Hello, I would like to introduce myself to the theory of quantization and noncommutative deformations of Riemann Poisson structures. In fact, I am familiar with Riemannian and Poisson geometry, but I cannot grasp the principle of the theory above.

As I understand, by reading some introductory texts on the subject, ideas come from physics. This involves replacing the coordinates of the phase space (the cotangent of $\mathbb{R}^3$ endowed with its canonical symplectic structure) by operators (on a Hilbert space) which do not commute with each other and is the description of quantum mechanics in a form similar to classical mechanics. Then there was conflict between the riemannian description (or pseudo-Riemannian) of space in general relativity and the space of quantum mechanics. On small scales, the innermost structure of space-time is broken: the position and velocity of a point do not make sense at the same time, if one is defined precisely the other will not defined (the Heisenberg uncertainty principle). This raises the question how to give a physical meaning to the concept of point?

The Gelfand-Naimark theorem first brought a solution to this "internal conflict" of physics, by establishing a bridge between topology and algebra. So we look at a point $x$ in a space $X$ be fixed, and considering its "shadow" consists of the values $f(x)$ taken by all continuous functions $f$ on $X$. This leads to replace the space $X$ by commutative algebra $C(X)$.

The fact that pseudo-Riemannian geometry is a sufficient description of space-time for most purposes, suggests that noncommutativity might be treated as the limit where Planck's constant $\hbar$ tends to $0$ then I understand it, how to become a constant variable and move towards $0$! hence the idea of deformation quantization, which is to construct noncommutative algebras $\mathcal{A}_\hbar$, by deformation of the Poisson algebra $\mathcal{A}=C^\infty(M)$ (formal series in $\hbar$ with coefficients in $\mathcal{A}$) that converges to $\mathcal{A}$ when $\hbar$ tends to $0$. The work most results in this direction is that of Kontsevich.

1) How Heisenberg's uncertainty principle reviews the classical definition of a point?

2) Why deformation always starts with a Poisson manifold? if it is to deform the phase space of hamiltonian mechanics it suffices to consider symplectic manifolds!

3) Why the deformation of the algebra of functions $C^\infty(M)$ of a Poisson manifold is a way of quantization? in this case what is the Hilbert space, in which the observable $f$ are replaced by a bounded operators $\widehat{f}$?

Alain Conne then directed the program algebraization of differential geometry, in order to then work on "noncommutative spaces". He considered riemannians manifolds with additional structure of Spin. Such structure is canonically attached to the triple $(C^\infty(M),L^2(M),D)$ ; what is $L^2(M)$? and $D$ is the Dirac operator, with some number of properties that can be generalized to spectral triples $(\mathcal{A}, H,D) $: $\mathcal{A}$ a noncommutative algebra, $H$ Hilbert space, and $D$ the Dirac operator.

1) Why in physics we need to replace the Laplacian (which is of order $2$) by a differential operator of order $1$ the Dirac operator (which is the square root of the Laplacian)?

3) How to define, precisely, a spectral triple and how to find the metric using the Dirac operator?

4) Why we deform only Spin manifolds?

On the other hand, in an article \url{http://arxiv.org/abs/math/0504232v2} Eli Hawkins gave some definitions that I not understand (not being familiar with the language of algebraic geometry). In particular, the definition 1.4 (page 4) and the definition of "metacurvature" (page 5). In particular, how an extension of the algebra of differential forms gives rise to a Poisson bracket!?

If $\mathcal{A}_0$ is an algebra, that means an extension of the form $$0 \rightarrow\hbar\mathbb{A}\rightarrow\mathbb{A} \rightarrow\mathcal{A}_0$$ where $\hbar$ is a central multiplier $\mathbb{A}$, and for all $a\in\mathbb{A}$ $$\hbar^2a=0 \Longrightarrow a\in\hbar\mathbb{A}$$