# Comprehensive book on group theory for physicists?

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 I guess it's a "classic", but I fear that its notation might be a bit outdated?

• 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/q/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
• 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

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 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
• As a slightly alternative opinion, I personally think Sternberg isn't the best introductory text on group theory (for physicists), and not because of its (sufficient) mathematical rigor. Although it is certainly rich, it is written in a way that only is internalize-able if you've already seen the material. Every section starts from very general and abstract grounds, making no reference whatsoever to the end goal, so every "final result" seems mysterious and confusing. A good introductory text, I think, sufficiently motivates each idea before it is presented, thus giving you the "big picture". – Arturo don Juan Feb 27 '18 at 14:54
• (continuing my above comment) With that said, I think a combination of H. Georgi with B. Hall would be best. The former offers physical motivation, employs physics notations, covers a humongous range of topics relevant to actual physics, but is a little off-hand and sloppy at times. The latter offers rigorous proofs with very elegant and down-to-earth reasoning, still very readable unlike many other math textbooks. – Arturo don Juan Dec 5 '18 at 21:06

Here is my extensively review of various books I had read. For meta discussion, see I have several book reviews. How should I answer in the book request?.

# Wu-Ki Tung, Group theory in physics

Its approach isn't go from general to specific, but from intuition to generalization. For example, many books explain isomorphism after homomorphism, because the former is a specific case of the latter. But in this book, the order is reversed, because we can imagine isomorphism better than homomorphism.

Together with many connections and discussions between chapters and subsections, it shows that the author has a pedagogical mind. In specific, the book:

• Boldly uses ' for mappings (see def 2.5 for example). I've never seen this kind of notation before, and at first I think using this will make more confusion. But turns out it's not
• Important theorems are named, not just numbered
• Avoids study all groups in detail
• Has many advanced example without proof, because they are just illustrations, not a topic for you to study
• Proofs are deferred after discussing significance

A trivial thing: theorems and definitions have different numbering systems. So when you are told to refer to Def. 1.3, then make sure you are not reading Theorem 1.3.

I highly recommend this book, even though it's quite old (50 years or so).

# A. Zee, Group Theory in a Nutshell for Physicists

The book is written in xkcd style: funny and lots of footnotes, with quotes and historic stories. However, most footnotes are at the end of the chapter (endnotes), so when an idea is noted, you can't read it immediately but have to turn to the end of the chapter. This is where the frustration starts: most of the notes are funny comments. Having to break the reading flow and spend more effort just to get a tiny detail or a funny comment is not fun at all. But some of the notes are actually serious and you don't really want to miss it, so every time I see a note I have a mixed feeling.

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 Oxford requires the title to have the bit "in the nutshell", which I think misleading.

Yet, I think you should take a look at the fruitful bits. They do give you new perspectives.

# Jakob Schwichtenberg, Physics from Symmetry

Its structure:

• It starts with special relativity,
• then the symmetry tools (Lie group and Lagrange formalism),
• then the basic equations (free and interaction theory),
• then their specific applications: quantum mechanics, quantum field theory, classical mechanics, electrodynamics and gravity.

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.

# Other books

Here are some books came after I had acquired a good understanding of group theory, so I didn't have much motivation to read them. But I think they are good, and you may want to take a look.

• Sadri Hassani, Mathematical Physics A Modem Introduction to Its Foundations
It has side column for notes and summaries; convenient for skimming. At some pages, there are many emboldened characters at a place, quite confusing to read. It also discusses about $$Endk$$, $$Lk$$.

• Pierre Ramond, Group Theory: A Physicist's Survey
The author gives this analogy at the preface: the universe today is like an ancient pottery, that it isn't as beauty as when it was produced anymore, but we can still feel that beauty.

Explanation of new notation is introduced after its appearance. There is no numbering; the author focuses on making it as fluid as possible.

• Sternberg, Group Theory and Physics
So condensed. I can't get through it. Not recommended.

During my study, I read and take notes on tablet. Most of the books are scanned. If you feel frustrated because the pages are not well split, or the PDF does not contain a table of content, or not having enough margin to take note, you can read this article: The ultimate guide to process scanned books.

• This should be much higher. Upvote, people! – psitae Aug 19 '19 at 11:06

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.

Although if you're looking for mathematical rigor, this may be the wrong book and I would recommend Naive Lie Theory by Stillwell.

In fact, my recommondation would be to read both. The first one to understand what concepts are important for physics and to get a first idea for the motivation behind them and then Stillwell's book in order to get an idea how mathematicians think about these subjects.

