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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