Quantum Field Theory from a mathematical point of view

I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view.

Are there any good books or other reference material which can help in learning about quantum field theory? What areas of mathematics should I be familiar with before reading about Quantum field theory?

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Before studying QFT itself, I would recommend at the very least getting comfortable with special relativity and quantum mechanics. Being a student of mathematics myself, I understand how frustrating it can be to learn physics from a physicist, but at the end of the day, it will make learning QFT (or any subject of physics for that matter) much easier if you understand the physical meaning of the subject and why you are doing what you are doing. In any case, it will certainly improve your appreciation of the subject. – Jonathan Gleason Oct 10 '11 at 14:46
A great question on a very similar subject over at MO: mathoverflow.net/questions/57656/… – Igor Khavkine Oct 10 '11 at 18:23

Let me add just a couple of things to what was already mentioned. I do think that the best source for QFT for mathematicians is the the two IAS volumes. But since those are fairly long and some parts are not easy for mathematicians (I participated a little in writing those down, and I know that largely it was written by people who at the time didn't understand well what they were writing about), so if you really want to understand the subject in the mathematical way, I would suggest the following order:

1) Make sure you understand quantum mechanics well (there are many mathematical introductions to quantum mechanics; the one I particularly like is the book by Faddeev and Yakubovsky http://www.amazon.com/Lectures-Mechanics-Mathematics-Students-Mathematical/dp/082184699X)

2) Get some understanding what quantum field theory (mathematically) is about. The source which I like here is the Wightman axioms (as something you might wish for in QFT, but which almost never holds) as presented in the 2nd volume of the book by Reed and Simon on functional analysis; for a little bit more thorough discussion look at Kazhdan's lectures in the IAS volumes.

3) Understand how 2-dimensional conformal field theory works. If you want a more elementary and more analytic (and more "physical") introduction - look at Gawedzki's lectures in the IAS volumes. If you want something more algebraic, look at Gaitsgory's notes in the same place.

4) Study perturbative QFT (Feynmann diagrams): this is well-covered in IAS volumes (for a mathematician; a physicist would need a lot more practice than what is done there), but on the spot I don't remember exactly where (but should be easy to find).

5) Try to understand how super-symmetric quantum field theories work. This subject is the hardest for mathematicians but it is also the source of most applications to mathematics. This is discussed in Witten's lectures in the 2nd IAS volume (there are about 20 of those, I think) and this is really not easy - for example it requires good working knowledge of some aspects of super-differential geometry (also disccussed there), which is a purely mathematical subject but there are very few mathematicians who know it.

There are not many mathematicians who went through all of this, but if you really want to be able to talk to physicists, I think something like the above scheme is necessary (by the way: I didn't include string theory in my list - this is an extra subject; there is a good introduction to it in D'Hoker's lectures in the IAS volumes).

Edit: In addition, if you want a purely mathematical introduction to Topological Field Theory, then you can read Segal's notes http://web.archive.org/web/20000901075112/http://www.cgtp.duke.edu/ITP99/segal/; this is a very accessible (and pleasant) reading! A modern (and technically much harder) mathematical approach to the same subject is developed by Jacob Lurie http://www.math.harvard.edu/~lurie/papers/cobordism.pdf (there is no physical motivation in that paper, but mathematically this is probably the right way to think about topological field theories).

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If you're a mathematician and you want to understand QFT, you're going to have to grapple with renormalization sooner or later. Your life will be easier if you understand from the beginning that the Wilson-Weinberg-etc 'effective field theory' philosophy is the essential organizing principle for the whole subject. In particular, you're going to need to know it to have any hope of understanding the intuition behind the existing rigorous constructions of QFTs. Unfortunately, the explanations of renormalization in the particle physics-oriented textbooks which mathematicians often consult first are not so great.

Maybe I can provide a little motivation, before adding to the list of recommended reading.

