Quantisation of Lie-Poisson manifolds

Details Quantisation of Lie-Poisson manifolds Quantisation of Lie-Poisson manifolds

2004-11-03

arxiv.org/abs/math/0411066v1

Mathematics

Differential Geometry

34 pages

Scientific paper

In quantum physics, the operators associated with the position and the momentum of a particle are unbounded operators and $C^*$-algebraic quantisation does therefore not deal with such operators. In the present article, I propose a quantisation of the Lie-Poisson structure of the dual of a Lie algebroid which deals with a big enough class of functions to include the above mentioned example. As an application, I show with an example how the quantisation of the dual of the Lie algebroid associated to a Poisson manifold can lead to a quantisation of the Poisson manifold itself. The example I consider is the torus with constant Poisson structure, in which case I recover its usual $C^*$-algebraic quantisation.

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