A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Book Description

Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.

Customer Reviews:

A fast introduction to mathematics in physics.......2006-01-02

The book does not assume prior knowledge of the topics covered. However, the reader will find use of prior knowledge in algebra, in particular group theory, and topology. Compared to texts, such as Arfken Weber, Mathematical Methods for Physics, A Course in Modern Mathematical Physics is different, and emphasis is on proof and theory. The text is reasonably rigorous and build around stating theorems, giving the proofs and lemmas with occasional examples. The style is not the strictest, although making the text more reader friendly, it is easy to get confused with which assumptions have been made, and the direction of the proof. Sometimes only the "if" part is proven.

Students familiar with algebra will notice that the emphasis is on group theory, interestingly the concept of ideals is left mostly untouched. For more on representation theory a good reference is Groups Representations and Physics by H.F. Jones where solutions to some of the exercises can be found, and examples of the use of the fundamental orthogonality theorem applied to characters of represenations.

The first 6 chapters are relatively straight forward, but in chapter 7 Tensors the text becomes much more advanced and difficult. Chapter 10 on topology offers some lighter material but the reader should be careful, these consepts are to re-appear in the discussion of differential geometry, differentiable forms, integration on manifolds and curvature. These are not the most simple subjects and it is clear that they deserve entire courses of their own.

The book has insight and makes many good remarks. However, chapter 15 on Differential Geometry is perhaps too brief considering the importance of understanding this material, which is applied in the chapters thereinafter. The book is suitable for second to third year student in theoretical physics.

Jumping over the Gap.......2005-12-30

Most physicists avoid mathematical formalism, the book attacks this by exposing mathematical structures, the best approach I've ever experience. After reading the first chapter of this books I can assure is a must for everyone lacking mathematical formation undergraduate or graduate.

It surely jumps over this technical gap experienced by most physics opening the gate for advanced books an mathematical thinking with physic intuition.

Unfortunately is very expensive, i hope i could have it some day.

A serious, wide spectrum introduction to modern mathematical physics.......2005-10-10

This book covers almost every subject one needs to begin a serious graduate study in mathematical and/or theoretical physics. The language is clear, objective and the concepts are presented in a well organized and logical order. This book can be regarded as a solid preparation for further reading such as the works of Reed/Simon, Bratteli/Robinson or Nakahara.

Not a review, only a little more information.......2004-12-11

Since I don't yet have this book, I cannot review it; however, I have found the contents of this book on the publisher's web site in case it would help anyone decide to purchase it or not.

Contents

Preface
1. Sets and structures
2. Groups
3. Vector spaces
4. Linear operators and matrices
5. Inner product spaces
6. Algebras
7. Tensors
8. Exterior algebra
9. Special relativity
10. Topology
11. Measure theory and integration
12. Distributions
13. Hilbert space
14. Quantum theory
15. Differential geometry
16. Differentiable forms
17. Integration on manifolds
18. Connections and curvature
19. Lie groups and lie algebras

I will return at a later date to properly review it in case I need to change the rating I gave it.

Average customer rating:

clear, concise book
A Group Theory book that allows students to learn
A much needed breath of fresh air!

Molecular Symmetry and Group Theory
Robert L. Carter
Manufacturer: Wiley
ProductGroup: Book
Binding: Paperback

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ASIN: 0471149551

Book Description

A Thorough But Understandable Introduction To Molecular Symmetry And Group Theory As Applied To Chemical Problems! In a friendly, easy-to-understand style, this new book invites the reader to discover by example the power of symmetry arguments for understanding theoretical problems in chemistry. The author shows the evolution of ideas and demonstrates the centrality of symmetry and group theory to a complete understanding of the theory of structure and bonding. Plus, the book offers explicit demonstrations of the most effective techniques for applying group theory to chemical problems, including the tabular method of reducing representations and the use of group-subgroup relationships for dealing with infinite-order groups. Also Available From Wiley:
* Concepts and Models of Inorganic Chemistry, 3/E, by Bodie E. Douglas, Darl H. McDaniel, and John J. Alexander 0-471-62978-2
* Basic Inorganic Chemistry, 3/E, by F. Albert Cotton, Paul Gaus, and Geoffrey Wilkinson 0-471-50532-3

Customer Reviews:

clear, concise book.......2003-10-09

this small book on symmetry and group theory is easy to understand and packed with examples. it is a small book (only about 300 pages) but everything in it is relevant and to the point. the writing is easy to grasp and the back includes character tables for all the common symmetry groups. buy it!

A Group Theory book that allows students to learn.......2000-10-27

The strength of this book is its many examples. Carter takes the concepts and applies them to simple inorganic or organic compounds. Very helpful to students. The end of chapter problems are nice as well. A great text and one that I would recommend to students as well as faculty.

A much needed breath of fresh air!.......1998-07-29

This is a text set apart from the pack. It clearly states what other books attempt to describe. There need to be more texts on the market like this. Dr. Carter has taken a subject that has historically been elusive, and presented it in a comprehensive, READABLE volume. This text is helping me through my thesis in chemistry. It is highly practical, easily read, and heavily referenced, all of the qualities, I believe, that make up an excellent text! Excellent work!

Average customer rating:

Review of Knapp's "Lie groups: beyond an introduction."

Lie Groups: Beyond an Introduction
Anthony W. Knapp
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Hardcover

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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Representation Theory: A First Course (Graduate Texts in Mathematics / Readings in Mathematics)
Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics)
Representation Theory of Semisimple Groups: An Overview Based on Examples. (PMS-36).
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ASIN: 0817642595

Book Description

From reviews of the first edition: "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future." -- Newsletter of the EMS "This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book." -- Mathematica "Each chapter begins with an excellent summary of the content and ends with an exercise section... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists." -- Publicationes Mathematicae Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups. Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond.

Customer Reviews:

Review of Knapp's "Lie groups: beyond an introduction.".......2002-08-13

The short version: this is a superbly written and conceived book; if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen. If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book. It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).

The long version, if you want more convincing or details:

I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.") Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory. Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious. Carefully chosen examples motivate and
clarify the general theory; consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view). But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure. Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding. Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured. This is a fantastic book, and exactly what I was
looking for. Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source.

Elements for Physics: Quantities, Qualities, and Intrinsic Theories
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Elements for Physics: Quantities, Qualities, and Intrinsic Theories
Albert Tarantola
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Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion (Cambridge Monographs on Mathematical Physics)

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Very useful for the student who practices renormalization for the first time

Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion (Cambridge Monographs on Mathematical Physics)
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Lie Groups, Lie Algebras, and Some of Their Applications

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Book Description

Most of the numerical predictions of experimental phenomena in particle physics over the last decade have been made possible by the discovery and exploitation of the simplifications that can happen when phenomena are investigated on short distance and time scales. This book provides a coherent exposition of the techniques underlying these calculations. After reminding the reader of some basic properties of field theories, examples are used to explain the problems to be treated. Then the technique of dimensional regularization and the renormalization group. Finally a number of key applications are treated, culminating in the treatment of deeply inelastic scattering.

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Very useful for the student who practices renormalization for the first time.......2007-01-18

It really shed light on some concepts that other books simply do not discuss. It is also the only one I know that gives a theoretical basis for understanding dimensional regularization.

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Foundations of Quantum Group Theory
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Group Representation Theory for Physicists

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Lie Groups, Lie Algebras, Cohomology and some Applications in Physics (Cambridge Monographs on Mathematical Physics)
Josi A. de Azcárraga , and Josi M. Izquierdo
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Book Description

Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fiber bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics.

Average customer rating:

Most gentle introduction
A nice introduction to modern methods
Poorly written
A welcome book

Symmetry in Mechanics
Stephanie Frank Singer
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Paperback

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Book Description

"Symmetry in Mechanics is directed to students at the undergraduate level and beyond, and offers a lovely presentation of the subject...The first chapter presents a standard derivation of the equations for two-body planetary motion. Kepler's laws are then obtained and the rule of conservation laws is emphasized. ..... Singer uses this example from classical physics throughout the book as a vehicle for explaining the concepts of differential geometry and for illustrating their use. ... These ideas and techniques will allow the reader to understand advanced texts and research literature in which considerably more difficult problems are treated and solved by identical or related methods. The book contains 122 student exercises, many of which are solved in an appendix. The solutions, especially, are valuable for showing how a mathematician approaches and solves specific problems. Using this presentation, the book removes some of the language barriers that divide the worlds of mathematics and physics..." ---- Physics Today

