# Quantum Field Theory from a mathematical point of view

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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.

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A great question on a very similar subject over at MO: http://mathoverflow.net/questions/57656/standard-model-of-particle-physics-for-mathematicians

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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://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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answered Oct 9, 2011 by (570 points)
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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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answered Oct 13, 2011 by (415 points)

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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.

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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](http://meta.stackoverflow.com/q/26964/156389) and gets money from any purchases.

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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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answered Oct 9, 2011 by (6,025 points)
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.

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I didn't mean to be polemical at all. Where do you sense polemics?

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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 :-) (http://golem.ph.utexas.edu/category/2009/10/structural_foundations_of_quan.html)

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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".

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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.

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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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answered Oct 9, 2011 by (2,405 points)
Now that Moshe mentions physics.SE, there is the related question http://physics.stackexchange.com/q/6530/2451

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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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answered Oct 9, 2011 by (550 points)
I second the recommendation of Folland's QFT.

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The only book on QFT that I can read without getting frustrated.

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Kevin Costello, a mathematician, has written a book on quantum field theory, particularly the perturbative aspects. The book used to be available from his web page, but it has now been published by the AMS. You can find the links in his webpage to which I linked above.

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answered Oct 9, 2011 by (2,315 points)
This is definitely a very important mathematical work but it is not good as an introduction to quantum field theory - the main purpose of that book is to set up the mathematical language which will enable to talk rigorously about perturbative renormalization, particularly for 4-dimensional gauge theory.

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E. Zeidler is writing a 6-volume treatise under the general title "Quantum Field Theory: A Bridge between Mathematicians and Physicists". So far 3 volume appeared and are available also via springerlink.com (if one has the right subscription). This work is quite accessible for a mathematician.

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answered Oct 11, 2011 by (70 points)
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In addition to the good references given in this post, in particular in the answer by user388027, an excellent introduction to the mathematics of QFT which is truly a textbook (which can for instance serve as support material in a 1st or 2nd year graduate course in mathematics) is "Quantum Mechanics and Quantum Field Theory, A Mathematical Primer" by Jonathan Dimock, Cambridge University Press, 2011. See this link.

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answered Oct 13, 2011 by (640 points)
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This is not an answer, I wanted to post as a comment but probably I don't have enough reputation for that.

I think first you need a good understanding of Classical Mechanics particularly the Lagrangian and Hamiltonian formalism. The best treatment of this in a mathematically rigorous manner is in Arnold's "Mathematical Methods of Classical Mechanics .

Also nobody mentioned that for Supersymmetry (which, as Alexander Braverman rightly said, is most important for mathematical applications), the book by Dan Freed "Five Lectures on Supersymmetry.

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answered Oct 11, 2011 by (270 points)
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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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answered Oct 18, 2011 by (140 points)

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