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 that this product has a front, which is described in terms of geodesic triangles in M. The quantization of 6-functions induces a family of symplectic reflections in T*M and generates a magneto-geodesic connection F 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 h h(2)-part of the quantum product contains the Ricci curvature of Gamma and a magneto-geodesic coupling tensor.