On the relation between geometric and deformation quantization

Details On the relation between geometric and deformation quantization On the relation between geometric and deformation quantization

2008-09-11

arxiv.org/abs/0809.1946v1

Physics

Mathematical Physics

20 pages

Scientific paper

In this paper we investigate the possibility of constructing a complete quantization procedure consisting of geometric and deformation quantization. The latter assigns a noncommutative algebra to a symplectic manifold, by deforming the ordinary pointwise product of functions, whereas geometric quantization is a prescription for the construction of a Hilbert space and a few quantum operators, starting from a symplectic manifold. We determine under which conditions it is possible to define a representation of the deformed algebra on this Hilbert space, thereby to extend the small class of quantizable observables in geometric quantization to all smooth functions, as well as to give a natural representation of the algebra. In particular we look at the special cases of a cotangent bundle and a K\"ahler manifold.

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