Which mathematical subjects must I learn to understand basic QFT?

[camel_jockey]

So I did QFT at university and didn't feel that I really understood what was being done. We just did some calculations, heuristic guesswork and dwelled on phenomenology.

I want to do it the way I personally understand things best: by learning the mathematics in detail, almost at the level of a pure mathematician. I am wondering which order this works best in.

I am attempting to do it in two stages. First, do quantum mechanical mathematics in this order

Finite-dimensional vector spaces including operators
Infinite-dimensional vector spaces
Hilbert spaces and their operators

I am then a bit lost what to do for the second stage, where I learn the mathematics underlying quantum field theory. I was wondering if someone on Physics forums could recommend in which order I should do the math topics. In general, many of the "QFT for mathematicians" assume to much foreknowledge of advanced mathematics, which my physics brain can't handle. I need a slow, gradual and thorough development of the math, even though that will take time.

So I remember Lie group theory being used, but what else and in what order? Any good books on the topic would be great, I am currently planning on using Sadri Hassani's "Mathematical Physics" for stage 1, and as much of stage 2 as possible (due to his enormous clarity and well-worked, relevant examples).

Edit: the best mathematical treatments of QM are Shankars text and Ballentines, so I will do these after stage 1. After completing stage 2, I hope to be able to understand most of the Spin-zero part of Srednicki's QFT text, though I have considered buying Ramond's text on QFT too.

 

[homeomorphic]

You need to describe what mathematical knowledge you have already, otherwise, we don't know where to begin.

Maybe you could try Mathematical Physics by Robert Geroch for a good start.

QFT, of course, hasn't really been made mathematically rigorous.

There's a lot of good stuff on this page, once you can figure out the prerequisites:

http://math.ucr.edu/home/baez/QG.html

For QFT, what's particularly interesting is the course on quantization and categorification. I'm hoping to develop the ideas started there further to have a better conceptual understanding of QFT. The basic insight seems to be that quantum mechanics is actually very combinatorial. You have discrete bits of energy that are going around and discrete bits of energy are something that can be counted. And combinatorics is the art of counting things. Things like Feynman diagrams are already fairly combinatorial-looking things, but here that's explored a little more deeply.

Another thing that is good to look into is developing an intuition for spinors and spinorial objects (that's what fermions are). Some of this can be found in Baez's notes, but another good source is Penrose's book, The Road to Reality (and perhaps his books on Spinors and Spacetime, to some degree, but they might be over-kill). I find Dirac's approach to the square root of the wave operator to be a little contrived. I prefer Clifford's approach to the square root of the Laplacian, which seems much better motivated.

I'm not sure you'll be able to avoid calculations, guesswork, and phenomenology if you want to learn QFT, but it's definitely possible to get a deeper understanding of it than what you'll get from the standard texts. Mathematical approaches can help with this, but more physical intuition might also help.