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What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties:

1- We should be able to associate to each (smooth (?)) function $f:M\rightarrow \mathbb{R}$ an operator $\hat{f}:H_{M}\rightarrow H_{M} $

2- We should be able to associate to each lagrangian submaniflod $L\subset M$ a vector (a wave function) $v_{L}\in H_{M}.$

I think that this procedure is called a Quantization. The language of classical categories seems to be useless ? What can we do mathematically to model such wish ?

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    $\begingroup$ In general, quantization is not a functor. There are several procedures of quantization, so this question is too broad. See for instance ncatlab.org/nlab/show/quantization $\endgroup$
    – user40276
    Commented Sep 17, 2015 at 8:39
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    $\begingroup$ @user40276 Thanks for the link, In some article of ncatlab I can't see what is really important about a subject, at the end I get more confused :). Anyway, I did notice that the functorial (categorical) approach looks useless.. I was looking for another opinion about this subject. $\endgroup$
    – Max
    Commented Sep 17, 2015 at 9:03
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    $\begingroup$ Well, I think you should point what exactly you want to do, so you can choose a suitable quantization. Until now, as I understand, the most formal non-perturbative method is deformation quantization, however in general (in infinite degrees of freedom) the star product does not exist (and so people work with power series instead of C^{*}-algebras, but this is perturbative, so I don't think this have any physical significance in reality). $\endgroup$
    – user40276
    Commented Sep 17, 2015 at 9:08
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    $\begingroup$ I'm not a physicist, what I wrote was explained to me by a physicist. He shared his wish. I do think that the written one look really to the one that the physicists wants. In a mathematical text book, it is more about what is convenient for mathematicians, and I'm not sure that is has something to do with what physicists are looking for. $\endgroup$
    – Max
    Commented Sep 17, 2015 at 9:10
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    $\begingroup$ Well, I think you should look first at Dirac axioms. There are three axioms (see here math.berkeley.edu/~alanw/GofQ.pdf in page 81). The point is that it's not possible, in general, to satisfy all the axioms, so we just require some of them and we get different approaches to quantization. Depending on the kind of quantum field theory of prefers one kind instead of the other. $\endgroup$
    – user40276
    Commented Sep 17, 2015 at 11:39

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