Quantum Mechanics and Representation Theory: talk and book progress

Last week I gave a colloquium talk at the Texas Tech math department, slides are here if you’re interested. One motivation for the talk was to advertise the book project I’m working on, which gives a lot more detail about these topics if you find something interesting in the slides.

The current state of the book is visible here. There are 31 chapters done, about another 5 to go. I also need to go through the entire thing again and reconcile various choices of convention that currently are not necessarily consistent. I plan to get back to work on this in a couple weeks after fall classes are over here, have something like a finished draft of a book done around February. The next project will be to get back to what I was writing long ago on Dirac cohomology and make some more progress with that.

For the next few days though, will be taking it easy, and eating turkey. Happy Thanksgiving to all!

Update: The volume of the Feynman Lectures on Physics devoted to quantum mechanics is now available freely online here. This is a masterful introduction to QM from the perspective of a great physicist. What I’ve been writing in some sense is intended to function best as a supplement to this and an explanation of how it is related to some basic concepts in mathematics.

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29 Responses to Quantum Mechanics and Representation Theory: talk and book progress

  1. The two slides on cohomology in the talk had no corresponding entries in the book. Where can I find some good references on this?


    Florin Moldoveanu

  2. Avattoir says:

    FM, he gave you a BOOK, for crying out loud — and you knitpick?

  3. Per says:


    This book looks really interesting. Thanks for putting the draft online,


  4. Peter Woit says:


    I should have provided this link:

    As I mentioned, next plan is to get back to work on this…

  5. Cplus says:

    The dual of your title, Representation Theory for Physicists, may claim equal validity.

  6. kashyap vasavada says:

    I would like to know if your book would clarify foundations of QM or help with the perpetual debate on its interpretations.

  7. Yair says:

    I think the use of the Feynman statement as the motivation for the colloquium is inaccurate.

    Feynman was not referring to a lack of understanding of the mathematical tools.
    He was rather referring to issues like the non intuitive nature of QM.

  8. Ray says:

    How come none of the equations in the book are numbered? Also you should replace hand drawn figures by figures produced in latex.

  9. Avattoir says:

    Book needs a sexed-up title to goose sales: one suggestion, to get the ball rolling about the hat: Where there’s a Weyl there’s a Woit. (Logic suggests those might be reversed; but this is about marketing, dammit!).

    Another – The Big Lie Updated; or – Quantum Mechanics: Lies, Embedded Lies and Mathematics.

  10. Thank you Peter, I will study the reference in detail.

  11. Adam says:

    I am currently an undergraduate studying physics, and am taking Quantum Field Theory for the first time. Through this class, I really have gotten a feel for how important representations of lie groups really are. As far as I can tell, however, most undergrad physics majors – unless also advanced math majors – seem to pick it up as they go, which is really odd. Representation theory arguably contains the most important mathematical tools used in modern physics, save for say calculus… Point is, this book will help fill a gaping void in the literature for young students of physics – just some words of encouragement.

  12. Chris Oakley says:

    I agree with Avattoir. How about this for an opening sentence: “For decades the worst kind of populist bullshit has been peddled in quantum mechanics s0-called text books.”

  13. Peter Woit says:

    Cplus and Adam,

    Thanks, I do hope this is as useful to physicists trying to understand representation theory and some of the mathematics behind quantum theory as it might be to mathematicians trying to understand quantum theory. I should find another title for it, although I’m not going to follow Avattoir’s suggestion…

  14. Attempting to be useful says:

    Your books seems to be pretty good. As a math person that took some grad physics courses, however, my recollection is that what I missed most was physical intuition. So, my suggestion is to add some remarks on the following:
    (1) at what scale do you need quantum mechanics to describe motion? (something like what is done in the Feynman lectures in physics)
    (2) what is a lot of energy or little energy – what sort of process and behavior you can see at different energy levels?
    (3) Which particles exist?

    Basically, I think mathematicians tend to focus too much on the math side of any given scientific theory. Hence, they will most likely look at the book as a small subset of representation theory – and as such, they might think that it would be more profitable to read a book on representation theory instead. It is crazy, of course, that many of them would think like that – so, if you give them a concrete grasp that there are some real objects out there, you may end up waking up some sleeping beauties.

  15. Peter Woit says:

    The book is still a long ways from a finished product, latex makes it look much more finished than it is. It definitely needs better and more graphics, and probably equation numbering.

    kashyap vasavada,
    There’s nothing really about interpretational issues, although there is somewhat of an argument that QM is really founded on certain kinds of mathematical ideas, so in that sense is making an argument about foundations.

