Geometric Pseudodifferential Calculus on (Pseudo-)Riemannian Manifolds

Abstract

One can argue that on flat space $\mathbb{R}$ the Weyl quantization is the most natural choice and that it has the best properties (eg symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization – we call it the balanced geometric Weyl quantization. Among other things, we prove that it maps square integrable symbols to Hilbert–Schmidt operators, and that it maps even (resp. odd) polynomials to even (resp. odd) differential operators. We also present a formula for the corresponding star product and give its asymptotic expansion up to the 4th order in Planck’s constant.