• Regarding "Physics from Symmetry": In the first edition, it has so many typos and errors, all books by Gerland Folland don't have that many... – DanielC Sep 28 '17 at 6:39
• Your answer reads as though Prof. Stillwell's book lacks rigor. John Stillwell strives for the simplest, clearest possible explanations, but never lacks rigor UNLESS he explicitly says so; sometimes his texts sketch a proof or give an intuitive discussion and then tell you the background you need to go and learn to achieve a rigorous understanding. He has been known, like all our fellow human animals, to make mistakes, but will most graciously and enthusiastically receive notification of these and act on them. – Selene Routley Oct 30 '17 at 1:08
• Oh no, I meant that Naive Lie Theory is the mathematical rigorous alternative to Physics from Symmetry – Jony Nov 11 '17 at 9:45
• @Jony I'd suppose Naive Lie Theory would be more rigorous than the physics book, but the 'naive' in front makes me think it's less rigorous compared to other math books, á la naive set theory. – John Clever Jan 29 at 16:41

Anthony Zee just came out with Group Theory in a Nutshell for Physicists - covers most of what a undergrad physics student needs including finite groups and representations, except Young diagrams.

• To be quite honest, I don't think most undergrad physics students even need to know much group theory at all. – Arturo don Juan Feb 27 '18 at 14:56
• Zee's book is not a valid recommendation from me. It fails to distinguish between real Lie algebras, complexified Lie algebras and real forms of complex algebras, particularly in the context of the representations of the Lorentz group in 4D – DanielC Nov 18 '18 at 18:34
• I have a mixed feeling about Zee's book. See more details in my answer – Ooker Dec 6 '18 at 15:29

A rather recent book is An Introduction to Tensors and Group Theory for Physicists. It also speaks of vectors and tensors at a good level.

In my opinion it clears up the confusion physicists tend to make when speaking of these topics. Moreover the book is disseminated with examples and applications from mechanics, EM and QM, so is a great introduction to these topics for an advanced undergraduate.

• I can second this. The book clears up a lot of confusion about tensors, upper and lower indices and has an enormous amount of very enlightening examples connecting a wealth of disparate topics one has seen throughout undergrad. The book also strikes a great balance between good explanations that seem informal the way a friend would explain it to you while still being rigorous on the proofs and statements with no handwaving taking place. – Zarathustra Apr 24 '19 at 20:53

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.

• +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
• +1 indeed, this is the most thorough book I know, especially with regards to unitary representations of noncompact groups like the Lorentz group. Although this is important for physics, typical treatments do not cover this in a truly satisfactory manner. However, this is for a reason: the theory is quite hard, and many questions about classifying unitary representations of such groups are still open, see: liegroups.org – Abdelmalek Abdesselam Jan 28 '20 at 16:49

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.

• 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

John Baez's "Gauge fields, knots and gravity" has a very illuminating chapter on lie groups and lie algebras, which is just at the right level of rigor for a physicist. His chapters on differential geometry are also pretty awesome.

• I love this book! In fact, anyone almost anything that John Baez writes is gold. There are a lot of great explanations on his blog – jak Jun 21 '15 at 5:02

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).

• 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
• This book is good if you are willing to believe some of the author's claims. If you want that everything is properly justified, then one finds that loose stated claims need previous knowledge in group theory. After studying group theory and reading this book, I was just remembering all the proofs I have seen before. – user2820579 Mar 31 '20 at 19:58

"Lie Groups: An Introduction through Linear Groups" by Wulf Rossmann gets my vote. It gets the elementary ideas really cemented. Then read "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" by Brian Hall.

I personally recommend Georgi's book with a particular focus on SU(3).

And there is also Ramond'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

I see almost all the classical recommendations, all except one. It's this book by Wu Ki Tung: https://www.amazon.com/Group-Theory-Physics-Wu-Ki-Tung/dp/9971966573. There's also the book by Willard Miller, but I find Wu Ki Tung's one more appealing. Check out the contents table on the Amazon preview. It should satisfy the needs of any college (under)graduate to supplement the QM and QFT courses.

• I highly recommend this book. See more details in my answer – Ooker Dec 6 '18 at 15:33

Just filling in some gaps. Generations of practitioners have used these books, so they underlie what you read about in many of your textbooks.

In order of quite subjective preference,

• 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.

• Lie Groups, Lie Algebras, and Some of Their Applications, by Robert Gilmore. Somewhat chaotic, but has lots of geometrical illustrations and examples, and tracks down nontrivial, non-hackneyed physics applications like few others. Invaluable in appreciating Wigner-Inonu contractions beyond name-dropping. Easy to develop reliance on.