    Feynman was contrasting QM to classical physics, including the fact that they seem to have radically different formalisms. I’d argue that representation theory does give some “understanding” of the QM formalism, giving a sort of answer to questions like “why are observables self-adjoint operators?”, which is the sort of question I think Feynman had in mind as “no one understands” the answer to.


    Thanks for the comment. I may try and add more material like you suggest (there will be some in the last chapter on the standard model). Giving some real physical insight though is difficult, I think what I’ve written would best be used in conjunction with a standard physics text, supplementing but not replacing it.

  16. Chris W. says:

    Loosely related to the question of what Feynman meant: See this new post by Steve Hsu.

  17. Emanuel Quant says:

    I am struggling with the understanding of the physical relevance of unitary inequivalent representations.
    You say:
    “In the case of quantum field theory one has an in finity of inequivalent irreducible representations of the commutation relations to consider, one source of the difficulties of the subject.”
    It seems to me that this issue is not fully settled yet. What is your opinion ?

  18. Peter Peterson says:

    Hi Peter,
    I liked your presentation of QM for mathematician – the version from couple of years back. This is a great continuation, as a physicist, I am a bit challenged by a very matter of fact presentation that expects basic mathematical knowledge, that physicists often try to get away without.
    Just a question – I was trying to search for word Clifford, just to verify how much in depth you go there (again, a great introduction here, but with expectation of having necessary mathematical background would challenge a number of physicists, and I suspect should be expected basic knowledge for a mathematician). My search did not work that well, because ‘ff’ in Clifford is using a special (I assume Unicode?) character (I cannot paste it here, edit box will not let me). It is probably intended.
    Again, I like your book a very much.

  19. Peter Woit says:

    Emanuel Quant,
    I don’t think that it’s a problem that can be “settled”, it’s a real aspect of the mathematics of qfts, partly responsible for making the subject much more difficult than quantum mechanics of a finite number of degrees of freedom.

    Peter Peterson,
    The treatment of Clifford algebras, spinors and their relation to fermionic quantization is one thing that’s in the notes, but not so well known (to physicists or mathematicians). I don’t know of a really good, easy to follow treatment of these topics elsewhere, so I hope what I’ve written will be useful.
    The “ff” issue is about the handling of “ligatures”, I’m not sure what source you are having the searching problem with. For more about this issue, see here

  20. Art says:

    Title contest? Avattoir sets a high standard, but how about “Giving the Lie to Feynman”? A marketer’s dream…

  21. Peter Peterson says:

    Hi Peter,
    You likely read the following books:
    1. Clifford Algebra to Geometric Calculus A Unified Language for Mathematics and Physics (Fundamental Theories of Physics) By D. Hestenes, G. Sobczyk http://www.fishpond.com/Books/Clifford-Algebra-to-Geometric-Calculus-D-Hestenes-G-Sobczyk/9789027725615
    2. New Foundations for Classical Mechanics (Fundamental Theories of Physics) by David Hestenes http://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148
    3. Geometric Algebra for Physicists by Chris Doran, Anthony Lasenby. http://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954/ref=sr_1_1?s=books&ie=UTF8&qid=1385925080&sr=1-1&keywords=doran+and+lasenby
    4. The Berezin Calculus by Paul Robinson – http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCsQFjAA&url=http%3A%2F%2Fwww.ems-ph.org%2Fjournals%2Fshow_pdf.php%3Fissn%3D0034-5318%26vol%3D35%26iss%3D2%26rank%3D1&ei=noybUtTIE4fxoATvqoDAAg&usg=AFQjCNE9kLrL24OG59Kvzf2ql7WLSjOxbg&bvm=bv.57155469,d.cGU&cad=rja
    There are a lot of shorter articles and studies if you search.
    Hestenes started applying and promoting Clifford/Grassmann framework as a consistent approach to all areas of physics a few decades ago. There were multiple attempts to teach physics based on this geometric (Clifford) approach, even though in my opinion, notwithstanding the claims by proponents that geometric algebra approach makes understanding physics easier, the prerequisite is understanding of geometric algebra, which I suspect adds a more challenging preliminary step than the traditional approach. Doran Cambridge group has a very nice gauge theory of gravity based on this approach (see: http://arxiv.org/abs/gr-qc/0405033v1), among many other things.
    In any case, you may already have an opinion on most if not all that I referenced, would be interesting to read your opinion, even just a short comment.

  22. Peter Woit says:

    Peter Peterson,

    I’ve always thought that the spinor/Clifford algebra point of view on geometry ultimately will have something to do with how to unify internal and space-time symmetries, but I don’t think any of the attempts to do this tried so far really work. Much of the “geometric algebra” stuff is purely classical, but what I think is most fascinating is the way it shows up in quantization of fermionic variables. Of the things you mention, I’m not specifically familiar with those sources, but I’ve looked at closely related things off and on over the years.

