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.