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I have asked this question in the physics.stackexchange.com forum too - sorry if that is a no-no; I feel it straddles the boundary between the two, and similar questions have been answered rather vaguely in the past, IMO. And perhaps that isn't surprising, since I am looking for the maths that lead up to quantum mechanics rather than QM itself, it that makes sense. So, if you wil forgive:
I come from a mathematical background, but would really like to understand the mathematical reasoning behind QM. I have been through a number of books about the subject already, but they all descend into matrix groups and coordinate representations, and what I really want to find is a pure mathematical treatment of the subject - Lie groups and Lie algebra are generally much cleaner when you don't have to feel your way through a mist of matrix elements, differential geometry even more so. As an example of a style I find very helpful, look for Frederic Schuller's lectures on youtube (eg https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic).
So, bearing all this in mind, are there any good, abstract, mathematical treatments of the foundations of QM, which go into the reasoning and intuition behind things like quantisation, Noether's Theorems and similar, and which throughout apply rigorous definitions and proofs (which I find actually helps rather than hinders intuition)?
OK, I have now taken some time to check out the other question that is allegedly an exact duplicate, but I don't think so. I'm not looking for an introduction to QM, I am looking for a through, mathematical treatment of the foundations of QM; take a subject like quantization - so far, what I have found introduces it as a ready-made procedure into which you plug a certain equation and out of it comes, magically, 'QM'. I suppose I could go and find Paul Dirac's doctoral thesis, but I had hoped for a modern treatment; I looked at Amazon for the books mentioned in the comments, and they are expensive, I think, so I'd rather be sure they have what I'm looking for.