Quantum Field Theory from a mathematical point of view

Question

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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Comments

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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Answer 1

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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Answer 2

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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Comments

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](http://www.springer.com/physics/theoretical%2C+mathematical+%26+computational+physics/book/978-0-387-98318-9)

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Answer 3

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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Answer 4

Along with the above answers, you might also find these links helpful:

• Formalizing Quantum Field Theory;

• Rigor in quantum field theory;

• CFTs and formalizing quantum field theory

For relevant textbooks, in addition to G. Folland's title, some others are:

• Quantum Mechanics and Quantum Field Theory: A Mathematical Primer

• Mathematical Aspects of Quantum Field Theory (Cambridge Studies in Advanced Mathematics

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Additionally, the following is an excellent "roadmap" / reading resource to modern physics:

The Road to Reality: a complete guide to the laws of the universe

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Answer 5

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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Comments

I have to say I find Tricciati style very different from what I expect from a book for mathematicians.

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Answer 6

QFT - By Prof. C N Yang & Prof. Kenneth Young

Thematic Melodies of 20th Century Theoretical Physics

Here is the youtube version.

http://www.youtube.com/view_play_list?p=D1044B877283883E

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Answer 7

QFT - By Sidney Coleman

http://www.physics.harvard.edu/about/Phys253.html

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Comments

Care to explain *why* you feel this is helpful? Goes a lot further than just a title and a link...

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The LaTeXed notes for this course were just recently posted on the arXiv: http://arxiv.org/abs/1110.5013

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Answer 8

QFT - By David Tong

http://pirsa.org/09090108/

QFT in a Nutshell - Zee

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Comments

While these are very good Introductions, they do not meet the requirements of this question. Especially Zee requires a "physics" mindset to understand it properly.

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