Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists 1st Edition by Robert Gilmore (PDF)

Description

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.

User’s Reviews

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⭐This is a book with class…..the first chapter gives a hint at what Sophus Lie was trying to do with differential equations as Galois was trying to solve algebraic equations with the power of group theory which by the way he also developed but that’s another story. It is very amusing to see the solution of the cubic, quartic and the proof that the quintic cannot be solved with radicals that’s what Galois achieved and is presented in the first chapter. Lie groups are presented in the second chapter it was in this and this only book that I realized what a Lie Group really is and this is due to the great notation employed for the operations that define a group but in the particular case of Lie groups. Basically a Lie group is a Group that depends on a continuous parameter which is also a Manifold. Once this is stablished examples of Lie groups are given in chapter three with the usual Matrix Groups. The fascinating thing is that being a manifold on each point you can define the tangent space to the manifolds with tangent vectors, these vectors form an algebra through the bilinear Lie bracket, so now we have an algebra for the Lie group and this algebra is named The Lie Algebra of the Lie Group, this vectors can be represented as matrices as well who fulfill the algebra this is shown first on chapter four for Lie Algebras and then on chapter five for Matrix algebra. Why the Lie algebra or the many operations that you can defined like action by the left and by the right are done near the identity element always? BECAUSE, Lie Groups are such fabulous animals that they are manifolds very symmetric which means that they look the same from every point of them so for this purposes all points are equivalent and we may as well choose the identity point as a conventional point to do all calculations! Chapter 6 is about Fermion and Boson operators algebras (I skipped) to get to very important chapter 7 about Exponentiation, this is how to go from the algebra to the group, for most of the times this procedure will work but as Elie Cartán showed it does not always works, what you can say for sure is that many different Lie groups have the same Covering simply connected Space, then all their Algebras can get you to this monster that’s for sure. The book then goes to talk about representation theory for Lie Groups and classification of Lie Algebras with roots and Dynkin diagrams The chapter about Symmetric Riemannian spaces is quite amusing as a metric made out of matrices are given like the Hilbert-Schimdt form and the Cartan-Killing form also a measure for integration is derived for integrating in these amazing manifolds. The finals i have not covered which is about the Hydrogen atom and Maxwells equations for electromagnetism, it ends with a chapter on differential equations and how to solve them using Lie groups, Sophus Lie initial goal and scheme. The best introduction for a Physicist, get yours!

⭐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.

⭐Okay book about lie groups, physics, and goemetry

⭐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 Inönü 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 Inönü-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 Poincaré 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.

⭐I bought this book from “Global Books” and there were a lot of troubles.First of all I got charged twice ! I got my money back without complaining (after a week), so that’s handeled. I think it might have been a double-booking error of amazon and Global Books.After a few weeks of waiting, I recieved the book at my place and the cover was all dented and deformed by moist. I’ve been able to get most of the damage out, but it’s still there.These two items are the ones costing the book two stars, because for the rest the book is really amazing, one of the best and most complete texts about Lie-Groups and algebra’s that I’ve ever seen !

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