Group Theory in a Nutshell for Physicists

by A. Zee

A concise, modern textbook on group theory written especially for physicists

Although group theory is a mathematical subject, it is indispensable to many areas of modern theoretical physics, from atomic physics to condensed matter physics, particle physics to string theory. In particular, it is essential for an understanding of the fundamental forces. Yet until now, what has been missing is a modern, accessible, and self-contained textbook on the subject written especially for physicists.

Group Theory in a Nutshell for Physicists fills this gap, providing a user-friendly and classroom-tested text that focuses on those aspects of group theory physicists most need to know. From the basic intuitive notion of a group, A. Zee takes readers all the way up to how theories based on gauge groups could unify three of the four fundamental forces. He also includes a concise review of the linear algebra needed for group theory, making the book ideal for self-study.

Provides physicists with a modern and accessible introduction to group theory
Covers applications to various areas of physics, including field theory, particle physics, relativity, and much more
Topics include finite group and character tables; real, pseudoreal, and complex representations; Weyl, Dirac, and Majorana equations; the expanding universe and group theory; grand unification; and much more
The essential textbook for students and an invaluable resource for researchers
Features a brief, self-contained treatment of linear algebra
An online illustration package is available to professors
Solutions manual (available only to professors)

Genres: Physics, Mathematics, Science, Textbooks

608 pages, Hardcover

First published March 22, 2016

Reviews

[Adam Lantos · Rating 5 out of 5]

Just to get this out of the way: this is a must-buy.

Group theory can be taught in a very boring way. See Georgi's book for example, where he starts off by giving definition after definition without truly motivating things; this becomes cumbersome and causes the reader's attention span to drop dramatically fast. In contrast, Zee approaches the subject in a much more accessible way, starting from very simple things (even has a nice review of Linear Algebra at the start of the book) and gradually and -unfortunately but inescapably- slowly builds up towards more advanced concepts.
The fact that everything, from the very simple to the very abstract, is motivated is simply amazing and gives a whole new touch to how this subject is taught. I enjoyed reading it; from going through the small bits of physics history to following each small proof to reading the amazing gems of unique insight that the author provides, everything is enjoyable. And the humor is also good (although if you did not like the style of Zee's other Nuts, then you won't like the style of this one too).
To give an example of something done really good, I mention that Zee is the only author of a Group theory book (at least from the popular ones) that has made me truly understand what exactly the quotient group is and why it is important. I mean, you can say that you totally get what the quotient group is through its definition, but besides that there's a much-needed intuition to be built in order to truly appreciate why we define it. Zee provides this in spades.

With these being said, this is no book for anybody wanting a rigorous exposition to this beautiful subject. Zee uses the slang of Group theory in most occasions, of course, but he omits doing everything with the painstaking accuracy of rigorous math. This leaves more space for the brain to deepen the understanding of the basics, so for me this is great. Bear in mind that I used this book for a Group theory course that is non-rigorous, so the course ties really well with this. I have to admit that sometimes I needed more rigor, but I just supplemented this with our lecture notes. The conclusion I draw from this is that this will probably serve you very well, no matter the approach you want to take, but I could definitely see some readers wanting to use this only as a supplement.

Note that this is also great for undergrads who want to learn Group theory. This is a super good intro. Graduate students can at least use this as a supplement. One thing is true though: this is a gem that everybody must buy.

Lastly, I want to also mention that I really like the breadth of the topics covered here. From the standard stuff of particle physics to cosmology and even condensed matter physics, the author provides us with really nice applications of Group theory in physics and the topics presented here showcase the author's deep and wide understanding that comes from his own wide research experience (lot's of condensed matter and high-energy physics).

So, to conclude, this has the trademark writing style of Zee (smooth and easy-going), the unique insights of the author, breadth of topics and paves the way to deep intuitive understanding of the topic. A must-buy.

[Brian Powell · Rating 5 out of 5]

Group theory is the mathematics we use to describe, and by which we ultimately understand, symmetries in physics; the modern theory of particle physics (so-called "gauge theory") is fully infused with group theoretic constructions and concepts. Virtually every textbook on graduate-level particle physics and quantum field theory therefore introduces the subject, and these introductions tend to be successful (e.g. Ryder and Cheng & Li come to mind as being particularly good).

But, eventually, you inevitably get to a point where these texts start to throw ideas around without much basis or background: for example, I don't think I can count how many times an author starts talking about "spinor representations" as if we're old friends, without providing a shred of introduction. Similarly when you, having built a modicum of confidence with the standard model gauge group SU(3)xSU(2)xU(1), attempt to study grand unification: at this point most texts start to do some seriously hardcore representation theory gymnastics: decomposing product groups into seemingly "obvious" direct sums of irreducible representations, as if this stuff is as easy as counting. Conformal symmetries? Forget about it. SL(2,C) and the Lorentz group? Seriously, WTF.

