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arXiv · 0809.1946

On the relation between geometric and deformation quantization

Christoph Nölle
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Abstract

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 Kähler manifold.

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Christoph Nölle. 2008-09-11. On the relation between geometric and deformation quantization. https://arxiv.org/abs/0809.1946

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