In a system with infinitely many degrees of freedom (such as field theory on a spacetime of dimension at least 2), you have to organize the degrees of freedom somehow, before you can even begin to talk about how they interact. In QFT, we frequently organize the degrees of freedom by asking how big they are, in comparison to some fixed distance scale. (The Fourier decomposition of the electromagnetic field is an example of this. We think of the electromagnetic field as a sum of sin/cos waves of various wavelengths.) So when we talk about a field theory, what we really have in mind is a sequence of approximations, which begins with a set of degrees of freedom whose characteristic scale is comparable to the reference scale and then systematically adds in new ones whose characteristic scales are further from our reference scale.

The basic idea of the effective field theory philosophy is that, instead of thinking of the degrees of freedom we use near the reference scale as being those that remain when we toss out all the other ones, we should think of these degrees of freedom as being an approximate 'effective' description of the system we obtain by averaging out those other degrees of freedom. If you take this point of view, you'll frequently find that the degrees of freedom at the reference scale do resemble the ones we would have gotten by blindly ignoring the shorter distance degrees of freedom, and their interactions have the same basic form, except that the coupling constants are all different. The renormalization procedure that shows up all over QFT is concerned with computing how interactions between the degrees of freedom at the reference scale are determined in terms of the interactions between the degrees of freedom appropriate to shorter distance, in particular with figuring out which interactions get stronger and which weaker.

This philsophy has its origins in statistical mechanics, the oft-neglected third leg of the QFT stool. (The path integral of QFT is closely related to the partition function computations which show up in the statistical mechanics of field systems.) If you want to understand QFT, you have to study QM, relativity, and stat mech. The stat mech isn't really optional.

A few references:

• Tim Hollowood's "Cutoffs & Continuum Limits: A Wilsonian Approach to Field Theory" is an excellent introduction.

• Kerson Huang's Statistical Mechanics has a nice treatment of the Ising model, which is pretty much the ur-example of the subject.

• Zinn-Justin's QFT & Critical Phenomena works through these ideas in a huge amount of detail.

• David Brydges "Lectures on the Renormalization Group" in the IAS/Park City volume Statistical Mechanics is pretty great.

• Battle's "Wavelets & Renormalization" does a thorough and mathematically rigorous treatment of the Euclidean path integral for 3d scalar field theory, very much in the spirit of the renormalization philosophy.

• Glimm & Jaffe's "Quantum Physics: A Functional Integral Point of View" explains a lot of the mathematical machinery like nuclear spaces and cylinder measures which can be used to make the effective field theory idea mathematically precise, and uses this machinery to constructive 2d scalar field theories and prove some non-trivial facts about them.

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Can you, where possible, linkify your references? – Simon Oct 25 '11 at 21:51
Hollowood's notes were online, but seem to have disappeared. The others are all published books. I've seen Huang, Zinn-Justin, & Glimm & Jaffe online, but I don't think stack exchange wants its users linking to dubious websites. – user1504 Oct 25 '11 at 22:12
Yeah, it was mainly Hollowood's notes that I was looking for. As for books, I've seen both Amazon and Google books links on other stackexchange sites... In fact, stackexchange automatically converts Amazon links to affiliate links and gets money from any purchases. – Simon Oct 25 '11 at 22:27

There are two aspects to this question:

1) Which sources try to communicate the usual vague and speculative physics story in a way that mathematicians are more likely to appreciate?

2) Which sources try to give an actual mathematical treatment of QFT, something that lives up to being maths?

For the first, Deligne et al's Quantum Fields and Strings is probably the best answer existing to date.

But there is a lot to be said about the second question, too. Much progress has been made here in the last few years. This December (2011) an AMS volume appears that collects surveys and original articles on this topic:

Sati, Schreiber (eds.) Mathematical Foundations of Quantum Field Theory and Perturbative String Theory AMS (2011) Proceedings of Symposia in Pure Mathematics, Volume: 83.

The introduction with more links is at arXiv:1109.0955

[Edit: in view of the discussion below, I should say that I don't mean "vague and speculative" in a pejorative way at all. It's just a fact that from the point of view of mathematics, much of physics, certainly much of quantum field theory and string theory, well established and robust as it may be, is vague and speculative. To get a sense of the truth of this it may help to go to a pure mathematician who is interested in learning about the subject but has no background in it and try to teach him or her. One learns from this that many texts written by physicists that claim to be "for mathematicians" are in fact not. There is quite a distance between a mathematically aware theoretical physicists and a pure mathematician without background in the usual physics lore. Many physicists are not aware of this distance.]

