Cotangent bundle quantization : Entangling of metric and magnetic field

For manifolds M of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space L 2 (T * M) and construct an irreducible representation of this algebra in L 2 (M). This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over T * M is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in M. The quantization of δ-functions induces a family of symplectic reflections in T * M and generates a magneto-geodesic connection Γ on T * M. This symplectic connection entangles, on the phase space level, the original affine structure on M and the magnetic field. In the classical approximation, the 2-part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor.