# Geometric Pseudodifferential Calculus on (Pseudo-)Riemannian Manifolds

June 05, 2018

One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the
most natural choice and that it has the best properties (e.g. 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.

Keywords:

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