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I feel that this is a good answer which is somewhat compromised by the polemical tone. There's certainly few independent things the question may refer to, perhaps the dychotomy is between foundations and applications of QFT. Both subjects can (but don't have to be) useful and interesting. – user566 Oct 10 '11 at 2:28
I didn't mean to be polemical at all. Where do you sense polemics? – Urs Schreiber Oct 10 '11 at 6:19
If I had to guess I'd say that "vague and speculative" may be considered to be "polemical". I remember writing something along these lines two years ago at the nCafe :-) (golem.ph.utexas.edu/category/2009/10/…) – Tim van Beek Oct 10 '11 at 8:33
Thanks. I have added a comment to the bottom of my answer above in an attempt to clarify this. I think it is important to realize that much of physics, even the most established theories, is "vague and speculative" from the point of view of actual mathematics, of actual precision of argument and certainty of truth. This is not at all to say that this physics is bad. But realizing this gap to the non-vagueness and non-speculation of maths is the necessary first step for appreciating what it means -- or would mean -- to genuinely have "QFT from a mathematical point of view". – Urs Schreiber Oct 10 '11 at 9:04
Thanks for the clarification. I'd probably prefer heuristic to vague and speculative, which implies degree of uncertainty that is not appropriate. I also think the two efforts are not distinct - if you want to make the heuristic structure of QFT (which nonetheless is efficient in producing true mathematical statements) more precise, perhaps the first logical step is learning what it is. – user566 Oct 10 '11 at 11:52

Moshe has already addressed many points. You might be interested in Folland's Quantum Field Theory: a tourist guide for mathematicians. He tries to do as much things as possible in a mathematically rigorous fashion, and points out those points where this cannot be done.

As for the mathematical background: some familiarity with partial differential equations and the theory of distributions will be convenient.

This concerns "conventional" quantum field theory. You might also be interested in topological quantum field theory which is much more of a mathematical nature.

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I second the recommendation of Folland's QFT. – Jonathan Gleason Oct 10 '11 at 14:39
The only book on QFT that I can read without getting frustrated. – MBN Oct 13 '11 at 10:11

QFT is a huge subject, underlying much of modern theoretical physics. I think by and large the Mathematics community has been interested in special simple cases (e.g. topological or rational QFT), so the standard caveat about the proverbial elephant is very much relevant here.

One good survey is the one year course given in the IAS for mathematicians, which covers a lot of ground. There is a two-volume book which is useful not only for mathematicians, and a website: http://www.math.ias.edu/qft. This will give you a survey of the central topics, and (depending specifically what you are interested in), the background needed.

As for attempts to formalize general QFT, there are many. Since in the modern (post-Wlison) treatment of the subject, the defining properties of QFT all have to do with the process of renormalization, I have asked a question to that effect here http://physics.stackexchange.com/questions/4068/formalizing-quantum-field-theory, the answers may give you a flavor of what is out there on that front.

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If you're looking for something easier and more pedagogical, then you should take a look at the wonderful book by Baez and Muniain called Gauge Fields, Knots and Gravity. This book develops the mathematical formalism of gauge theory in a friendly and entertaining way, and it requires very little background to read. If you want to learn about the physical aspects of quantum field theory, you might want to look elsewhere, but this book gives a completely self contained mathematical introduction to Chern-Simons theory, a quantum field theory with important applications in pure mathematics.

Another very friendly book on quantum field theory for mathematicians is Frobenius Algebras and 2D Topological Quantum Field Theories by J. Kock. This is a great place to start if you want to study the recent work of Jacob Lurie on the classification of topological quantum field theories. The only problem with this book is that it doesn't say much about how quantum field theories are used to compute invariants of topological spaces. I therefore think it's best to supplement this book with something else -- perhaps the classic paper of Atiyah.