• Group Theory and Its Application to Physical Problems (Dover Books on Physics) by Morton Hamermesh. A classic, yeomanly, solid, responsible Lie Group resource; heavily relied on by boomers. This actually means it is useful in illuminating their universally shared "you know"s.

• Unitary Symmetry and Elementary Particles (2nd Ed 1978), D. B. Lichtenberg. Universally shared bare minimum background on SU(3), again a "live in the background" boomer mainstay resource. If your teacher throws something on the eightfold way you are unsure about, this one is by far the most likely to resolve it. A second best on this is Quantum Mechanics - Symmetries (Springer, 1989) by W Greiner and B Müller. Explicit, albeit somewhat ponderous; but beware of the odd actual stereotypical misconception: do not use unthinkingly.

• Lie Algebras and Applications (Springer 2006) by F Iachello, delightfully tabulates Lie algerbas and their standardized features. A superb starting point (beyond Patera & McKay's phone-books) for identifying or dialing your Lie Group and irrep, indices thereof — you name it.

• Semi-Simple Lie Algebras and Their Representations by Robert N. Cahn (Benjamin 1984). Well logically organized, it provides proofs and arguments for the mathematically exigeant physicist, at just the right level: no hidebound pedantic drivel here.

Parting notes: Michael Stone's Mathematics for Physics is a pearl—boy, would I have loved it, had it been available in my college years. For informed grad student work, R Slansky's classic 1981 Physics Reports 79 sourcesbook review GROUP THEORY FOR UNIFIED MODEL BUILDING can hardly disappoint.

Finally, a worker's book, not a student's, which I am only adding here because I'd be remiss if I did not point out how truly important and accessible it is for theoretical physicists. Really. The three volumes of N Vilenkin & A. Klimyk's Representation of Lie Groups and Special Functions I, II, III, (Kluwer 1991). Truly, as they quote Hadamard, "The shortest path between two truths in the real domain passes through the complex domain".

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 like Hall's book quite a lot. – joshphysics Feb 11 '13 at 20:43
• I'm confused. This book is a graduate text for mathematics, and the first chapter jumps right into Lie group without explaining what group means. How can this be easier then Sternberg's book? – Ooker Oct 25 '17 at 15:22
• @Ooker Have you tried reading both? Sternberg is definitely harder, or at least less readable (as a pedagogical text) than Hall. Sternberg essentially moves at much faster pace, giving little motivation although technically assuming less. Hall, on the other hand, moves much more slowly and carefully, giving lots of motivation but technically assuming a little more. – Arturo don Juan Dec 5 '18 at 21:11
• @ArturodonJuan unfortunately they were both too advanced to me (at that time). I'll take note this and see if Hall's book is good for Lie group – Ooker Dec 6 '18 at 15:28
• @Ooker It might help to try out this online lecture series. – Arturo don Juan Dec 6 '18 at 18:57

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.

The books by J.F. Cornwell are well written and a mix of formalism and examples. There are several different editions but "Group Theory in Physics vols 1 and 2" are excellent choices containing well-chosen examples.

• I would also recommand books by J.F.Cornwell. Also, there are lecture notes from my prof at our nature science faculty at Zagreb but these are in croatian :-). – Žarko Tomičić Jun 21 '15 at 6:51

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:(

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

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.

For those who only care about Lie groups and representations (i.e. not the OP), you can read Quantum Theory, Groups and Representations - An Introduction | Peter Woit | Springer

Systematically emphasizes the role of Lie groups, Lie algebras, and their unitary representation theory in the foundations of quantum mechanics

For erratas, reviews and other posts check out Peter Woit's Home Page

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.

• 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". – Selene Routley 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. – rainman Mar 12 '14 at 17:55

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.

These only cover point group and space group symmetries for solid state physics. For the next semester, I may use also this paper:

But it would be nice to complement these with a paper that uses Lie algebras to solve a simple but interesting and illustrative problem (undergrad level). Any suggestions?

From the list of new books listed in the other Answers, I like "Anthony Zee - Group Theory in a Nutshell for Physicists". I'll add to the list these two:

1. A. W. Joshi, Elements of Group Theory for Physicists
2. Zhong-Qi Ma, Group Theory for Physicists
• why don't you use traditional books for teaching? – Ooker Dec 6 '18 at 15:35
• I do use Tinkham, Hammermesh, Joshi and Zhong-Qi Ma above, and a Brazilian one. However, my experience is that the students get more involved if they study these books while following some papers. My approach is to follow those papers above paragraph-by-paragraph, and go for the books to understand what the paper is doing, and them complement with a deeper discussion on each topic. The students get much more focused and interested in the class. – Gerson J Ferreira Dec 7 '18 at 17:18