# Group Theory in a Nutshell for Physicists

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.

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…

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### 30 Responses to Group Theory in a Nutshell for Physicists

1. edmeasure says:

So when (and where) will we see Peter Woit’s version in print?

2. Peter Woit says:

With a little bit of luck, late this year or early next year from Springer. The current version available online is mostly complete, except for ongoing work on a few of the last chapters.

3. Fernando says:

All finite groups? No, only all finite simple groups.

4. Peter Woit says:

Fernando,
Thanks. My mistake, I didn’t look carefully enough, he does get this right. Will fix.

5. anon says:

At least as of year 2008, the number of (non-isomorphic) groups of order 2048 was not precisely known (the number is known for all smaller orders): see http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf

6. Peter Woit says:

anon,
Thanks. I was being quite unfair to Zee about this, my misreading, due to not looking all that carefully at the finite group chapters. This is a topic he covers in quite a bit of detail, in a manner closest to the way mathematicians teach the subject.

7. ronab says:

I actually found Zee’s QFT book extremely hard to read: Repeatedly, I’d be sinking my teeth into some interesting physics, when I’d be suddenly distracted as I hit upon yet another of his pointless anecdotes about some physicist’s personal habits, completely breaking my concentration. These are scattered everywhere so you really can’t avoid them. If this had been an online version I’d have suggested that at least a “Show Bullshit / Hide Bullshit” button might have helped.

8. Peter Woit says:

ronab,
There’s a fair amount of that in the new book also. You can however miss a lot of it by ignoring the footnotes…

9. theoreticalminimum says:

I saw that this is Zee’s 3rd contribution to PUP’s “Nutshell” series, his 2nd being a book on general relativity, wherein he even has a small intro to twistors. I think what’s generally interesting about reading Zee’s books is the beautiful subtle insights he provides on different topics that one has been exposed to via standard courses supported by standard textbooks in graduate school. All this to say that I am eagerly waiting for his group theory book to show up at my doorstep soon!

10. Ryan says:

Although there are already several very good books on group theory for physicists like Robinson’s “Symmetry and the Standard Model” or Schwichtenberg’s “Physics From Symmetry”, I think, it’s great that Zee took the time to write this book. However, as with his QFT book, I think, it is better suited as a complementary book. In my humble opinion, he covers too many advanced topics, which aren’t essential for beginners. Especially many beginners may feel the need to understand all these advanced topics before reading on and thus give up before they read about the most important topics, such as spinors etc. .

That said, after reading a quicker introduction to the essentials, it’s great to read Zee’s special perspective and his enjoyable explanations of advanced topics. As you already noted, exactly as the other books mentioned above, it lacks mathematical rigor and that’s why I’m eagerly waiting for your book Peter, to fill this gap!

11. simplicio says:

Anyone have thoughts on Zee’s Relativity book? I used his QFT book in conjunction with my class text (Peskin and Schroder) for my QFT class and found that it made the going a lot easier. I was thinking I might try and go through his Relativity book this summer, since I never had a class on the topic and wanted to learn.

12. David says:

His relativity book is excellent. I’ve tackled several other intro books on the subject (Hartle, Carroll, Schutz), and I’m not a physicist, just an amateur. Zee takes a unique approach, building up the mathematics of curvature in conjunction with variational principles and just enough group theory. The sections in which he derives classical mechanics from variational principles and symmetry considerations helps consolidate your math skills before diving into curved spacetime. He gives an outstanding exposition of how the field equations follow from the Einstein-Hilbert action. Overall, this book added a lot to my level of understanding.

Some caveats: the modern trend is to develop GR using index-free forms; Zee’s book is almost entirely based on index notation. That’s both a strength and a weakness. The last quarter of the book is a less-than-rigorous survey of some frontier topics that I found to be pretty worthless. If you don’t like the dialogues with Confusio, you will find minor annoyances scattered throughout (though they do serve the purpose of letting the author point out common misconceptions and their antidotes).

13. David Metzler says:

A bit OT: your readers might be interested to know that the LHC achieved its first stable beams of the year yesterday, albeit with a tiny intensity.

https://indico.cern.ch/event/506406/

It seems that they are more or less on schedule, if I’m reading things correctly. Peter, do you know if last year’s problem with the CMS detector is still around and likely to seriously degrade its performance, or not?

14. Peter Woit says:

David Metzler,
As far as I know, CMS has successfully dealt with that problem, see a relatively recent report at
http://science.energy.gov/~/media/hep/hepap/pdf/201603/Olsen_CMS_HEPAP_April_1.pdf

15. This review is a prime example of the kind of thing that reviews ought to do, but in my view the conclusioon that the book is “highly recommended” is in conflict with the bulk of the review.

Zee is a good writer — and good writers need to be *extra* careful to get things right, because what they say is more memorable and more persuasive.

