Pseudo-differential operators and deformation quantization

Abstract

Using the Riemannian connection on a compact manifoldXwe show that the algebra of classical pseudo-differential operators onX generates a canonical deformation quantization on the cotangent manifoldT*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization.