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 I guess it's a "classic", but I fear that its notation might be a bit outdated?
This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Lagerbaer

Comments

Do you really want general group theory? I.e. theory of abstract groups, multiplication tables, classification of finite groups (using Lagrange's, Fermat's, Sylow's theorems, etc.), theory of presentations, uses of groups in number theory, etc.? Because if you just want to use group theory in physics then in my experience you won't need anything besides representations. See this question of mine over at MO: mathoverflow.net/questions/56304/…

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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Well, since I don't know too much about group theory, I therefore also don't exactly know what I want. Seems that representations is the thing to look for.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Lagerbaer

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Should this be CW? I guess not, but I wasn't sure if we should run it as a list question for book recommendations. There isn't just one correct book to use.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user David Z

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.

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."

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user luksen

Comments

This is what I would've recommended :) +1

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user dbrane

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This book has been suggested to me by one of my (physicist) teachers, so I am giving +1 in his sake :) For some reason, I've never looked at it though... should check it out.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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Sounds like something I should look into. Thanks.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Lagerbaer

Answer 2

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.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

Comments

+1 This is a very very nice book, but sadly out of print.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Heidar

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Out of print suggests that many people liked it.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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+1 It's a good book, but extremely dense. Not recommended as an introductory book (which is what the OP asked for)

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Simon

Answer 3

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).

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user dmckee

Comments

Dover Press reprints include a lot of good books on group theory for physicists. Unfortunately, I have not seen any such book that meets ALL the requirements the OP is asking for. But I think he could do well either with Georgi's (expensive) book mentioned below, or with Hamermesh AND Heine AND Lipkin from the Dover Reprints. You can even sample these books on Google Books with the Preview feature.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Matt J.

Answer 4

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.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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A very good book for everyone, although the main portion of it is structure theory and representation theory of semisimple Lie algebras.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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@MBN: good point. Some people might wonder whatever happened to Lie groups. And I am not sure which book would I recommend to such people. Probably Goodman & Wallach but I'd be reluctant to call it "for physicists" :)

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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Yes, but my impression is that algebras are more important to physicists than groups. I may be wrong. Goodman and Wallach is for mathematicians, but if physicists find it useful then I would too recommend it. It is quite lengthy though.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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agreed, this is a great book, but I think is more on the mathematical side.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user luksen

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@MBN: I am not sure it is for mathematicians (mainly because I am not one :)) but its content is definitely for physicists (at least I find basically everything very useful). On the other hand, I know many people would dislike the theorem/proof composition and algebraic geometry approach also need not be to everyone's liking. On the third hand, it was this book that gave me motivation to learn some algebraic geometry.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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@Marek: Well, I don't know what their intention was, I am guessing it is meant for mathematicians (and students in mathematics). The content may be useful to physicists but what do you mean by "it is for them"! It is mathematics, so it is for mathematicians and anyone who can find use of it in his work.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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@MBN: it's just that I got an impression that the range of topics (e.g. spinors, branching rules, harmonics, etc.) is what a physicist would be interested in quite a lot more than a regular mathematician (unless he's a mathematical physicist). But I didn't in any way mean to imply that mathematicians wouldn't find this book useful.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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@Marek: I didn't think that you were implying that. I am just saying that all those things that you list are things that some mathematicians are interested in. And many of them have no knowledge nor interest in physics. I know that it may be surprising for you but it is the case.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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@MBN: it is indeed surprising. I'd actually like to know what mathematicians work on and what tools they use. But I realize this question is so broad that it makes no sense :)

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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@Marek: You could narrow it down a bit and ask it on MO. The thing is that any mathematics that is used in physics is of interest to some mathematicians which are interested in just for its own sake and not the applications to physics.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

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@MBN: is that really the case? What about e.g. special functions and systems of orthogonal polynomials? These are used heavily in physics but I doubt mathematicians would find them that useful. In my opinion there are lots of math topics like this one, which mathematicians would consider just as applications to physical problems and so not very interesting. But perhaps I should try MO, hard to get a definitive answer without asking mathematicians themselves.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Marek

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@Marek: There are many mathematicians that are doing research in those areas and know nothing about physics.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user MBN

Answer 5

I personally recommend Georgi's book.

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

Also online there are some notes available from Grossman, 't Hooft, and Slansky

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user DJBunk

Answer 6

A supplementary source may be my online book 

Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras, 2011. http://lanl.arxiv.org/abs/0810.1019

Answer 7

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.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Andrea Amoretti

Comments

I like Hall's book quite a lot.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user joshphysics

Answer 8

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.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user mmdanziger

Answer 9

I am surprised no one has mentioned Lipkin yet. His "Lie Groups for Pedestrians" uses notation that is not too out of date, since it was written in the early 60s. He covers the use of group theory in nuclear physics, elementary particle physics, and in symmetry-breaking theories. From there, it is only a small jump to more modern theories.

Georgi's book (mentioned above) may be even better, but it is awfully pricey: as a Dover Press book, Lipkin's is quite cheap and easily available. It can even be downloaded as a PDF file from 4shared. Or bought as an e-book from Google. Even the Preview on Google is not bad, being surprisingly close to complete.

Lipkin does assume the readers knows quantum mechanics at about the sophomore physics major level, since the quantum-mechanical angular momentum operator is basic to his whole presentation; he also assumes familiarity with Dirac's bra and ket notation. But I am sure that is not asking too much.

Heine's "Group Theory in Quantum Mechanics" and Weyl's "The Theory of Groups and Quantum Mechanics" are also classics, but their notation really is old. And both books are too old to cover use of group theory with QCD or symmetry breaking. But both these books explain the philosophy of the use of groups in QM, which later authors seem to usually assume you already know. Heine also includes a lot more than most about the application of finite and 'point' crystallographic groups. But he does still seem to take a more mathematically abstrat approach than most physicists need: as Lipkin points out, the interests of a physicist and those of a mathematician in group theory really are different: as an example of the difference, Lipkin even mentions the rank of Lie algebras without ever defining it:(

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user Matt J.

Answer 10

There is a recent textbook which gives a fairly complete and concise presentation of group theory, covering both structure and representations of both finite and continuous (Lie) groups, with a brief discussion on applications to music (finite groups) and elementary particles (Lie groups). The target level is advanced undergraduate and beginning graduate. It is freely available at

http://www.scribd.com/doc/207786199/Group-Theory-A-Physicist-s-Primer http://www.scribd.com/doc/209840863/Group-Theory-A-Problem-Book

The author has also co-published texts on contemporary particles and elementary particle theory, some parts of which discuss real life applications of group theory.

This post imported from StackExchange Physics at 2014-05-04 07:46 (UCT), posted by SE-user el vieto