Examples of quantisation of Poisson manifolds

In quantum physics, the operators associated with the position and the momentum of a article are unbounded operators and C∗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∗C∗-algebraic quantisation.