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Here's a question that I thought I had asked already, but perhaps not here: I remember being introduced to quantum mechanics, long ago, when I studied maths and physics. It went something like "there is this Hamiltonian, and you replace these part with differential operators and twist it like this; then solve the differential equation" - and you were left with a handful of bras and kets, and a head full of woolly rituals. It was one of the things that put me off quantum mechanics, in many ways: the lack of real explanation.

I have always loved mathematics for its devotion to explaining, as thoroughly as possible, the whys and hows of its results, and it frustrates me that I still haven't been able to find a real explanation of the mathematical reasoning that lead to the methods employed in quantization. Is it really the case that there is no solid, mathematical treatment of this subject? I am somewhat conversant with higher, modern maths, like differential geometry, category theory, abstract algebra etc - is there a good (ie. mathematically thorough) treatment somewhere?

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Firstly, thanks to those, that have attempted to answer my rant; I appreciate that.

Let me try to narrow the question down. So, the discussion in the article, kindly provided by David Bar Moshe in the comments, start by listing a few axioms, then goes on to describe how one takes the Hamiltonian energy from classical mechanics and transforms it by substituting parts of that with linear operators in a certain Hilbert space.

Now, I can understand why that particular formulation of classical mechanics is interesting, and I can understand why early quantum physicists saw some inspiring possibilities in the spectral theorems from linear algebra, but I still find difficult to fathom is, where do these particular axioms and quantization rules come from - why not some other rules?

I have an uterior motive: as a mathematician, my ideas about what is 'right' rests on whether I believe in the axioms, so to speak, whereas a physicist is more pragmatic: if the theory fits the known observations and survives the experiments, then it gets to live another day, even if the fundamentals aren't necessarily well understood. But I think that is probably why we are stuck with the standard model, which is disappointingly perfect within its own sphere, but simply has nothing to say about anything else; there is something incomplete about the foundations of QM, the axioms, possibly. Sorry, this may be on the way to becoming too broad again.

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  • $\begingroup$ You can always have a look at TQFTs; they've been rigorously defined - however Atiyah admitted it only just captured the 'most primitive part of QFT' which is it's vacuum structure. $\endgroup$ – Mozibur Ullah Jun 20 '18 at 10:38
  • $\begingroup$ To reopen this post (v1), consider to only ask one subquestion per post. $\endgroup$ – Qmechanic Jun 20 '18 at 11:03
  • $\begingroup$ Please see arxiv.org/abs/math-ph/0405065 $\endgroup$ – David Bar Moshe Jun 20 '18 at 11:13
  • $\begingroup$ There are more than one "solid treatment" of quantization from the mathematical point of view, at least for particle systems (non-relativistic quantum mechanics - QM). For QFT, the situation is more complicated but still many things can be said. QM quantization is studied in analysis under the name of "Microlocal analysis" and "Pseudodifferential Calculus". From a more algebraic/geometric point of view, it is studied in Deformation Theory, e.g. introducing Moyal brackets (the work of Kontsevich on the deformation quantization of Poisson manifolds gave him a very wide recognition). $\endgroup$ – yuggib Jun 20 '18 at 14:17

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