Lie Groups, Physics, and Geometry

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Describing many of the most important aspects of Lie group theory, this book presents the subject in a 'hands on' way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

Lie Groups, Physics, and Geometry Review

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 non-equivalent real forms. The material of this section is crucial for applications, since many important models are based on non-compact Lie algebras. Being a quite delicate question, I agree with the author in the decision of leaving out the notions of inner and outer involutive endomorphisms used in their classification.
These first eleven chapters cover the main facts about Lie theory that any student in either physics or chemistry should master for a full comprehension of more technical. Chapter 12 reviews Riemannian symmetric spaces, a very important type of manifolds. Here the geometrical role of the exponentiation map is exploited, helping to understand the implications of the choice of real form and its consequences in the geometry and topology of the corresponding manifold. The material is again presented and commented using important examples, instead of developing cumbersome theoretical argumentations, which can be found in the cited literature. The results are complemented by carefully chosen problems of physical nature, pointing out the relevance of symmetric spaces in applications. Chapter 13 introduces a more sophisticated technique, the contractions of Lie groups. This procedure, of essential importance in physics, is developed following the classical method of In�n and Wigner. How to use contractions in limiting processes of other objects is illustrated in the different sections. However, I believe that focusing only on In n -Wigner contractions gives a quite restrictive view of this technique (even if this constitutes a very important class of contractions, as shown by their applications to symmetry breaking).
Chapter 14 constitutes an introduction to the study of symmetry in physical systems. This is an important part, since many textbooks usually assume the reader is aware of the different notions of symmetries used. To this extent, the author chooses a classical and vital example, the hydrogen atom. The different types of symmetry (geometrical, dynamical, spectrum generating algebra) not only point out the different physical properties to be described by means of symmetry, but also the importance of how to embed a Lie group into another. The detailed description made by the author will surely clarify some aspects that are generally quickly reviewed, and therefore constitute a difficulty for the unexperienced reader.
The Maxwell equations are derived in chapter 15 using the properties of two fundamental groups in Physics: the Lorentz group SO(1,3) and the Poincar group. Although it may appear that this chapter is disconnected from the rest, it actually has been placed in the right place. On one hand, the Maxwell equations are connected to the most important physical groups,.and further, these are closely related to the conformal group previously introduced, being a natural link to justify the importance of symmetries of differential equations.
The last chapter connects with the first in the sense that Lie groups are used to determine whether a differential equation can be solved by quadratures or not. Since this is a large and complicated theory, only the basic elements that show how Lie groups are used to simplify the integration of differential equations are presented.

This book constitutes a very comprehensive introduction to Lie theory in physics, dealing with the most important features that students will encounter. The problems help not only to understand the material presented, but also exhibit real physical situations where Lie groups are used This book further solves some difficulties encountered by beginners in other books, usually written at a more specialized level.

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