Lie Groups for Pedestrians

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Courier Corporation, 2002 - Science - 182 pages

According to the author of this concise, high-level study, physicists often shy away from group theory, perhaps because they are unsure which parts of the subject belong to the physicist and which belong to the mathematician. However, it is possible for physicists to understand and use many techniques which have a group theoretical basis without necessarily understanding all of group theory. This book is designed to familiarize physicists with those techniques. Specifically, the author aims to show how the well-known methods of angular momentum algebra can be extended to treat other Lie groups, with examples illustrating the application of the method.
Chapters cover such topics as a simple example of isospin; the group SU3 and its application to elementary particles; the three-dimensional harmonic oscillator; algebras of operators which change the number of particles; permutations; bookkeeping and Young diagrams; and the groups SU4, SU6, and SU12, an introduction to groups of higher rank. Four appendices provide additional valuable data.

 

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I am slowly working my way through this after having been frustrated by very cursory references to the theory of Lie groups/algebras in various physics books. I have got something valuable out of it, such as the very helpful clarification that physicists refer to the name of the group when it is the corresponding Lie algebra they are really interested in. That one comment cleared up a lot of confusion!
But Lipkin has too many of his own obscure references, such as explaining that Lie groups (really, Lie Algebra commutators) used in physics tend to all look like angular momentum because there is only one Lie Algebra of rank one: he never explains what 'rank' is (it is the dimension of the maximal Cartan subalgebra). Nor does he ever explain WHY the rank is significant.
Despite that failing, this is a LOT more up to date than Weyl's "Group Theory and Quantum Mechanics" or Heine's "Theory of Groups in Quantum Mechanics", neither of which cover the use of Lie groups for strong interactions (except for a very brief section on nuclear physics in Heine).
Still, for a more up to date reference with more practical examples of how group theory is really used in calculations in physics, if I were a serious physicist with an adequate budget, I would prefer "Lie Algebras In Particle Physics: from Isospin To Unified Theories (Frontiers in Physics)" by Howard Georgi.
So in summary, I don't consider the book entirely useless, and it is certainly a bargain at the Dover price (or as en eBook on Google). But it is not self-contained, you will find the need to turn to other sources to make sense of it, and the range of problems you can actually solve based on the group theory in this book is still too limited.
 

Contents

INTRODUCTION
1
ISOSPIN A SIMPLE EXAMPLE
19
THE GROUP SU AND ITS APPLICATION TO ELEMENTARY PARTICLES
33
THE THREEDIMENSIONAL HARMONIC OSCILLATOR
57
ALGEBRAS OF OPERATORS WHICH CHANGE THE NUMBER OF PARTICLES
69
PERMUTATIONS BOOKKEEPING AND YOUNG DIAGRAMS
96
THE GROUPS SU4 SU6 AND SU12 AN INTRODUCTION TO GROUPS OF HIGHER RANK
103
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