This Time It’s For Real

Several months ago I was advertising a “Final draft version” of the book I’ve been working on forever. A month or two after that though, I realized that I could do a more careful job with some of the quantum field theory material, bringing it in line with some standard rigorous treatments (this is all free quantum fields). So, I’ve been working on that for the past few months, today finally got to the end of the process of revising and improving things. My spring break starts today, and I’ll be spending most of it in LA and Death Valley on vacation, blogging should be light to non-existent.

Another big improvement is that there are now some very well executed illustrations, the product of work in TikZ by Ben Dribus.

I’m quite happy with how much of the book has turned out, and would like to think that it contains a significant amount of material not readily available elsewhere, as well as a more coherent picture of the subject and its relationship to mathematics than usual. By the way, while finishing work on the chapter about quantization of relativistic scalar fields, I noticed that Jacques Distler has a very nice new discussion on his blog of the single-particle theory.

There’s a chance I might still make some more last-minute changes/additions, but the current version has no mistakes I’m aware of. Any suggestions for improvements/corrections are very welcome. Springer will be publishing the book at some point, but something like the current version available now will always remain available on my website.

Update: After writing to someone to answer a question and what is and isn’t in the book, and other things to read, I thought maybe I’d post here part of that answer:

For the main topics about QM and representation theory that I cover in the book, I don’t know of a better reference, even assuming an excellent math background. That’s one of the main reasons I wrote the thing… The problem with other books on QM for mathematicians (e.g. Hall, which is very good on the analysis point of view) is that they don’t do much from the representation theory point of view. Weyl’s book was written very early, when a lot of what was going on was still unclear, I don’t think it’s a very good place to try and learn this material from. One topic that is in there that I don’t cover at all is basically Schur-Weyl duality, but even for that arguably Weyl is not a good place to learn that theory.

One thing to keep in mind about my book is that the early chapters are deceptive. I wanted to start out with very simple things, make the simplest examples accessible to as many people as possible, mathematicians or physicists. If you know basic facts about Lie groups, Lie algebras, finite dim unitary representations, Fourier analysis and how to use it to solve e.g. the heat equation, then the first quarter of the book is only going to be of interest in telling you about some applications of math you know. Mathematicians generally should be learning the basic rep theory elsewhere (lots of good books on these topics, and the main reason I’m doing many things in a mathematically sketchy way is that doing them in full would take too long, and has been done better elsewhere). In early chapters, all I’m really doing is working out very special cases of Lie groups/algebras that are rank one or products of rank one, and the irreps of sl(2,C). I never touch higher rank or general semisimple theory (would argue this actually doesn’t get used much in physics, other than some simple SU(3) examples).

Around chapter 12 though, things get much more non-trivial. From a high mathematical level, a lot of what’s going on in the middle part of the book is the representation theory of the Heisenberg group (over R and C) and the implications of the action by the symplectic group by automorphisms (e.g. the metaplectic representation). This is done in a very detailed and concrete way, together with the relation to QM, although for some of the trickier parts of the mathematics (especially the analysis, e.g. the proof of the Stone-von Neumann theorem) I just give references. This is followed by discussing Clifford algebras, the orthogonal group and spinors (over R and C), in a very parallel way (interchanging symmetric and antisymmetric). I wish I knew of a good pure mathematics source for this material aimed at students, stripped of the quantum mechanics apparatus, but I don’t. It (as well as material about reps of the Euclidean groups) is not covered in any conventional rep theory textbook I’m aware of.

Much of the last third of the book, on quantum field theory, I think is just inherently quite challenging, for either mathematicians or physicists. From the representation theory point of view, the basic framework is that of an infinite dimensional Heisenberg group or Clifford algebra, but this is a difficult mathematical subject, and I think the physics point of view helps make clear why. For this stuff the rigorous treatments are quite specialized, I try and do some justice to what the main issues are and give references that provide the details.

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37 Responses to This Time It’s For Real

  1. CIP says:

    I looked at a couple of sections dealing with issues that I’ve previously found difficult and was impressed by your directness and clarity. Looks like a winner to me!

  2. RP says:

    I think the book looks very, very impressive. I look forward to delving into it more deeply!

    One typographical pet peeve of mine, which I also notice in the current draft: ‘digressions’ and ‘examples’ are printed in cursive. This is distracting to me, since I (and most readers I think) tend to interpret cursive text as a form of emphasis. This is certainly appropriate for theorems and propositions, and perhaps for definitions, but clearly not for something like a digression.

    Anyway, many congratulations on your achievement!

  3. G. S. says:

    I’m not exactly sure how copyright works, but how can something be copyrighted 2016 but dated 2017? I get that portions of the document were previously released and copyrighted, but wouldn’t the addition of new material extend the copyright of the document as a whole?

    Not sure if you want to get into a discussion of copyright law right before your trip to Death Valley, but I was curious. The book looks interesting and I’m looking forward to reading it. Thank you for posting it. Enjoy your break.

