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 L2(T*M) and construct an irreducible representation of this algebra in L2(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 that 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 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 ℎ2-part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor