Customer Reviews

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

5 star:		(5)
4 star:		(0)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

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.

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.

I started this book with very little mathematical background (just an electrical engineer's or applied physicist's exposure to mathematics). By the end of this book, I had an advanced exposure to foundational modern mathematics. Now, I am planning to start on "Differential Topology and Quantum Field Theory" by Charles Nash (with other mathematics reference books to complete the proofs in it).

This book also provides a good amount of material showing the application of mathematical structures in physics - Tensors and Exterior algebra in Special relativity and Electromagnetics, Functional Analysis in Quantum mechanics, Differentiable Forms in Thermodynamics (Caratheodory's) and Classical mechanics (Lagrangian, Hamiltonian, Symplectic structures etc), General Relativity etc.

For the intended audience (advanced students in theoretical physics) this is by far the best book available on the material it covers. The text is clear, the topics covered are presented in a logical sequence, and the student who works through the book will acquire a good background for understanding more advanced texts. Of course, not every area of mathematics used in theoretical physics is included; that would be impossible in a single book; but it is usable as a reference. In my opinion, however, this is mainly a book to study and work through, rather than a reference book. Exercises in the text help readers to check and solidify their understanding of the material.
The author has obviously taken great care in the preparation of the book. There are very, very few typographical errors. Sadly, it is rare nowadays to find a book which has been as carefully proofread as this one.
If you are a physicist and need to come up to speed on any of the topics covered by this book (one of the other reviews has helpfully listed the table of contents), waste no more time searching, just buy it.

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.

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.