Quantum Theory and Representation Theory, the Book

For the last few years most of my time has been spent working on writing a textbook, with the current title Quantum Theory, Groups and Representations: An Introduction. The book is based on a year-long course that I’ve taught twice, based on the concept of starting out assuming little but calculus and linear algebra, and developing simultaneously basic ideas about quantum mechanics and representation theory. The first half of the course stuck to basic non-relativistic quantum mechanics, while the second introduced free quantum field theories and the relativistic case. By the end, the idea is to bring the reader to the point of having some appreciation of the main elements of the Standard Model, from a perspective emphasizing the representation theory structures that appear.

The discussion of quantum field theory I think is rather different than that of other textbooks, taking a Hamiltonian point of view, rather than the Lagrangian/path integral one in which most physicists are now trained (myself included). One basic idea was to try and work out very carefully the quantization of a finite-dimensional phase space in all its representation-theoretic glory, with the idea that free quantum field theories could then be developed as a straightforward extension to the case of taking solutions of a field equation as phase space. While this point of view on quantum field theory is fairly well-known, writing up the details turned out to be a lot more challenging than I expected.

As part of this, the book attempts to carefully distinguish mathematical objects that usually get identified by physicist’s calculational methods. In particular, phase space and its dual space are distinguished, and the role of complex numbers and complexification of real vector spaces receives a lot of attention.

At the same time, the book is based on a relatively simple philosophical take on what the fundamental structures are and how they are grounded in representation theory. From this point of view, free relativistic quantum field theories are based on starting with an identification of irreducible representations of the space-time symmetry group using the Casimir operator to get a wave-equation. This provides a single-particle theory, with the quantum field theory then appearing as its “second quantization”, which is a metaplectic (bosonic case) or spinor (fermionic case) representation. These are some of the specifics behind the grandiose point of view on how mathematics and physics are related that I described here. For some indications of further ideas needed to capture other aspects of the Standard Model, there’s this that I wrote long ago, but which now seems to me hopelessly naive, in need of a complete rethinking in light of much of what I’ve learned since then.

The manuscript is still not quite finished, and comments are extremely welcome. While several people have already been very helpful with this, few have been willing to face the latter chapters, which I fear are quite challenging and in need of advice about how to make them less so. The current state of things is that what remains to be done is

  • A bit more work on the last chapters.
  • A rereading from the start, bringing earlier chapters in line with choices that I made later, and addressing a long list of comments that a few people have given me.
  • An old list of problems (see here) needs to be edited, with more problems added.
  • I need to find someone to make professional drawings (there are some funds for this).
  • An index is needed.

Optimistically I’m hoping to have this mostly done by the end of the summer. Springer will be publishing the book (my contract with them specifies that I can make a draft version of the book freely available on my web-site), and I assume it will appear next year. To be honest, I’m getting very tired of this project, and looking forward to pursuing new ideas and thinking about something different this fall.

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13 Responses to Quantum Theory and Representation Theory, the Book

  1. tulpoeid says:

    Impressive. Congratulations and may you pursue new ideas very soon 🙂

  2. Marcus says:

    Hi Peter,

    The content of your books look really good. Congrats for (almost) finishing!

    I’m interested in what you think is the unique difference of your book to existing books that seem to cover similar ground (e.g. Zee’s new Group Theory book, Schwichtenberg’s Physics from Symmetry or Robinson’s Symmetry and the Standard Model Reference)? At a first glance I would guess maybe mathematical rigor?

    Best wishes,

  3. Asaint says:

    Any disclosures as to what those “new ideas” are?

  4. StrangeRep says:

    Hi Peter,

    1) On p240 you seem to say that Kepler’s 3rd law is associated with the LRL vector. I believe this is not correct. Rather, the 3rd law arises from a dilation-like symmetry. In coordinate form, it is t -> kt, q -> k^{2/3} q. See section 10.2 of Stephani, i.e.,

    H. Stephani, “Differential Equations…”, CUP 1989C, ISBN 0-521-36689-5

    (I find this extra dilation-like symmetry interesting because it is not associated with a conserved quantity. Not sure how/whether it’s relevant in the quantized version.)

    2) What units are you using in the Hamiltonian and LRL vector on p240? Your expressions seem dimensionally invalid, at least superficially.

    3) Your expression W on p241 for the quantized LRL vector seems to have a typo. Your denominator |Q|^2 should be just |Q|, shouldn’t it?Also, you don’t seem to mention that the main reason why Pauli chose that modified version was in order to have an Hermitian operator. The GvH thm is less of an issue here, since one just looks for Hermitian versions of the classical quantities.

