I studied group theory for the first time around 15 years ago at the beginning of my PhD. There were six of us in the class, and I found it both a magical, as well as a mysterious subject. We had a great lecturer, but the way that the course was set up, and as a course designed for theoretical physicists, where the tools were more important than the construction of the tools, a lot of ideas were left as mysterious boxes where the right answers were guaranteed so long as the algorithm was correctly followed.

Tony Zee is known for his incredible ability to lead the student on a path from little knowledge, to an intuitive understanding of a topic in a seemingly painless process. His books are not necessarily the most technically rigorous (note that this doesn’t mean that they are wrong, but that the appropriate level of detail is chosen for the new learner such that the overarching ideas aren’t fogged in unnecessarily complication), but they are, in my opinion some of the best texts for taking a learner from nothing, to a working knowledge with which they can perform calculations that I’ve ever come across.

Zee has created here the book that I wish I had had when learning group theory for the first time.

At around 600 pages, it is far more in-depth than the texts that I used over a decade ago, but Zee makes it enormously readable. In fact, on receiving the book, I sat down for a weekend and worked my way through the first 200 pages in detail, going through exercises and seeing how he was building up intuition in the appropriate places. In these sections we take a parallel route going over the mathematics of groups as well as the applications to their physical manifestations. After going over the details of representation theory and the most important theorems we are led to the link between group theory, quantum mechanics and classical mechanics and then on through the necessary group theory to understand the standard model. Dynkin diagrams are introduced beautifully having spent a lot of time getting an intuition into roots. The full contents of the book can be found here and the first chapter can be found by clicking on the picture above.

Perhaps the one topic that I think is really missing from this exposition (which Zee mentions explicitely is left out for the sake of keeping the book within a reasonable volume) is that of Young Tableaux, though perhaps that is simply my own bias at having used these extensively in my own research, as well as finding them to be one of the most mysterious pieces of mathematical machinery when I first came across them. In particular, the algorithm for finding the dimension of a representation using hooks and weights seemed to have been pulled out of thin air on first sight. Zee’s clarity on such a topic would also be hugely valuable, and perhaps this may find its way into a future volume. This omission aside, I think that what is covered in the book is absolutely spot on for a modern theoretical physics student.

Zee has done a truly remarkable job here to create what I think may be the definitive text on the topic of group theory aimed at this specific audience. This is not a book for mathematicians (though they may be interested in the physics applications), but for physics students of just about all theoretical varieties, this book is going to be an invaluable resource for decades to come. I highly recommend anyone with an interest in group theory as applied to physics to get this beautifully written, carefully expounded book.