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I am interested in mathematically rigorous treatment of quantum field theory, constructive QFT in particular.

I have read 'QFT, A Tourist Guide for Mathematicians' and am going to read "Quantum Physics, A functional integral point of view'. But I want something encyclopedic to refer to.

And I have come across the series of books by Zeidler. They are expensive to purchase.. so I ask opinions on the books in terms of the rigor achieved.

Are the books as rigorous as math books and papers? And would they be good reference for reading the book "Quantum Physics, A functional integral point of view'?

I welcome any feedback!!

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    $\begingroup$ I am not a physicist, but QFT is full of situations where one at first gets infinity as an answer, and then does some kind of regularization to get a finite answer. It is in my opinion fascinating, since it agrees with experiments, but not rigorous. There have been attempts to make QFT as rigorous as possible, but I am not sufficiently familiar with that. I know that some axioms have been proposed. Maybe someone else will comment. $\endgroup$ – Malkoun Jul 17 '20 at 7:34
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    $\begingroup$ This isn't really an appropriate question for this site, but having bought these books I'll give you my opinion: they're interesting if you want to learn a lot of things on various subjects, but if you want to get a clear picture of the mathematics of QFT, they're very disappointing. They read more like a “brain dump” of the author's mathematical culture (which is indeed far-reaching) than like a focused story. $\endgroup$ – Gro-Tsen Jul 17 '20 at 9:34
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    $\begingroup$ Now if only there were some underground web site (named, maybe, shall we say, like a place where books are stored, followed by the title of the first book of the Bible) whence one could download just about any science book whatsoever, I could advise you to check it out and build your own opinion before buying; but, of course, the implacable Laws of copyright, which we all follow because the Law is always Right, prohibit the crime of freely disseminating information, so I would never dream of telling you to check such a site (even if it existed, which it surely does not!). $\endgroup$ – Gro-Tsen Jul 17 '20 at 9:42
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    $\begingroup$ Also imagine if there were a website where one can download almost any paper on mathematics or science for free, perhaps the first three letters could be evocative of 'science' and the last three letters of the name of the website could be the same as a well-known space telescope. Needless to say, this website also definitely does not exist and I would be against you using it even if it were. $\endgroup$ – Hollis Williams Jul 17 '20 at 10:05
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    $\begingroup$ I am sure there are lots of interesting things to learn in the multivolume book by Zeidler, but if you want to learn about "constructive QFT, in particular" this is not a helpful reference. If you want to learn about constructive QFT, given the difficulty of reading research articles in the area on your own, I think it would be best to ask specific/focused questions about constructive QFT here on MO. There are experts who can try to answer. $\endgroup$ – Abdelmalek Abdesselam Jul 17 '20 at 17:01
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In general reading entire textbooks in QFT is not the best way to learn QFT (or any subject in science and mathematics). I have not heard of the books, but my guess is that they are not going to be as rigorous as math books and papers as there is not a mathematically rigorous formulation of QFT (I'm sure you are aware that the basic problem of Yang-Mills existence and the mass gap is a Clay Institute Millennium Problem).

Also to be honest it has been a while since anyone has done anything in algebraic or axiomatic QFT which the wider community of field theorists really cares about. This is my personal opinion on the matter, but one which is also expressed by some other theoretical physicists (see for example Modern Quantum Field Theory by Tom Banks).

Edit: If I remember rightly, you can find Steven Weinberg's lecture where he reflects on the fiftieth anniversary of his classic paper 'A Model of Leptons' and at the end he expresses similar sentiments about a mathematically rigorous formulation of QFT (ie. that it is important and desirable, but that he does not particularly care as such a formulation is unlikely to inspire any actual new physics).

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    $\begingroup$ Sorry but I believe it is necessary to add to your last sentence that it expresses your personal opinion. $\endgroup$ – მამუკა ჯიბლაძე Jul 17 '20 at 9:47
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    $\begingroup$ Added as requested. $\endgroup$ – Hollis Williams Jul 17 '20 at 10:00
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    $\begingroup$ I am even more sorry but how about adding "some" before other physicists? Do you know for sure, say, opinion of Witten on the recent work of Costello? $\endgroup$ – მამუკა ჯიბლაძე Jul 17 '20 at 10:13
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    $\begingroup$ Many thanks! I am quiet now :) $\endgroup$ – მამუკა ჯიბლაძე Jul 17 '20 at 10:21
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    $\begingroup$ For what it's worth, Axiomatic QFT and Algebraic QFT refer to very specific approaches rather than the general approach of doing things rigorously. The axiomatizations of TQFT via categories and subsequently $\infty$-categories, has been very important, but people wouldn't generally refer to that as "axiomatic quantum field theory" due to the history. $\endgroup$ – Aaron Bergman Jul 17 '20 at 21:48
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In my opinion, Zeidler's books are good for getting a detailed overview of a lot of the different mathematical ideas and concepts undergirding modern efforts in QFT, if one already knows a fair bit of QFT. They are (again in my opinion) no good at all for learning QFT.

One major problem with the books is that they are often rather disorganized (the same concept is discussed twice in different chapters, different notation for the same thing is introduced in different places, etc.). The other major problem is that there are a lot of errors that indicate that proofreading and copyediting did not take place at all (e.g. some of the Feynman diagrams for QED are hilariously wrong). There are also lots of more minor problems. So unless you have a good idea of what precisely you want to learn, I'd advise you to stay away from these books.

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