Quantization of the sphere with coherent states

Quantization with coherent states allows to " quantize " any space X of parameters. In the case where X is a phase space, this leads to the usual quantum mechanics. But the procedure is much more general, and does not require a symplectic, or any kind of structure in X, other than a measure. It is simply considered as a different way to look at the system, the choice of a resolution, in analogy with data handling, where coherent states (e.g., under the form of wavelets) are very efficient. Here, we present the complex coherent states quantization of the 2-sphere, with emphasis on the links with group representation. We show how this procedure leads naturally to the fuzzy sphere and to non commutative geometry.