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Differential geometry over the structure sheaf: A way to quantum physics. (English) Zbl 0945.58008
Slovák, Jan (ed.) et al., Proceedings of the 17th winter school “Geometry and physics”, Srní, Czech Republic, January 11-18, 1997. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 54, 45-51 (1998).
An idea for quantization by means of geometric observables is explained, which is a kind of the sheaf theoretical methods. First the formulation of differential geometry by using the structure sheaf is explained. The point of view to get interesting noncommutative observable algebras of geometric fields is introduced. The idea is to deform the algebra \(C^\infty(M, \mathbb R)\) by suitable interaction structures. As an example of such structures the Poisson-structure is mentioned and this leads naturally to deformation quantization.
For the entire collection see [Zbl 0904.00040].
58B34 Noncommutative geometry (à la Connes)
53D55 Deformation quantization, star products
53D50 Geometric quantization