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 $a\in\mathbb{A}$
$$\hbar^2a=0 \Longrightarrow a\in\hbar\mathbb{A}$$
Best Answer
Well, a lot of questions, some of which Theo already answered in a very nice way. Let me just give some additional remarks and hints how I think about DQ and Poisson geometry in relation to quantum physics.
Concerning the first question:
the good replacement (in view of Gel'fand duality) of a point on phase space is a (pure) state on the quantum algebra. While for $C^\ast$-algebras this is standard lore, in formal DQ things are slightly more tricky: of course you can argue that a formal star product yields not yet a quantum observable algebra as $\hbar$ does not have a "value" (say $1$ in your favorite unit system), so you should postpone the question of states till when you reach a "convergent/strict" DQ. This is often possible but in general completely unknown. Surprisingly, there is a good notion of states already for formal star products: essentially the same definition applies, take positive functionals of the algebra $C^\infty(M)[[\hbar]]$ which are $\mathbb{C}[[\hbar]]$-linear and take values in $\mathbb{C}[[\hbar]]$. To define positivity you make use of the fact that $\mathbb{R}[[\hbar]]$ is an ordered ring. Then many techniques of $C^*$-algebra theory can be carried over to this entirely algebraic framework. In fact, we have worked out many things like the GNS construction of representations etc.
Now the point is that a classically positive functional $\omega_0\colon C^\infty(M) \longrightarrow \mathbb{C}$ (which is a positive Borel measure with compact support by a smooth version of Riesz' Theorem) may no longer be positive with respect to a given star product $\star$. Thus you need to add higher order corrections $\omega = \omega_0 + \hbar\omega_1 + \cdots$ in order to gain positivity. It is a (quite non-trivial) theorem that this is always possible, even in a "differential" sense that all the higher orders are of the form $\omega_0 \circ D_r$ with some differential operator $D_r$.
You can apply this now to your favorite classical state, the delta-functional at a given point. The corresponding (non-unique) deformation is then the quantum analog of what a point can be, in some sense the best thing you can get.
The uncertainty principle can be understood as the reason why positivity fails for $\omega_0$ itself and why higher orders are necessary...
The second question: of course, for hard physical applications you only need $\mathbb{R}^{2n}$, maybe a cotangent bundle but that's it. Even a generic symplectic manifold is hard to motivate from this point of view.
But there are also reasons from physics why one should take care of DQ of more general Poisson manifolds:
a) Symmetries: whenever you have a classical symmetry encoded by a momentum map, then $\mathfrak{g}^\ast$ is a Poisson manifold. Quantizing a symmetry then amounts to quantize the momentum map in an appropriate way. There are several competing definitions but essentially all involve a DQ of the linear Poisson structure on $\mathfrak{g}^*$.
b) Aesthetics: to have a general framework in which you can discuss your relevant examples might be useful and open your view, even though the examples might be very very special inside this bigger framework.
c) Applications in NCG: many models of noncommutative space-times require more general Poisson structures to be quantized than just symplectic ones. It is even not clear that space-time allows for a symplectic structure at all, but it certainly carries interesting Poisson structures. In serious models of NC space-times, the Poisson structure itself should be treated as a dynamical quantity, i.e. a field. Then there is no reason why it should be non-degenerate everywhere. These models are of course all still very speculative...
d) Toy models: one can view complicated Poisson/symplectic manifolds as toy models for the infinite-dimensional phase spaces of classical field theories with gauge symmetries. Here the true phase spaces are sort of Marsden-Weinstein quotients (in ugly infinite dimensions) which can have quite generic geometry. So one tries to learn something about their quantization by looking at finite-dim models having at least also a complicated geometry.
Third question: Where is the Hilbert space...
After what I said for the first question, this is now pretty clear and follows the same line of argument as in AQFT: Having the algebra of observables, one takes a look at all $^*$-representations on, say pre-Hilbert spaces, by means of a GNS construction. The notion of pre Hilbert space works very much the same for ordered rings like $\mathbb{R}[[\hbar]]$. This has been worked out in detail in many places and gives indeed physically interesting results. The main advantage is now that one can take a look at different representations which can encode different physical situations...
OK, sorry for such a long blurp. I hope it gives some inspiration.