10 of 10 people found the following review helpful:
5.0 out of 5 stars
An excellent practical guide, November 6, 2003
This review is from: The Lie Algebras su(N): An Introduction (Paperback)
This short book covers an important aspect that has been neglected by most textbooks on Lie algebras written for physicists, namely providing a comprehensible introduction for undergraduates based on detailed examples, computations and precise motivations, without having to develop the formal theory. This is not a textbook on Lie algebras in the usual sense, but a practical guide whose intention is to provide a solid comprehension of the main facts on (finite dimensional) Lie algebras used in physics. This justifies the choice of the objects analyzed, the compact real form su(N) of the Lie algebras sl(N,C), which constitute an essential tool in the study of the interacting boson model and nuclear rotational states. The topics covered by this book are quite modest (there are no general proofs and no development of classical problems like the classification of simple Lie algebras), and focuses on a detailed comment on the properties of simple algebras using mainly three Lie algebras, su(2),su(3) and su(4), before ennouncing the general case in the last chapter. However, this should not be understated, specially because the book explains carefully the usual notations (which change in the literature from author to author) and tries to clarify the reasons that justify the study of the formal theory.
The book is divided into six chapters, which we comment separately. The first chapter is a quick and effective overview on the basic properties of simple Lie algebras, namely the adjoint representation, the Killing form, representations and their reducibility. For the inner product the Dirac bracket notation is used. The concept of multiplets, which plays an essential role, is introduced at the end of this chapter. Chapter 2 begins with a short discussion of hermitian matrices, and introduces the Lie algebra su(N) in the usual way. The complexification of this algebra is shortly commented, as well as the generation of the algebra by means of operators. The structure constants over the standard basis are obtained, and as application the Killing form for su(N) is computed. It should be said that the notations used in this chapter have in mind the Gell-Mann matrices, which will be introduced later. Chapter 3 studies the fundamental facts concerning the rank one algebra su(2), and which will be central to later developments. The topics commented are generators of su(2), that is, the Pauli matrices, the quantum mechanical operators J of angular momentum, the su(2) multiplets and the irreducible (complex) representations. Further the tensor products (called "direct products") of these representations and their decomposition into irreducible components is commented. Many very detailed computations are presented, which illustrate clearly the procedure and its significance. Moreover the graphical method for the tensor product decomposition is developed,
The fourth chapter, devoted to the Lie algebra su(3), which cosntitutes in some sense the core of this book, actually develops the main aspects necessary to the description of global symmetry schemes for hadrons (without deeping into the actual classification, for this would require a basic knowledge of quantum field theory). The Lie algebra su(3) is introduced according Gell-Mann's notation. The step operators and states of su(3) are introduced, and the individual states and multiplicities are carefully constructed using graphical motivation (which actually corresponds to the standard application of the su(2)-triples). In order to formalize the construction, the Young tableaux are used (these constituting an essential tool for the analysis of the su(N) algebras). Special attention is devoted to the fundamental su(3)-multiplet (the quark representation 3) and its dual. This leads naturally to the introduction of the hypercharge Y (however no reference to the Gell-Mann-Nishijima formula is made). The (quadratic) Casimir operator of su(3) and its eigenvalues are analyzed, with explicit examples that point out the main properties of this invariant. The next section focuses on the tensor products of su(3)-multiplets, and develops also the graphical method to deduce the decomposition. A table presents some of these tensor products (for highest weights lower or equal to (2,1)). Again, this motivation is used to present the Young tableaux. Chapter 5 presents more or less the same topics for the rank three algebra su(4), and discusses the charm C (as a natural consequence of the quantum numbers discussed for su(3)). The multiplets and tensor products are reviewed (the diagrams are of exceptional quality and clarity), and the chapter finishes commenting on the standard Weyl basis (that is, the basis obtained from the root system of the corresponding algebra; this is the presentation that will be found in almost any book on Lie algebras). These facts are presented without proof, but serve to illustrate fundamental facts like the Cartan integers or the presentation by generators and relations that the interested reader will find in any standard text. Chapter six gives a recopilation of the basic facts of the su(N) algebras for arbitrary values of N (hermitian generators and multiplets, quadratic Casimir operator, etc). The bibliography presents some texts to profound the study. A little remark: the reference to Cornwell's book refers specifically to volume II, which deals with the theory of finite dimensional Lie algebras.
On balance I think this book is an excellent first contact with Lie algebras for those using them in physics, because of the lucid style and the clarity in the exposition. The very detailed calculations and step by step introduction of the material allow the readers not familiar with Lie algebras to become confident with the main facts they will find in any standard textbook, and which often discourages because of notational problems or implicit assumption of knowledge concerning the fundamental properties. Although the notation is mainly that used in physics literature, the examples and motivations introduced in this text will help the reader in the transition to other books using alternative notations. This work is a welcome reference for both beginners in Lie algebras for physics, as well as for instructors.
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2 of 2 people found the following review helpful:
5.0 out of 5 stars
Lie algebra demystified, July 6, 2005
This review is from: The Lie Algebras su(N): An Introduction (Paperback)
A practical introduction to an esoteric topic which frightens many physics students.
This book presupposes little background mathematics and begins by defining lie alegebras and providing adequate examples. He then details some basic properties of finite dimensional lie algebras and offers several ways of "representing" them including the adjoint representation. From the beginning there is an emphasis on applications to quantum mechanics and I especially enjoyed the section on SU ( 2) and it's application to angular momentum operators. SU ( 3) and SU ( 4 ) are developed in due time in a logical and easy to understand format.
He also shows, in a simple way, how the tangent space of the identity of a lie group has a lie algebra structure which is useful in studying the group's local properties.
A very handy reference for those studying advanced quantum mechanics and particle physics yet basic enough for undergraduates to grasp the concepts.
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1 of 1 people found the following review helpful:
5.0 out of 5 stars
Excellent Text!, July 7, 2011
This review is from: The Lie Algebras su(N): An Introduction (Paperback)
This book is fantastic and written very clearly. My peers and I were very appreciative of this straightforward text!
Check out the author's website, where this and many of his other books are freely available in PDF form!
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