  4. Anonyrat says:

    Page 4: After calculating a final result, insert appropriate factors of ℏ can be inserted to get answers in more conventional unit systems

  5. Anonymous says:

    I’m really happy for you. What a magnificent piece of work.
    I have one little nitpick about the usage of \left and \right.
    This image explains it — http://1.1m.yt/S2j8hi.png
    Not really an issue, because both have the same meaning. But I think you will agree the usage of \left( and \right) are prettier.

    Congratulations.
    This book is a milestone.

  6. Congratulations from me too Peter! This is a great work and I’m very happy you finally pulled it off!

  7. a reader says:

    Congratulations !
    And thank you for making the final draft freely available.

  8. vmarko says:

    This kind of book has been missing in the landscape of QM textbooks. I’ve already recommended it to a few mathematicians who wanted to learn QM efficiently. Great work, congratulations!

    🙂
    Marko

  9. Just started giving it a quick read. Found this on page four:

    “After calculating a final result, insert appropriate factors of (h-bar) can be inserted to get answers in more conventional unit systems.”

    I’ve written a few EE texts and I know it’s nearly impossible to catch all of these minor things no matter how many times you read it yourself. Will note any other issues I happen to spot, but it looks great overall.

  10. DRLunsford says:

    I like it. When I get the hard copy it will sit next to Weyl 🙂

    -drl

  11. Anonymous says:

    I was curious to see the Tikz pictures and noticed one minor error in one of the captions. The caption of Figure 8.2 says “Cylindrical Coordinates”. This should be “Spherical Coordinates”. I also agree with the comment about the (lack of) use of \left and \right, especially when taking Lie brackets of items which involve larger delimiters, it looks much better if the outside ones are at least the same size as the inside ones. My guess is that an editor from Springer would make those changes anyway.

  12. Richard Ferguson says:

    Great news, Peter! Your book looks great, nothing really to nitpick on the presentation of the material. It’s a very interesting and novel way to go about introducing QFT.

    By the way, now that you’re finished the book, do you have any research projects in Langlands or such that you’re working on? Any plans for publishing papers? I would definitely be interested in reading them.

  13. DRLunsford says:

    Peter, don’t know if you’ve ever been, but the journey up to Palomar Observatory from LA south toward San Diego is a great drive with a better destination. It’s a strange feeling to be sitting there in the presence of the building surrounded by scrub and forest, but mindful of the Virgo Cluster..

    -drl

  14. Peter Woit says:

    Thanks to all who have pointed out typos/typographical improvements. I’ve started fixing/implementing them, in particular glad to have pointed out the issue about delimiters, where I’m for now fixing the most egregious cases.

    Any help of this kind is greatly appreciated, you can send me comments here or by email. If you’re up for the more advanced topics of the book, you’re encouraged to start looking at those, since I’m getting a lot more help with the early chapters than with the later ones. It’s expected that many readers will get lost before they get to the end, but I hope some will make it all the way…

  15. Peter Woit says:

    DRLunsford,
    I made a trip up to Palomar once, quite a long time ago and it was impressive. This trip may include a stop by Anza Borrego, which supposedly is in the midst of an unusual wildflower bloom. Palomar is nearby, maybe another visit there is indicated.

    Richard Ferguson,
    Definitely several research projects I now want to get to work on. The first is something I’ve wondered about for years, but working on this book has finally given me what looks like a way to finally get somewhere with it. We’ll see soon. Another project is to finish something I started writing about Dirac cohomology and BRST. I will try and get something written and done about both of these first, since I now in principle have the background I need for them. For anything about number theory, first I have much more to learn…

  16. Martin says:

    With respect to paragraph 3.4 an interesting reference would be the Jaynes-Cummings model, dating from 1963, which features a two-level atom in a cavity and interacting with a single bosonic radiation mode.

  17. theoreticalminimum says:

    Peter,
    Thanks very much for sharing a finalised version! I sure hope to find the time to read through, and learn.
    In November last year, I noticed a book by Hayashi titled “Group Representation for Quantum Theory” published by Springer. Did you know about it, and would you recommend it?

  18. Cobi says:

    On page 573 you write that “The Hamiltonian with Einstein’s equations as equations of
    motion is however not of the Yang-Mills form. Applying standard perturbation
    theory and renormalization methods to this Hamiltonian leads to problems with
    defining the theory (it is not asymptotically free)”.
    Two things struck me about this and i hope you can clarify.
    First of all the notes by Schwartz (http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf, p. 245) explain that general relativity leads to a perfectly sensible quantum field theory, at least as sensible as the standard model without gr.
    On the other hand, even without coupling to general relativity the standard model is not asymptotically free due to the U(1) factor.
    Thus “it is not asymptotically free” can not be the crucial obstruction.

  19. Peter Woit says:

    Cobi,
    The topics you bring up have nothing at all to do with what I wrote, which is just pointing out that the pure Einstein theory Hamiltonian has a very different form than the pure Yang-Mills Hamiltonian, not sharing its good UV properties.

    If you’re looking for the usual tedious debates about quantum gravity, you really need to look elsewhere, that’s not what this book is about, it’s intent is to explain some very different things.

