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.
\"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.\"
Here and there there are some insights or unexpected facts (mostly in the introductions and appendices of each chapter), but the rest are verbose and can be reduced, especially when math is involved, so you may want to have good foundation before skipping them. The author explicitly states that he tends to \"favor those are not covered in most standard books, such as the group theory behind the expanding universe\", and his choices reflect his own likes or dislikes. So if you want to have a standard knowledge in standard book, this is not your choice. The contract of the author with Princeton requires the title to have the bit \"in the nutshell\", which I think misleading.
While the physical meanings of mathematical objects are emphasized, mathematical meanings of mathematical objects are underconsidered. Trace is only a sidenote thing, not the character of equivalent irreducible representations. Schur's lemma is mentioned only in one sentence. The whole representation theory is discussed very fleeting (only one subsection in the Lie group theory section), before going straight to important groups: $SU(2)$, Lorentz group, Poincaré group.
There is a new book called Physics From Symmetry which is written specifically for physicists and includes a long, very illustrative introduction to group theory. I especially liked that here concepts like representation or Lie algebra aren't only defined, but motivated and explained in terms that physicists understand. Plus no concepts are introduced which aren't needed for physics, which was always a big problem for me when I read books for mathematicians. Group theory is a very big subject and mathematicians find a lot of things interesting that aren't very relevant for physicists.
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.
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.
Classical Groups for Physicists , by Brian G. Wybourne (1974) Wiley.Has the most usable Lie Group theory beyond monkey-see-monkey do SU(2) and SU(3). Is addressed to readers who habitually illustrate and attempt understand abstract mathematical notation (a rare species). Once one learns how to use it, one may spend a lifetime doing just that. Dynamical group treatment for solvable systems a veritable classic.
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.
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.
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.
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:(
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
Instead of following the books, I've been teaching group theory for physicists by following these papers below. The idea is to study the papers from top to bottom, and use a traditional books (e.g. Tinkham, Hammermesh, Dresselhaus, Joshi) to fill the gaps. 1e1e36bf2d