In my experience, teaching at the undergraduate level in a different field, it is *not* at all straightforward for the classroom instructor to try to correct the book. Patently bad books make this easier. Books that are well written and well packaged gain, thereby, an increment of credibility and auctoritas. For all these reasons, this sounds like a book I would stay away from.

16. John McAllison says:

Personally, I love the way Tony Zee adds colour to his texts via his interesting and sometimes hilarious anecdotes which creates a link to what’s being learned. This is from his General Relativity book concerning the teaching of tensors:

Long ago, an undergrad who later became a distinguished condensed matter physicist came to me after a class on group theory and asked me:

‘What exactly is a tensor?’
I told him that a tensor is something that transforms like a tensor.

When I ran into him many years later, he regaled me with the following story. At his graduation, his father, perhaps still smarting from the hefty sum he had paid to the prestigious private university his son attended, asked him what was the most memorable piece of knowledge he acquired during his four years in college. He replied:

“A tensor is something that transforms like a tensor.”

17. Peter Woit says:

John McAllison,
The same story is in the new book.
One of the things that does grate about the book to a mathematician’s sensibilities is that Zee uses the tensor product symbol for just about every kind of product, except the tensor product. Given the kind of confusion indicated by the story he likes, I don’t see why he (and other physicists), don’t just define the tensor product.

18. Ryan says:

@Peter Do you have a reference where the correct usage of the tensor product $\otimes$ vs. the direct product $\times$ is explained for physicists? Or maybe some quick example, where Zee gets this wrong and should use another symbol instead?

19. Peter Woit says:

The tensor product is a product of vector spaces, the things Zee does that I was referring to are
1. use the tensor product symbol for a product of groups (SU(3) times SU(2) times U(1))
2. use the tensor product symbol for the cross product of vectors (angular momentum is x times p). This is an unusual usage.

This distinction between types of product can become pedantic. The more substantive question in my mind is that of why he avoids use of the tensor product. I can see why physicists don’t want to use the abstract, coordinate invariant definition, but the definition in terms of a basis is straightforward.

20. Johnny says:

I think that the fact that mathematically minded readers disagree with the choices he makes would only please Zee. He writes for physicists and is proud of the way physicists deal with pedantic issues.

21. Peter Woit says:

Johnny,
I’m sympathetic with Zee on avoiding pedantry, but things like distinguishing a Lie group from a Lie algebra are not pedantic distinctions, and one really confuses students by acting like they are by using the same notation for both.

22. Johnny says:

I haven’t seen the book so I can’t really judge, but I would be very surprised if Zee doesn’t explain clearly that they are separate things, adding that it should be clear from context which is which in every case he discusses.

23. Peter Woit says:

Johnny,
That’s exactly what he says. Of course he’s aware that they are different things. This doesn’t change the fact that using the same notation to mean quite different things and expecting your reader to figure out for themselves which one you’re talking about is a bad idea. Especially when you’re dealing with students trying to learn the subject for the first time.

24. Jeff M says:

He really uses the same notation for the Lie algebra and the Lie group? Pedantic mathematician that I am, I am appalled…

25. A.J. says:

Et’s leke useng the same symbol for ‘i’ and ‘e’ when spelleng Englesh words. Weth suffeceent effort, your readers can fegure out what you meant. But there’s no benefet to doeng et, et makes errors more lekely, and et makes you sound leke an edeot…

26. Conducator says:

“Et’s leke useng the same symbol for ‘i’ and ‘e’ when spelleng Englesh words. … et makes you sound leke an edeot…”

27. Justin says:

I’m a layman, with no physics. I understand algebra at the level of Fraleigh from my undergraduate days. I’m interested in learning the representation connection to quantum mechanics, and was wondering how I could go about studying the prerequisites to Zee’s book.

28. cthulhu says:

Anybody have opinions on how Zee’s books compare to Susskind’s “Theoretical Minimum” books? As an engineer who agonized between majoring in physics vs aerospace engineering in college, I am still quite interested in physics, and have enjoyed the Susskind books (although I’m not all the way through the quantum mechanics book, and there’s no GR book yet). So I’m always on the lookout for stuff that can be understood by an engineer who made through tensor calculus and calculus of variations, even if I’m a bit rusty 🙁

29. Johnny says:

A.J. I don’t think your example is good (it’s not even funny). A closer example (not perfect though) would be to use the same symbol “x” to multiply numbers and vectors. Truly appalling, ha?

30. Peter Woit says:

Justin,
Zee’s book is aimed a people with a significant background in physics, only real math background needed is linear algebra (which he reviews at the beginning). I don’t think it’s that useful if you don’t already have a course in quantum mechanics.

Cthulhu,

Zee’s is quite a bit more advanced that Susskind’s, with Susskind explicitly trying to write for people with minimal background, Zee’s book developed out of a course for advanced undergrads and beginning graduate students, and is more at that level.