    What I’ve tried to put into this book is a good reflection of what I see as the important mathematical ideas, together with how they show up in basic ideas about quantization. There will be some more of this in the chapter I’m in still writing about quantized spinor fields. Once I’ve finished this, there are some ideas there I think worth pursuing, will try and write more about them at some later point.

  23. vmarko says:

    Peter Peterson:

    “[...] http://arxiv.org/abs/gr-qc/0405033v1 [...]”

    I don’t know about the whole geometric algebra approach, but what I see from the above article is the following:

    (1) The authors introduce the geometric algebra by essentially rewriting the tensor calculus in the spinorial representation of SO(3,1). They just use a somewhat unconventional abstract notation to hide spinor indices. Apparently no mention is being made that this is equivalent to an already existing body of knowledge in a different notation.

    (2) The authors do not appear to understand the difference between symmetry localization and introduction of interactions. They claim that localization of Poincare symmetry should be enough to generate gravity, criticize conventional (Kibble/Hehl/others) approach for having an extra step of saying “spacetime is curved”, and then go on to perform both of these steps in their own approach, without recognizing what they do. In the end they advocate that their approach is better and that it leads to Einstein-Cartan gravity. However, from what I saw, it is quite equivalent to the standard approach.

    (3) The authors make some explicitly incorrect statements regarding the localization of Poincare symmetry, such as that there is no way to disentangle translations and rotations, criticizing the conventional localization approach. They even cite Kibble and Hehl papers (from 1961 and 1976 respectively) to back that up. However, they seem to be unaware of the more recent literature where it was shown that this disentangling can indeed be done and that there is no problem with the conventional approach.

    I don’t want to sound like a referee here, but IMO the authors have not done their homework, and I don’t trust any conclusions of that paper. Moreover, I don’t see any benefit of introducing the geometric algebra language for this whole ordeal.

    My advice on reading this paper: nothing interesting, move on.

    HTH, :-)

  24. D R Lunsford says:

    Thanks for the book. Fills a gap. Will sit next to Weyl when I have the print edition :)


  25. Mtor says:

    Hello Prof. Woit
    You have a very beautiful book there. With all the theory you could easily add a chapter of “applications ” containing elegant solutions of QM problems using representation theory. I´m thinking, for example, about the Onsanger solution or the Kauffman solution of the 2D Ising model or in general problems that can be stated clearly in a mathematical way and can be solved neatly using representation theory.

  26. lun says:

    OT but IT for this blog, perhaps you can comment on the latest stringy PR fad,
    The hyerarchy between the quality of the evidence (very thin if you read both the original paper by Maldacena and Polchinski and these PRLs) and the “splash” in the academic press is remarkable even by string theory standards

  27. Peter Woit says:

    Thanks! Unfortunately there are many, many such applications, and I want to keep the length of this book to a reasonable size (below 400 pages).

    Arguments about entanglement/holography/black holes seem to be one of the major industries today among particle theorists. None of this seems to address any issues I find interesting so I’m mostly trying to ignore it, and have no intention spending the time necessary to follow these arguments well-enough to comment on them here knowledgeably. I suspect there’s plenty of hype to be deflated there, but someone else will have to do it….

  28. Peter Peterson says:

    Thank you for reviewing the Arxiv post. You are much better qualified than I am to evaluate the merits of the approach, and reading the books I quoted, I noticed a number of ‘leaps of faith’, mostly in Doran’s case. Lasenby, if I am not wrong, cooled down to the idea over time. What was appealing to me though if one could come up with formalism as the authors claimed, where they could operate in flat spacetime it would lead to some new insights.

    However, I agree that yes, Doran and group try to apply the geometric algebra formalism to everything that is already known (but not much, if anything, new) just as another mathematical tool that would lead to the same result in a more straightforward way. Which opens a question that is never answered: what spacetime property makes macroscopic interactions behave to follow this mathematical abstraction. Meaning, if mathematics can indeed be formalized to include all types of interactions, which was not proven yet. The beauty of geometric (Clifford/Grassmann) product application and scalability to any number of dimensions in a way consistent with Lie group symmetries is aesthetically pleasing, if anything else.

  29. Lowell Boggs says:

    Thanks for posting the reference to the Feynman lectures. Your book is beyond my abilities, but you regularly post lots of interesting information that I actually can appreciate and learn from. Thanks!

    Best of luck on the success of the book.