Lie Algebras in Particle Physics

Lie Algebras in Particle Physics (Frontiers in Physics), by H. Georgi. Perseus Publishing; 2nd edition (October 1999)


Intended as an introduction to Lie theoretic methods in Particle Physics, the book starts with the operator treatment of angular momentum, the analysis of SU(2) and SU(3) [the eighfold way of Gell-Mann and Ne'eman (1964)] in order to introduce the semisimple Lie algebras, via the direct deduction of the Cartan form. Having in mind these important physical examples, and the d-harmonic oscillators, the authors develops the standard topics on Dynkin diagrams and representation theory of (classical) Lie groups. This book does not pretend to develop an exhaustive study of Lie algebras and its representation theory, but a practical and manipulable introduction for physicists to appreciate how useful group theory becomes to describe a physical system. The stars of the text are the groups SU(5) [Georgi and Glashow introduced the SU(5) theory] and SU(6), which are fundamental for unified theories and the quark model and which form excellent representatives to analyze the different pathologies appearing in Lie theory. More advanced topics like spontaneous symmetry breaking [specially important for the group SU(6) and the branching rules (Judd 1971)] and lepton number as fourth color are also introduced. The main objective is to explain and describe the techniques which are necessary to understand the modern particle physics, without leading the reader to a mountain of formal facts whose direct application are beyond their scope.

Physicists and mathematicians at the beginning graduate level. Physical knowledge is presupposed, and some manipulation capability in representations of symmetric groups is recommended [this refers to Young tableaux]

To provide a practical introduction to representation theory methods applied to physics, and to provide a solid background to attack more formal and difficult texts.

Any step is carried out having in mind the immediate physical significance, mainly the unitary groups [for obvious physical reasons]. It comments important facts like the unified theories and the quark model directly, avoiding very lenghtly treatments which usually lead to confusion and insecurity of the studied material.

As a consequence of the previous points, the lack of rigour is the most remarkable objection, mainly referring to some questions of wave functions and the weight theory underlying semisimple Lie algebras. This text will not provide a profound understanding of Lie algebras/Groups and its representation theory, since it avoids the development of the structure theory and some specialized topics indispensable for a full comprehension of the topic. Maybe its reading should be simultaneous to the probably best short book on general Lie algebras [J. P. Serre, Algèbres de Lie semi-simples complexes, Benjamin Inc 1966, english translation by Springer, 1987].
The price is also excessive for an introductory text.

Each of the 27 lectures in which the book is divided (~1 hour per lecture) concludes with some nice problems in the spirit of the book. The author tells the necessary material in a highly attractive way. In short, it is a direct experience of group theory written with much practical sense.
The book can be found at:
Rating: 4/5

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