Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists Reviews
March 28, 2011 by Actaphysica
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Best introductory text on Lie Groups so far.,
I am so happy with this book that I could not wait to finish one chapter and then post a review. This is my initial review but maybe I will extend it.
I am a theoretical physics student and so far I have read one section on lie algebra and the approach is very clear and not at all In mathematical language and notations ( that you are required to master first to understand the underlying mathematics generally).
Robert Gilmore done a very good job on this introductory book which fits with the title. He explains the ideas in very clear and concise way for non mathematical students. First he explained lie groups briefly and then came to lie algebra and explain why this is done. All most all authors forget to mention why they introduced lie algebra. I have many other books on group theory and lie groups e.g. Sternberg, Fuchs & Schweigert , Wu-Ki Tung, Georgi etc and the main point to be noted is that many authors do a good job explaining ” how” but they forget to mention “why”. This is where Robert Gilmore comes in. It is a pity that he did not write a book on Group theory as a whole including other topics in group theory as well.
My advice is if you need an introduction to lie groups and lie algebra and tired of authors who only try to impress other authors instead of the student then invest on this book.You won’t be disappointed and maybe this one goes into your collection.
PS: for student of particle physics, try also Lie algebras from Howard Georgi.
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Very competent and realistic introduction,
This book is intended as an introduction to the topic for students in physical and chemical disciplines. It should not be thought that this book is an abbreviated version of the previous one. The structure of this text is radically different from the 1974 book, which was more a compendium of group theoretical techniques, and presented very actual topics used in physics. This book, preserving the essential motivations, has been written to develop, step by step, the techniques and methods used when groups are applied to describe physical phenomena, with details and explanations that are usually omitted in most textbooks.
The book consists of sixteen chapters containing a large number of problems to be worked out by the reader. The results are presented in a very direct way, avoiding too technical developments and extracting the main facts. This philosophy is very convenient at a first level, because it focuses on the most important points and does not confuse the reader with involved proofs.
In the first chapter, the author presents the historical motivation that led S. Lie to develop the theory of continuous groups, the Galois theory. This serves mostly to motivate the study of Lie groups, but presents no specific interest to the physicist. Chapters 2 and 3 are devoted to the main properties of matrix Lie groups, which are the main object of study in this book and correspond to the types usually encountered in applications. In this sense, the different classical groups are presented as those subjected to different constraints, motivating the geometrical interpretation of these groups.
In chapter 4 the discussion of Lie algebras begins. First of all, it is illustrated why these structures serve to simplify a lot the analysis, since Lie algebras correspond to the linear approximation to the group at a given point. The exponentiation map is shortly introduced, without going yet into more involved questions like the local determination of the group from the Lie algebra. Important facts like the adjoint representation, the Killing and invariant metrics are introduced. This leads to a first insight into the structure of Lie algebras. This analysis continues in chapter 5, where the Lie algebras of the classical groups are derived from the corresponding constraints. The role played by the Killing form is studied in these examples, constituting a first approximation to the well known characterization of semisimple algebras. Chapter six is devoted to the usual techniques to deal with Lie algebras in physical applications, namely, the realizations by creation and annihilation operators and the realizations by vector fields. Although a very short section, the problems illustrate important topics like the angular momentum by means of Schwinger representations used in Quantum Mechanics. The seventh chapter reconsiders the problem of exponentiation in a more technical way. The limitations of the procedure and the isomorphism problem are developed having in mind the important su(2) case. The main result, the covering theorem, is presented graphically, illustrating quite well the general pattern of the theory. The Campbell-Hausdorff formula is introduced motivated by the non-trivial reparameterization problem. The informal way chosen to present this deep result is quite adequate, since it focuses on the meaning of the theorem instead of presenting a technical proof that it not trivial. Once the basic material has been presented, chapter 8 begins with the systematic study of the structure of Lie algebras. The main types of algebras, abelian, nilpotent, solvable, simple and semisimple are defined using the properties of the adjoint representation. Although not explicitely stated, this corresponds actually to the Levi decomposition. One important point should be clarified here: in section 2.3, the “canonical” form of solvable algebras is presented, according to the well known flag space technique of the Lie theorem. However, upper (respectively lower) triangular matrices are the model for solvable Lie algebras only for the complex base field (the Lie theorem being false in general for real solvable Lie algebras). At no point this crucial point is mentioned, which could lead to confusion to the non-expert. Chapters nine and ten concentrate on the classification problem of complex semsimple Lie algebras. This part is a shortened version of the material contained in the previous book of the author, presenting only the indispensable facts. The graphics of root systems help a lot to understand the general situation and the motivation of the classification of Dynkin diagrams. I miss however some comments on the Cartan matrix, which is the natural link between the (fundamental) roots and the corresponding diagram. The next chapter focuses on the real forms of simple complex Lie algebras. The main idea of its obtention is studied, as well as the main steps of the Cartan method to determine the…
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