  20. R. Rosenfelder says:

    Dear Peter Woit,

    I was surprised that you do not mention the Maslov “correction” when treating the time-dependent harmonic-oscillator propagator, i. e. the phase, which arises when the particle goes through a focal point.

    Thus your eq. (23.12) in chapter 23 on page 271 is only valid for T < omega/pi; otherwise the propagator gets an additional phase.

    This is covered in the standard textbook by H. Kleinert: "Path Itegrals in…", 3rd edition, p. 100 or in arXiv:1209.1315. Since the latter source may be not easily accessible to you, I recommend P. A. Horvathy's paper in quant-ph/0702236 as a detailed treatment of this caustic phenomenon.

    Best regards

    Roland Rosenfelder

  21. Peter Orland says:

    Hi Peter,

    A very good beginner’s treatment of the Maslov index from conjugate points is in Larry Schulman’s book on path integrals (from 1981). He also has a nice elementary discussion of how a LEAST-action/arc-length principlea become an EXTREMAL principle, after passing a conjugate point.

    You have plenty of other topics treated already, so this is something you could mention, without reviewing it.

  22. Peter Orland says:

    Hi Peter,

    I just took a look at your discussion. I think you can forget about the Maslov index.

    The result (23.12) is exact (the Maslov index is hidden in the answer), for the problem you consider (the simple harmonic oscillator). There is probably no need to mention Morse theory or the Maslov index to your intended audience, unless you study more general physical systems.

  23. Peter Orland says:

    Scratch that… the Maslov index is definitely there in the SHO. I just checked it.

  24. Peter Orland says:

    … by which I mean it has to be inserted into (23.12) past the first conjugate point.

  25. Peter Woit says:

    theoreticalminimum,
    I hadn’t seen that. From the table of contents it looks like there’s some overlap with what I’ve written, but the books are mostly pretty different. I’m traveling now, when I get back will take a closer look.

  26. Peter Woit says:

    Roland Rosenfelder/Peter Orland,

    Thanks for pointing this out! When I get back to New York soon I’ll think a bit about how much I can sensibly say about this in the book, but the existence of this phenomenon certainly needs to be mentioned.

  27. Maximillian G. Tresmond, Esq. says:

    Congratulations! What a great accomplishment!

  28. Austin says:

    At a first glance this is an impressive display of author’s scholarship. The writing is clear. However, I don’t know what is the purpose of a book of this size? Is is supposed to be used for a semester long material? The book seems diffuse on what chapters are considered important knowledge for the subject. What is the idea behind chapters that are just a couple of pages long?

  29. Peter Woit says:

    Austin,

    The book roughly corresponds to a year-long course I’ve taught a couple times, assuming roughly a typical advanced undergraduate in math background. The later parts are much more challenging than the earlier ones. The size of chapters very roughly corresponds to a lecture (the class meant twice a week).

    The goal of the book was to get to the point of explaining the basics of the quantum field theory ideas that make up the Standard Model, with the constraint of dealing just with free fields, not interactions, all in a way that emphasizes symmetries and how to think of quantum theory in terms of representation theory. There’s a lot of material in early parts of the book which is not normally covered in courses at that level, but it’s there because it’s the simplest example of ideas and calculations which normally only are first explained in the much more complicated context of relativistic quantum field theory. I hope that this material will be useful as background for anyone studying relativistic QFT.

  30. Anon says:

    Hi Peter,

    This is off topic but looks like Jester has won his SUSY bet with Lubos? CMS and Atlas reported their 36 fb-inv search results at Moriond 2017.

    Surprised to see that blogs havent picked this up yet.

    https://indico.in2p3.fr/event/13763/other-view?view=standard

    cornering natural susy with 13 TeV… https://indico.in2p3.fr/event/13763/session/4/contribution/89/material/slides/0.pdf

  31. Shantanu says:

    OT : Peter, very curious to know your take on https://arxiv.org/abs/1703.05331
    It says that no BSM physics below QG scale will be seen as pulsar timings severely constrains such bounds.

  32. Peter Woit says:

    Shantanu,

    That claim seems highly implausible. They’re making an extraordinary claim, that we have experimental access to just about any physics at arbitrarily short distance scales. That would be wonderful, but I don’t see any extraordinary evidence so will wait until someone with more time on their hands figures out whether they really have something.

  33. Peter Woit says:

    Anon,

    Will write very soon about the new LHC results.

  34. Peter Woit says:

    Roland Rosenfelder,

    Now back from vacation and one thing I’ve gotten done is to add something about the issue you mentioned. From my point of view the source of this choice of phase factor is determined by analytic continuation from, so I’ve referred to a source that discusses it in that way. Thanks for pointing this out!

  35. Anonymous says:

    Why keep an empty Acknowledgements section?
    I would remove it.

  36. Peter Woit says:

    Anonymous,

    That will be the last thing to be added. I’m still getting help that I’d like to acknowledge, some from this blog…

  37. Georges E. says:

    Peter Woit, thanks for this wonderful book: so clear, and yet it doesn’t leave anything out! A tremendous achievement.
    Congrats!

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