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Let ($X,\omega$) be a symplectic manifold (phase space of some physical system). Consider the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ of smooth functions on $X$ and $[\omega]\in \textrm{H}^{2}_{\textrm{de Rham}}(X,\mathbb{R})$. If I'm not wrong, any element of the second Hochschild cohomology $\textrm{HH}^{2}(\mathcal{C}^{\infty}(X,\mathbb{R}))$ gives rise to a deformation of the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ i.e. we obtain a noncommutative algebra $\mathbf{A}$ "extending" the commutative algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ .

Question 1 How can we associate to each element of $[\omega]\in \textrm{H}^{2}_{\textrm{de Rham}}(X,\mathbb{R})$ an element of $\textrm{HH}^{2}(\mathcal{C}^{\infty}(X,\mathbb{R}))$ is a natural way ? Saying it in another way, how can a symplectic structure on a manifold generates a deformation of the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$ ?

Question 2 (Vague) The quantum mechanics is noncommutative theory in nature, In which sense (if there is) the algebra $\mathbf{A}$ is a quantum version (or a quantization) of the physical system described by phase space $(X,\omega)$

Edit January 7. I don't know what are the reasonable hypothesis on $X$ that we should consider. Vaguely, the idea I had in mind is that the $\textrm{HH}^{\ast}(\mathcal{C}^{\infty}(X,\mathbb{R}))$ is the algebra of polyvectors $\oplus_{n} \Gamma (X, \wedge^{n}TX)$. If there is a canonical identification (e.g riemanian geometry) between $TX$ and $T^{\ast}X$ in such way that we can associate to the symplectic structure $\omega$ an element of $\textrm{HH}^{2}(\mathcal{C}^{\infty}(X,\mathbb{R}))$ via $\Gamma (X, \wedge^{2}TX)$. Hence we obtain a deformation $\mathbf{A}$ of the algebra $\mathcal{C}^{\infty}(X,\mathbb{R})$... The question is: That is make sense what I wrote ? is it reasonable from purely physical perspective ?

Edit January 8. As it was mentioned in the comments, it seems that the symplectic form $\omega$ gives rise to an element in the second cohomology $\mathrm{HH}^{2}$. The symplectic manifold $X$ can be taught as the phase space of some physical system or as $X=TY$ for some smooth manifold $Y$ where the total tangent space $TY$ is seen as a symplectic manifold in a natural way. One way of interpretation of the second question is understand the algebra representation $\mathbf{A}\rightarrow \mathrm{End}(L^{2}(X))$ where $L^{2}(X)$ is the Hilbert space of square integrable functions on $X$.

Edit January 8 bis. as starting point, I will be happy if someone could provide a solution for the standard example when $X=\mathbb{R}^2$ and $\omega=dx_{1}\wedge dx_{2}$.

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    $\begingroup$ Is your question basically asking to describe deformation quantization? $\endgroup$
    – Igor Khavkine
    Commented Jan 6, 2017 at 23:52
  • $\begingroup$ @IgorKhavkine I have edited my question. The point is that I don't know really how to formulate the question. I hope that someone can provide a nice interpretation. $\endgroup$
    – mathphys
    Commented Jan 7, 2017 at 9:54
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    $\begingroup$ The inverse $\Pi = \omega^{-1}$ of a symplectic form is a Poisson bivector. A Poisson bivector, via deformation quantization, gives rise to a non-commutative deformation of $C^\infty(X)$ and hence a cocycle in $\mathrm{HH}^2$. But I don't know if this correspondence respects de Rham cohomology classes. $\endgroup$
    – Igor Khavkine
    Commented Jan 7, 2017 at 10:44
  • $\begingroup$ @IgorKhavkine I think your comment clarifies my first question. It seams that we need a riemanian metric also to make things canonically, right ? After all in classical mechanics the symplectic form is needed and not just its corresponding element in the second de rham cohomology. I hope my seconde question will get answers $\endgroup$
    – mathphys
    Commented Jan 7, 2017 at 13:55
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    $\begingroup$ The standard $(\mathbb{R}^2, dx_1 \wedge dx_2)$ case is answered by the Wigner-Weyl-Moyal formula for the deformed "star" product, which then essentially gives the CCR algebra for $\mathbf{A}$. The CCR algebra has a known representation theory, though its irreducible representations are not of the form that you have guessed. This is all standard in the literature on deformation quantization. Reading up on that might answer many of your questions, or at least help you sharpen them. $\endgroup$
    – Igor Khavkine
    Commented Jan 7, 2017 at 23:45

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