    And btw, do you want further comments delivered here on your blog, or would email be more convenient?

  5. fg says:

    To follow up on the point 1 in StrangeRep’s comment:

    The existence of the LRL conserved vector is responsible for the fact that all the bounded orbits a 1/r potential are closed. The conservation of LRL is specific to this potential, and is not related to the usual geometrical symmetries (such as angular momentum, from rotational invariance). It may (but I am not sure) be in fact related to the scaling symmetry pointed out in StrangeRep’s comment.

    At the quantum level, I think it is related to the degeneracy of the energy levels of the hydrogen atom (all states with the same n and different l have the same energy).

    The only other rotationnally invariant potential whose orbits are all closed is the harmonic oscillator. Again, this property can be attributed to the existence of an extra conserved quantity, besides the geometrical ones.

  6. Peter Orland says:

    Hi Peter and Commenters,

    After the comments above, I took a look at page 240. It is never actually stated which of Kepler’s laws follows from which property of the Coulomb problem. It probably would be good to clarify this.

    It’s nice to have an explicit solution of the hydrogen atom done this (Pauli’s) way in a textbook. I always preferred Fock’s method of stereographic projections (it’s somewhat more elegant and quicker, esp. for Coulomb scattering), but what you are presenting is probably more suitable for a course at this level.

  7. Peter Woit says:

    Thanks! Such comments are greatly appreciated, in general better to send them to me via email.

    Thanks to others for comment about the Coulomb potential calculation. One thing one realizes when writing something like this is that even quite a few hundreds of pages is not enough to do more than scratch the surface of the topics one wants to explain. The main goal in this case was the Lie algebra calculation of the hydrogen spectrum, which is both very important physics and a great demonstration of representation theory techniques.

    I think there’s a detailed answer to your question in the introduction. I’m trying to cover different material not so readily available elsewhere, and also to develop a certain unifying point of view about the relationship of QM and representation theory.

    The level of rigor is higher than typical physics books, less than that of typical math textbooks. I’ve described it jokingly as a level of rigor of the kind one sometimes sees in Russian math textbooks. Unlike many math textbooks, the aim is generally to develop specific central examples, rather than to develop theory and make general statements. The goal is to simultaneously be precise enough to keep mathematicians happy, while still writing something that physicists might be willing to read and try to follow

    One project is to get back to work on Dirac cohomology. The book actually develops a lot of the basic elements needed for going farther with that.

  8. Justin says:


    What exactly are the prerequisites for working through your book? I understand linear algebra from Serge Lang’s famous text, but it does not mention anything about Hilbert spaces and I’ve never studied them. I also know some algebra from Fraleigh’s classic text. I have a very minimal amount of physics just now learning some quantum mechanics from a text geared towards freshman undergraduates so it’s below the level of Griffith’s text.

  9. Peter Woit says:

    You should be able to read the book. It avoids getting into the serious analysis of Hilbert space theory. A warning to anyone though is that more mathematics and more physics come into it as the book goes on, while the treatment gets sketchier, so the later parts will be much more challenging.

  10. D R Lunsford says:

    Book looks like a valuable addition to the canon, thanks for the effort.


  11. Chris Oakley says:

    Roger Penrose writes the wave equation for a free helicity s particle as ∂^{AA’}Φ_{AB…} where there are 2s LH SL(2,C) indices on the field. It may have been obvious to him why this followed from Wigner’s irreducible reps of the Poincare group, but as it was not obvious to me, I did the analysis in section 3.3 here: maybe there’s a quicker way. This equation is the massless Dirac equation for spin 1/2, source-free Maxwell’s equation for helicity 1 and linearised GR for helicity 2. So it gets you from your chapter 42 to chapter 46 without needing to invoke pre-quantum physics, and I would have thought, especially if you are aiming the book at the mathematical end of the physicist market, it would be a welcome addition.

  12. Peter Woit says:

    Thanks for the pointer to the notes. I wanted though to do something different than the SL(2,C) spinor formalism, for various reasons (one is to keep separate the different ways complex numbers occur). As in a lot of other places in the book, I also wanted to focus on the structure of specific examples, as opposed to a more general formalism. So, while the SL(2,C) formalism is of great generality, covering all spins, I wanted to instead concentrate on the specific cases of field equations for fields occurring in the Standard Model.

  13. Auntiegrav says:

    Do you have a notification list for when the book is published? I look forward to it. Sounds like just what this old nutjob needs.

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