Pseudodifferential Weyl Calculus on (Pseudo-)Riemannian Manifolds

TL;DR

This paper introduces a balanced geodesic Weyl quantization for pseudo-Riemannian manifolds, extending Weyl quantization with properties like mapping symbols to Hilbert-Schmidt operators and providing a star product expansion.

Contribution

It proposes a new quantization method on pseudo-Riemannian manifolds that generalizes Weyl quantization with desirable mathematical properties.

Findings

Maps square integrable symbols to Hilbert-Schmidt operators

Polynomials are mapped to differential operators with parity correspondence

Provides a star product formula with 4th order asymptotic expansion

Abstract

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 geodesic Weyl quantization. Among other things, we prove that it maps square integrable symbols to Hilbert-Schmidt operators, and that even (resp. odd) polynomials are mapped 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.