There’s a new book out in the Princeton “Nutshell” series, Tony Zee’s Group Theory in a Nutshell for Physicists. I liked his Quantum Field Theory in a Nutshell quite a lot, it’s packed with all sorts of insights into that subject. Both books are written in a very light, chatty and entertaining style, full of various sorts of worthwhile digressions. Long ago I tried to use the QFT book in a course I was teaching, and there found that it functions better as a supplement to a standard QFT book. Zee’s treatment is just too short, covering too much, to provide the level of detail most students need to learn the subject for the first time. For people who already have gone through a standard book or course, Zee’s QFT book is great as a follow-on, likely to explain a lot of things they found confusing the first time around.

The new book on group theory has a length much better matched to the amount of material (it’s longer than the QFT book, and the material covered is much less complicated). The level of detail for most topics should be a good amount for students encountering the subject for the first time. The main topics covered are:

- Finite groups and their representations.
- Unitary and orthogonal groups, their representations, and applications to quantum mechanics.
- Classification of simple Lie algebras.
- The Lorentz and Poincare groups and their representations, with a discussion of the Dirac equation and Weyl and Majorana spinors.
- A grab-bag of some other topics, including a little bit about conformal symmetry and grand unified theories.

While for each of these topics there are other good textbooks out there, this is a great selection for an advanced undergraduate/graduate physics course. I expect this to justifiably become a popular choice for such courses.

While I liked a lot about the book, I have to confess that there were things about it that did put me off. Some of this likely has to do with the fact that I’ve been working for the last few years on a book (see here) that covers some of the same topics, so I’m hyper-aware of both the technicalities involved, and the issues that arise of how best to approach these subjects. In addition, much of these topics is standard core mathematics, but Zee seems to have consulted few if any mathematicians (at least I didn’t recognize any in his acknowledgements). Unlike some others, this is a subject where mathematicians and physicists really can communicate and teach each other a lot.

Some of the choices Zee makes that I don’t think are good ones are things that input from mathematicians probably would have helped with. Maybe the most egregious is his decision to use the same notation for a Lie group and its Lie algebra, on the grounds that physicists sometimes do this, and to notationally distinguish the two in the usual way (upper vs. lowercase letters) is “rather fussy looking”. Using the same notation for two very different things is just asking for confusion, and I remember struggling with this as a student. Zee is well aware of the problem, on page 79 having his interlocutor “Confusio” say:

When I first studied group theory I did not clearly distinguish between Lie group and Lie algebra. That they allow totally different operations did not sink in. I was multiplying the Js together and couldn’t make sense of what I got.

Please, if you’re using this book to teach students about this subject, discourage them from following Zee in this choice.

~~There are places in the text where Zee gets things wrong in a way that just about any mathematician could likely have saved him from~~. ~~One minor example is a footnote saying “Mathematicians have listed all possible finite groups up to impressively large values of n” (actually, they’re classified for ALL values of n)~~ *(my mistake, I misread and wasn’t looking at the finite group chapters carefully enough. Zee does get this right)*.

One place Zee gets things wrong is when he writes down the Heisenberg commutation relations, and says this is an “other type of algebra”, off-topic “since this is a textbook on group theory, I talk mostly about Lie algebras”. Actually those are the commutation relations of a Lie algebra, the Heisenberg Lie algebra, and there’s a group too, the Heisenberg group.

This gets into my own prejudices about the subject, with the story of the Heisenberg group to me (and I think to most mathematicians), a central part of the story of quantum mechanics, something little appreciated by most physicists. Another place where I think Zee goes wrong due to current physics prejudices is in ignoring Hamiltonian mechanics in favor of Lagrangian mechanics. As a result, instead of being able to tell the beautiful story of the Lie algebra of functions on phase space and what it has to do with conservation laws, he just mentions that Noether’s theorem leads to conservation laws and refers elsewhere for a discussion. The connection between symmetry and conservation laws is one of the central parts of the connection between Lie groups and physics, and deserves a lot more attention in the context of a course like this.

So, in summary, the book is highly recommended, with the caveats that you absolutely shouldn’t use the same notation for Lie groups and Lie algebras, and you should supplement Zee’s treatment with that of a certain more mathematically-minded blogger…