Self-learning QFT for mathematicians

[joseph0086]

Hi all,

I am pretty sure this post is repetitive. But I am asking from a mathematician's point of view. I am currently a final-year grad student in a respectable university. I will get my phD next summer. My research interest is mainly in symplectic geometry and some operator algebras.

I want to learn some QFT since my work seems to be very intimately related to some original ideas in QFT.

When I was undergrad, on the physics side, I have learned classical mechanics (symmetry, lagrangians, Hamiltonians), a little bit of classical EM and special relativity(by self-study, must have forgotten some key points). I tried to read QM and general relativity but I could not quite understand at that time. So I hope to come back to these topics too, if I have enough time. On the math side, I am comfortable with differential geometry, algebraic topology, functional analysis, ODEs, dynamical systems. So I believe my math training is enough. It's just that I want the physical ideas from the physicists.

I am 26 and I am worrying that it's too late for me to read these physics stuffs. Or maybe I am too stupid to understand QFT. I am not sure but I just want to give it a try.

What suggestions can you guys provide me?

 

[joseph0086]

Hi all,

I am pretty sure this post is repetitive. But I am asking from a mathematician's point of view. I am currently a final-year grad student in a respectable university. I will get my phD next summer. My research interest is mainly in symplectic geometry and some operator algebras.

I want to learn some QFT since my work seems to be very intimately related to some original ideas in QFT.

When I was undergrad, on the physics side, I have learned classical mechanics (symmetry, lagrangians, Hamiltonians), a little bit of classical EM and special relativity(by self-study, must have forgotten some key points). I tried to read QM and general relativity but I could not quite understand at that time. So I hope to come back to these topics too, if I have enough time. On the math side, I am comfortable with differential geometry, algebraic topology, functional analysis, ODEs, dynamical systems. So I believe my math training is enough. It's just that I want the physical ideas from the physicists.

I am 26 and I am worrying that it's too late for me to read these physics stuffs. Or maybe I am too stupid to understand QFT. I am not sure but I just want to give it a try.

What suggestions can you guys provide me?

I am also quite familiar with Lie groups and representation theory.

Can you guys give me some suggestions on how to learn QFT by self-learning? Which book is good? etc..

 

[Kevin_Axion]

Here are video lectures on the subject provided by PIRSA at the Perimeter Institute: http://pirsa.org/C09021/5. There are also updated versions: http://pirsa.org/C10017/7.

 

[Fredrik]

You might want to start with a standard intro to QM, like Griffiths. You might actually be able to skip that, and instead use Isham to learn or relearn what you need to know about QM. It's a very nice book that will help you understand the theory better than the standard introductory texts can. On the other hand, it won't teach you much about how to calculate stuff, e.g. the energy levels of an atom. You would need a standard intro for that sort of thing (but you don't need to know these things to study QFT).

You need to understand the basics of special relativity. If you don't, read something about SR. Taylor & Wheeler gets the most recommendations here. (I haven't read it). I like Schutz myself. (It's a GR book, but it has a better presentation of SR than most SR books).

When it's time to move on from basic QM to QFT, I think you should start with chapter 2 of Weinberg. It explains the concept of non-interacting particles in terms of representations. Then you need to learn about quantum fields and interactions. You can keep reading the rest of that book, or check out one of the standard introductory texts. I think you should do both. I don't think any QFT book is so good that you don't need another point of view. Srednicki seems to be the most popular introductory text. Mandl & Shaw is nice too. If you want to see an intro based on the path integral approach, check out Zee.

You might also be interested in "Quantum field theory: A tourist guide for mathematicians" by Folland, and "Quantum field theory for mathematicians" by Ticciati. I don't know much about them other than that the titles sound appropriate.

If you're interested in the algebraic approach to QM and QFT you should check out Araki, and maybe also Strocchi's book "Introduction to the mathematical structure of QM". (I don't think the latter mentions quantum fields). Even if you're not, you should still read the first few pages of Araki, where he explains how you should think about "states" and "observables".

Streater & Wightman is a standard reference for the axiomatically oriented.

 

[arkajad]

If you know some operator algebras and functional analysis then a good start can be the old good "Algebraic methods in statistical mechanics and quantum field theory" by Gerard G. Emch (Wiley 1972). It explains the concepts and gives a lot of references.

 

[DrFaustus]

Before giving you my suggestions I need to ask you what are you looking for in the books. Very simply, physicists' books can give mathematicians headaches. So my question is if you're looking to learn some physics like a physicist would or if you want to know the physics but prefer some mathematical rigour? Do you want to learn to calculate stuff or do you want to know the mathematical structure of a theory? I guess that depends on what your future plans are...

 

[fzero]

You might want to have a look at the notes from the IAS QFT program in 1996:

http://www.math.ias.edu/QFT/fall/index.html
http://www.math.ias.edu/QFT/spring/index.html

There was a book published based on the program: Quantum Fields and Strings: A Course For Mathematicians (P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds.,), 2 vols., American Mathematical Society, Providence, 1999. Unfortunately I've never looked at it, but the authors have a deep understanding of how to explain physics topics to mathematicians.

 

[Meir Achuz]

For a mathematician, I would recommend Dirac's QM text, which has no fuzzy math.
Then I would recommend Halzen & Martin's text to start into QFT