Groups and Symmetries: From Finite Groups to Lie Groups

Groups and Symmetries is a short, concise book (8 chapters, 128 pages of material, with an addition 66 pages of problems and references) that provides an introduction to the subject of Lie Groups, Lie Algebras, their representations, and their uses in theoretical particle physics. The book is a revised and adapted translation of Groupes et Symetries: Groupes finis, groupes de Lie, representations. (2006). In the author’s words, the book’s goal is to “introduce and illustrate the basic notions of finite group theory and, more generally, compact topological groups, to introduce the notions of Lie algebras and Lie groups, at least in the case of linear Lie groups, to study the Lie groups SO(3) and SU(2) in detail, and then the Lie group SU(3) and its application to the theory of quarks. The idea of a group representation, an action on a vector space by linear transformations, plays the most fundamental role in this study.” The author notes that this book arose from the notes for a nine-week course to mathematics and physics majors at the advanced undergraduate/beginning graduate level and is designed to serve as an introduction to more advanced texts.

There are many introductory books in this area (with at least one new one appearing each year, it seems), but what sets this book apart, in my opinion, is its concentrated focus. In its final chapter, Chapter 8, Groups and Symmetries relates the representation theory of SU(3) and the theory of baryons, mesons, and quarks. The fact that representation theory predicted the existence of the last (and eight meson) and the three final resonances (a type of extremely-short lived elementary particles), discovered in the early 1960s demonstrated the importance of representation theory in particle physics. Anyone interested in understanding the connection between quarks and representation theory is an expeditious fashion should read this book.

Groups and Symmetries is organized about explaining this prediction in a streamlined fashion and this focus explains both its strengths and weaknesses. To achieve its brevity, many proofs have been omitted or made an exercise. Worked examples are also few in number, with many examples needed to be worked out in the exercises. Such editing brings out the unity of the subject and provides an enjoyable read for readers with some previous background in Lie theory who would like to better understand its use in particle physics.

I am less sure how well the text would work as an introduction to Lie theory for someone new to the subject because the brevity and lack of examples would seem to present a steep learning curve. For example, the representation theory of finite groups is covered in 24 pages, the theory of Lie groups and algebras in another 24 pages, and spherical harmonics in 14 pages. This text is not slow-paced! However, the text has worked for Kosmann-Schwarzbach’s classes. Perhaps lectures and classroom discussions are sufficient to supplement the text and ease the learning curve for those new to the subject. To help students who wish to find other texts to consult, at the end of each chapter, Groups and Symmetries has a helpful reference section that provides readers with historical comments and additional references to the topics covered.

Tom Hagedorn is Associate Professor of Mathematics at The College of New Jersey.