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Pseudodifferential Weyl calculus on (Pseudo-)Riemannian manifolds. (English) Zbl 1436.81072

Summary: 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 fourth order in Planck’s constant.

MSC:

81S10 Geometry and quantization, symplectic methods
53D50 Geometric quantization
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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