Recent years have seen the appearance of several books bridging the gap between mathematics and physics; most are aimed at the graduate level and above. Symmetry in Mechanics: A Gentle, Modern Introduction is aimed at anyone who has observed that symmetry yields simplification and wants to know why. The monograph was written with two goals in mind: to chip away at the language barrier between physicists and mathematicians and to link the abstract constructions of symplectic mechanics to concrete, explicitly calculated examples. The context is the two-body problem, i.e., the derivation of Kepler's Laws of planetary motion from Newton's laws of gravitation. After a straightforward and elementary presentation of this derivation in the language of vector calculus, subsequent chapters slowly and carefully introduce symplectic manifolds, Hamiltonian flows, Lie group actions, Lie algebras, momentum maps and symplectic reduction, with many examples, illustrations and exercises. The work ends with the derivation it started with, but in the more sophisticated language of symplectic and differential geometry. For the student, mathematician or physicist, this gentle introduction to mechanics via symplectic reduction will be a rewarding experience. The freestanding chapter on differential geometry will be a useful supplement to any first course on manifolds. The book contains a number of exercises with solutions, and is an excellent resource for self-study or classroom use at the undergraduate level. Requires only competency in multivariable calculus, linear algebra and introductory physics.

Customer Reviews:

Most gentle introduction.......2006-03-22

Stephanie Singer's book is a service to all students who wish to learn modern aspects of mechanics for the first time.

A nice introduction to modern methods.......2005-02-20

I think the previous review is a bit harsh, and that the book's intents are not what this reviewer expected. I don't think it was the author's intent to write a comprehensive treatise on the subject. The book simply aims at introducing undergraduate students to the use of symmetry in simplifying the analysis of classical mechanic problems, nothing more. If you want a comprehensive treatise, you probably want to read V.I. Arnols's "Mathematical methods in classical mechanics". If what you want is a simple introduction where all the steps are worked out in details, then this book is a good starting point, and I think this is what the author intended. At any rate, the cost ($$$) is quite reasonable.

Poorly written.......2003-12-12

This is one of the very few books which I returned for refund.
The subject is intrinsicly interesting, and there is a need for a serious introductory text addressing the subject of geometry and physics. This one badly falls short, - carelessly written, with numerous irrelevant asides. She seems even to fail to realize that there exist three distinct geometric solutions to the Kepler problem. The bound, elliptic case is only one.
This book has supposedly been written for high-undergraduate students or early-year graduate students. It serves neither adequately.

A welcome book.......2002-01-26

There are a number of books available on the "geometric" view of physics (Classical Mathematical Physics, by Thirring, The Geometry of Physics, by Frankel, and Foundations of Mechanics, by Abraham & Marsden). The size, level of sophistication and extensive background assumed by these books can be very intimidating. On the other hand, the subject "looks" beautiful, and the benefits of using geometric intuition are desirable to many people.

Singer's book stands class of its own in these respects. All the basics of the geometrical "machinery" are there, in a book that is only 224 pages in length. Chapter one starts with a standard derivation of the equations of the "two-body planetary motion" problem; subsequent chapters proceed to introduce the necessary modern geometrical and mathematical concepts (differential geometry). The final chapter then revisits the "planetary motion" problem using the modern concepts previously introduced. Excellent!

There are some misprints, but the author has a Web page of errata. The book has numerous exercises, with many solutions included. I find myself rereading parts of this book over and over.

Reader be warned; the concepts are new, and it does take work to internalize them. However, this is the most accessible book on the subject available, and also one of the most affordable. The author references many other books, for the reader who wants to go further in the mastery of this subject (one excellent book which is not mentioned, however, is "Differential Forms: A Complement to Vector Calculus", by Weintraub). Enjoy!

Group Theoretical Concepts and Methods in Elementary Particle Physics
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Group Theoretical Concepts and Methods in Elementary Particle Physics

Manufacturer: Gordon and Breach
ProductGroup: Book
Binding: Hardcover
ASIN: B000BK7JO4

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