You can go searching in the mathematics literature for an understanding of some of this stuff, but that's a hard journey. Books dedicated to Lie Groups and Lie Algebras and representation theory, apparently the place to go looking for answers, are anything but a sure bet. Invariably you and your faithful steed will get stuck in the algebraic analog of the Swamp of Sorrows, a hopelessly, crushingly abstract incarnation of these allegedly familiar ideas. You might survive, and if you do, you'll be much better for it; but for us mere mortals, this is a dangerous road indeed.

Now, I am often wary of "such and such math for physicists" type books, because they are prone to dispensing with the meat (the true stuff of sustenance) for the cheaply-won, hollowed-out applications or computational tricks. Everything, though, is a balance when it comes to learning advanced physics: there is only so much time and there is a seemingly endless train of ideas to master. Surely, one cannot learn all the relevant math from the canonical textbooks. Often, mastering an application provides the motivation to appreciate and learn the mathematical basis for an idea, and, for me, this is true of group theory and its role in particle physics and grand unification.

Zee's book is the only one I've found (long since I allegedly learned this stuff in grad school) that settled some festering confusions for me. I'm proud to say I truly understand the representations of SO(N) and SU(N); Zee approaches these topics as if the reader is seriously intellectually challenged with great patience and pedantry. Sometimes, this can be annoying and even a little confusing because he can be a little *too* casual and familiar. But he succeeds, and the journey is fun. (In the specific case of reps of SU(N), btw, he doesn't discuss Young's tableaux, which I've never understood despite trying multiple times).

Zee's book is long and expansive. He covers tons of stuff, including discrete groups useful in solid state physics, and all the group theory one needs for high energy physics. He writes in an engaging, friendly style. He covers the formal basics, but it is deeply practical. I hope you enjoy it as much as I did.

[Theodor · Rating 4 out of 5]

Anthony Zee has a very personal and fun writing style. I often found myself eagerly flipping through the sections and notes, searching for his intriguing and humorous anecdotes. Anthony is undoubtedly a scientist with a good sense of humor, which is a noteworthy achievement in itself.

As for the textbook itself now, I believe it serves as a valuable introductory resource for physicists studying group theory. If you are doing research on theoretical physics though, you surely need more than this, but not a great deal also. The book covers all the fundamental concepts and ideas. For me, it was a really enjoyable read and I created a bunch of notes by solving the exercises. However, it was also this part of the textbook that I didn't really like. Many times, the exercises felt disconnected from the content of the section, and often, one had to flip to the end of the book to get a better idea what the problem is about and how to solve it. If the book featured clearer exercises, I would certainly give it a 5-star rating.

Overall, I definitely recommend this book as an introductory book for group theory, and only for physicists. It may not be as much appreciated by mathematicians.

[Kirk · Rating 5 out of 5]

This is by far the best book on group theory I've ever read. The review on matrices is excellent if, like me, you had a very bad introduction to them previously. Most books on group theory I've tried to read either go incredibly sparing on the details for layman, or go incredibly technical and dry. This is as it says: Group Theory for physicists.

[Ellen · Rating 1 out of 5]

I think that this book tries to do two things and falls short of both. I think that the book tries to be informative, but is written cryptically, so is hard to understand, and is difficult to use to as a book you just find the formulae you need. It also tries to have a fun tone, but given that I find it hard to understand the informative text, I find the fun tone annoying. Whenever I have a hard time understanding a subject or an exercise, I feel lost as to where to look for instructions or understanding. Furthermore, The use of footnotes, endnotes, and appendices are wack as hell. I see no clear distinctions as to what is put in footnotes and what is put in endnotes. To me it seems that both places are used for "fun trivia" or mathematical notes, and appendices here are sometimes used for things that are essential for understanding the subject, as opposed to what an appendix should be, which is supplementary reading.

[Thorbjørn · Rating 2 out of 5]

Some good things: It is a book that brings together the most important group theory in physics, and attempts to put perspective on why they are important. This is some undertaking, and credit is where credit is due for such an undertaking.

Otherwise: At best confusing, at worst misrepresenting (some times wrong). A friend of mine put it well; "It is a math book for people who don't like to read math books." But math books are written like they are for a reason, and Zee's attempt to compromise between this and ... whatever else he is trying at, degrades the quality in all aspects.

Credits to writing a book that touches on the things it does. However, it is a frustrating situation that courses are taught with this book, since it doesn't seem there are any alternatives.

[Ciren · Rating 5 out of 5]

Excellent, this kind physicist do want you could learn something by his brief and clear writing.