Comprehensive book on group theory for physicists?

Question

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that covers "all" the basics, and then, in addition, talks about the specific subjects of group theory relevant to physicists, i.e. also some stuff on representations etc.

Is Wigner's text a good way to start? I guess it's a "classic", but I fear that its notation might be a bit outdated?

Answer 1

there is a book titled "Group theory and Physics" by Sternberg that covers the basics, including crystal groups, lie groups, representations. I think it's a good introduction to the topic.

http://www.amazon.com/Group-Theory-Physics-S-Sternberg/dp/0521558859

to quote a review on Amazon (albeit the only one)

"This book is an excellent introduction to the use of group theory in physics, especially in crystallography, special relativity and particle physics. Perhaps most importantly, Sternberg includes a highly accessible introduction to representation theory near the beginning of the book. All together, this book is an excellent place to get started in learning to use groups and representations in physics."

Answer 2

Well, in my dictionary "group theory for physicists" reads as "representation theory for physicists" and in that regard Fulton and Harris is as good as they come. You'll learn all the group theory you need (which is just a tiny fragment of all group theory) along the way.

Answer 3

I would recommend A. O. Barut and R. Raczka "Theory of Group Representations and applications". It is about Lie algebras and Lie groups, and you are asking for general group theory, but this book, in my opinion, would be useful to a physicists. The applications are to physics, mainly quantum theory.

Edit: Forgot to comment on the last part of the questions. I think Wigner is a good read. You'll not learn much about general group theory, but you will learn about representation theory of the Poincare group and some general techniques from representation theory like the Mackey machine for induced representations.

Answer 4

Sternberg's book is excellent and illuminating but perhaps a bit hard for a beginner. I recommend as a first reading Lie Groups, Lie Algebras, and Representations. The book deals with representation theory of Lie groups of matrices. After reading this I also recommend the Sternberg's book for physical applications and the topological point of view of group theory.

Answer 5

Morton Hamermesh's Group Theory and Its Application to Physical Problems is a Dover Press book, so quite inexpensive (though the price seems to be up a bit since I bought it in the '90s).

Answer 6

There is no good book aimed at physicists. Robert Hermann, Lie Groups for Physicists is worth reading, but you didn't want something only about Lie Groups. Gelfand, Graev, and Vilenkin, Les Distributions, vol. 5 or, in English, Generalized Functions, vol. 5 is good for Fourier analysis on a group closely related to the Lorentz group, but not aimed at physicists, but is eminently readable and has some mistakes which don't really matter. Representations of finite groups are covered in Boerner, Representations of Groups: With Special Consideration for the Needs of Modern Physics an old classic written for physicists. None of these books are good, but they are the best I can think of. Strichartz has written about harmonic analysis on the actual Lorentz group, perhaps it is worthwhile, perhaps I will look at it some day...

A famous mathematician once told me no one had ever understood Weyl, The Classical Groups. I think much of it is covered by Boerner.

Answer 7

I took a course on group theory in physics (based on Cornwell) and even though I followed all of the proofs, I had no idea how it might help me solve physical problems until I picked up Tinkham's Group Theory and Quantum Mechanics. Literally just reading 5 pages (the introduction) made a tremendous impact on my understanding of why group theory is important to physical applications and what sort of group/representation properties I should be looking for. After almost every major group/representation result, he shows how it relates to a quantum calculation. His approach and examples might be considered dated (not much on Lie groups and a lot on crystallography) but if you're just getting acquainted with the field, I think it's the best around.

Answer 8

I personally recommend Georgi's book:

http://books.google.com/books/about/Lie_Algebras_in_Particle_Physics.html?id=g4yEuH5rBMUC

And there is also Rammond's book, which look's along the same as Georgi's textbook:

http://www.amazon.com/Group-Theory-A-Physicists-Survey/dp/0521896037

Also online there are some notes available from:

Grossman:

http://www.lepp.cornell.edu/~yuvalg/p7661/Lie.pdf

't Hooft:

http://www.staff.science.uu.nl/~hooft101/lectures/lieg07.pdf

and Slansky

http://bolvan.ph.utexas.edu/~vadim/Classes/11f/slansky.pdf