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This was meant to be a comment, not an answer, but I don´t have enough reputation. Basically, I did a masters in maths (pure maths), then a masters is physics (qft), then a phd in maths (pure, algebraic geometry stuff). So, I had to grapple with the issue you are trying to solve. I think it will be hard to get a good answer since you don´t specify for what reason you want to learn QFT. Some comments then:

If you are going to work on things like Seiberg-Witten equations from a math perspective, then I suppose the book of Baez and Muniain called Gauge Fields, Knots and Gravity (mentioned by Bob Jones above) is great since you will not need to quantize things anyways.

If you actually want to get an understanding of the subject that includes the physics perspective (which is what I tried to do), then I suggest developing some physics background. So, I suggest reading the book of Sakurai in quantum mechanics (which, from my pure math background of the time, was a good book), together with books that are for the laymen: Feynman´s QED and Weinberg´s The discovery of subatomic particles. I used these books with Peskin and Schroeder´s An Introduction To Quantum Field Theory.

Actually, I tried to follow at the same time a more "mathematically precise" approach to QFT - but in the end I thought this was harder than the physics approach - because, I think, you end up spending an enourmous amount of time to get anywere, and risk the change of being burried in a pile of math formalism before being able to do simple computations.

A last comment. In my experience, it was great to talk to physicists (they tend to be more chatty and tell more stories about their subject than mathematicians). So, I believe that it is highly profitable to hang out around a group of physics students/professors while studying qft.

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I forgot to mention Landsman´s book Mathematical topics between classical and quantum mechanics, which furnishs a good complement on the mathematical side of the more physics side approach to qft I mentioned above. link – JJ O Oct 26 '11 at 11:09

In addition to these great answers, I would like to recommend the books,

1. A Mathematical Introduction to Conformal Field Theory by M. Schottenloher
2. Supersymmetry for Mathematicians: An Introduction by V. S. Varadarajan
3. Mirror Symmetry by C. Vafa, E. Zaslow, et. al
4. Some Physics for Mathematicians by L. Gross

The first book develops some of the analysis necessary for CFTs (Chapter 8) as well as the theory of conformal compactifications (Chapters 1, 2) and the theory of the Witt and Virosoro Algebras (Chapters 4-6). The book ends with a discussion of the fusion rules and how to formally construct a CFT (starting from something analogous to the Wightman axioms). I believe that Schottenloher is an analyst, so you can get a more analytical feel [read: know some functional analysis and basic representation theory] from this book.

The second book is written from the perspective of someone who is a functional analyst with a heavy representation theory background. The first two chapters give a decent mathematical introduction to QFT as well as some of the more representation theoretic results that one might find interesting. The author also introduces some of the algebraic geometry that one might find in a formal analysis of QFT (which of course is elucidated in its full glory in Quantum Fields and Strings.

The third book is from a summer school for both mathematics and physics graduate students. As such, it introduces a large variety of topics and does provide a somewhat formal introduction to QFT.

Finally, the lecture notes from Leonard Gross's class on Quantum Field Theory is a good formal introduction for mathematicians with an a) analysis background and b) no physics greater than classical mechanics. It is a easy to read set of notes with good historical references. While I studied both physics and mathematics, I found these notes to be my favorite reference for QFT (perhaps because I prefer analysis and differential geometry to algebra and algebraic geometry).

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As an amateur mathematician, I found Franz Mandl & Graham Shaw's Quantum Field Theory a quick and concise introduction. However, one will need to have covered some Quantum Mechanics previously. The book was originally recommended to me.

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A good introduction is "Quantum Field Theory for Mathematicians" by Ticciati. It's great in the sense that it is quite rigorous and self-contained, and yet quite broad in its presentation.

A bit more engaged and lengthy presentation with specific topics is "Quantum Fields and Strings: A Course for Mathematicians". This is a 2-volume set filled with lectures by people in the field. Quite technical though.

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I have to say I find Tricciati style very different from what I expect from a book for mathematicians. – MBN Oct 13 '11 at 10:09

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