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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 I guess it's a "classic", but I fear that its notation might be a bit outdated?

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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/… –  Marek Feb 28 '11 at 17:04
    
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. –  Lagerbaer Feb 28 '11 at 17:29
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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. –  David Z Feb 28 '11 at 18:24
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Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

11 Answers

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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."

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This is what I would've recommended :) +1 –  dbrane Feb 28 '11 at 17:11
    
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. –  Marek Feb 28 '11 at 17:46
    
Sounds like something I should look into. Thanks. –  Lagerbaer Feb 28 '11 at 19:46
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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.

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+1 This is a very very nice book, but sadly out of print. –  Heidar Feb 28 '11 at 17:20
    
Out of print suggests that many people liked it. –  MBN Feb 28 '11 at 17:28
    
+1 It's a good book, but extremely dense. Not recommended as an introductory book (which is what the OP asked for) –  Simon Mar 1 '11 at 11:05
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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).

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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. –  Matt J. Jul 18 '13 at 1:29
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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.

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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. –  MBN Feb 28 '11 at 17:07
    
@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" :) –  Marek Feb 28 '11 at 17:13
    
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. –  MBN Feb 28 '11 at 17:15
    
agreed, this is a great book, but I think is more on the mathematical side. –  luksen Feb 28 '11 at 17:16
    
@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. –  Marek Feb 28 '11 at 18:06
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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

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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.

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I like Hall's book quite a lot. –  joshphysics Feb 11 '13 at 20:43
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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.

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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:(

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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.

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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.

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I believe, although I can't find a reference, that when Dirac was once asked by a journalist whether there was anyone whose thinking was over Dirac's head, Dirac answered "Hermann Weyl". –  WetSavannaAnimal aka Rod Vance Jul 17 '13 at 1:56
    
The entire interview is included in the memorial volume edited by Kursunoglu and Wigner –  joseph f. johnson Jul 18 '13 at 15:13
    
arxiv.org/abs/0810.3328 Along with it study arxiv.org/abs/math-ph/0005032 . Life will be beautiful inshaallah. –  Ome Mar 12 at 17:55
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