelve chapters.
The first two introduce the basic elements of group theory. The second of them presents Lie groups and Lie algebras. The following chapter studies the group of Poincaré and its representations, paying special attention to the transformation of wave functions and fields. The theory of spinors is developed in Chapter IV; the group SL(2,C) is studied in detail.
So far all the material presented can be found in many other books. Nevertheless, from Chapter V on the approach of the book is more original. The authors deal with concrete physical applications of the Poincaré group, following in some cases ideas developed by them in papers mainly published in the American Journal of Physics of recent years.
Firstly the harmonic oscillator in a covariant formalism is studied. This is discussed in the following chapter where hadronic theory is presented dealing with Dirac's ideas of relativistic quantum mechanics. Later, in the last two chapters of the book, some applications of this oscillator formalism to hadronic phenomenology are given, for example, to give an explanation of the mass spectra of hadrons and the study the deformation properties of relativistic hadrons.
Chapter VIII deals with massless particles, making a detailed analysis of the little group of such particles (isomorphic to the twodimensional Euclidean group E(2)). The next chapter is dedicated to contraction of groups. The idea is to show the relationship between the little groups of massive and massless particles (isomorphic to O(3) and E(2), respectively). Finally, the two remaining chapters deal with the representations of the groups SO(2,1) and SO(1,1) and SO(3,1).
Note that E. P. Wigner's noteworthy paper [Ann. Math. (2)40 (1939), 149204; Zbl 20, 296] was the source of inspiration for the authors when writing this book. There is also a remarkable trace of some of Dirac's papers in this book.
The book presents a very good collection of problems places at the end of each chapter.
 Mariano A. del Olmo (Valladolid).


