Pseudodifferential Weyl Calculus on (Pseudo)Riemannian Manifolds
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 chartwise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudoRiemannian 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 HilbertSchmidt 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.01572
 Bibcode:
 2018arXiv180601572D
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 38 pages, 2 figures