Question

Breaking into quantum field theory mathematically?

Answer 1

If you want a really good treatment of it, David Tong has great notes and some old lectures (480p era..) up on YouTube. The basics are not that hard, but it gets very advanced and difficult very fast. The thing to keep in mind with non-interacting field theories is that everything boils down to creation and annihilation operators which are really just elements of a Lie algebra. If you feel like you need a treatment of quantum mechanics beforehand, Shankar has a good, roughly mathematically rigorous introductory text on the subject. QFT is basically functional analysis and Lie algebra (and similar structures like Lie superalgebras) put together in a fairly messy way haha.

Answer 2

This might not be exactly what you mean, but TQFTs are an extremely active area of research in math, and ideas from N=2 supersymmetric field theories and even string theory have been essential to a lot of enumerative geometry.

EDIT: Idk how I forgot this one but geometric langlands is very, very closely related to N=4 super yang mills. Also, dynamical systems are closely related to other N=2 supersymmetric field theories.

Answer 3

Multiple mathematicians have told me you shouldn't learn quantum field theory until you have tenure.

Answer 4

I think, from interactions with physicists, there's a lot of work needed around formalizing QFT and resolving some mathematical difficulties, that could even be related to physical insights (like quantization appeared as a trick by Planck). I think you could use interaction with a physicist or at least develop physical intuition if you want to work at research -- maybe those insights could lead to better understanding or even new physical theories as well.

Answer 5

Check out Tobias Osborne’s courses on YouTube. I find he does a good job teaching the subject in a fairly standard way, but also making it explicit when something is non-rigorous and pointing out how you should think of things mathematically. More so than other authors.

*Quantum Fields and Strings: A Course for Mathematicians* is probably the most "QFT for mathematicians" book I've ever seen, but as it's a compilation of lectures from IAS by people like Witten and Deligne, it very much violates the "undergrad physics knowledge" requirement. It's basically for working mathematicians working near the subject already.

Answer 6

I'm just curious: how would a mathematician use QFT for their research? In what math field would QFT be useful?

Answer 7

>How should a mathematician most effectively learn and use Quantum Field Theory for their research?

Why would quantum field theory be useful to a mathemacian?

To be clear, this is not a rhetorical question meant to indicate that it couldn't be, but it seems like a rather presumptuous question to ask without any sort of use case in mind.

Answer 8

The core of QFT and physics models in general is math so you’re well off. So long as you have the appropriate math background Id say search grad level classes online and just start reading through the textbook. I say grad level mainly because QFT probably isn’t covered in any undergrad program but if you want to start with more basic quantum first my classes used textbooks by Griffiths or another by Zetilli

Answer 9

Are you looking to do mathematically rigorous work on QFT? If so, I can recommend some resources on that.

Edit: Also, what is your math background?

Answer 10

Depending on your background and intentions I can recommend Factorization algebras in quantum field theory by Costello and Gwilliam. It doesn’t really assume much in the way of physics but gives you a nice way to work with QFTs mathematically.