On the Dequantization of Fedosov's Deformation Quantization

Abstract

Abstract. To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of T * M to the formal neighborhood of the diagonal of the product M × M , where M is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we "dequantize" Fedosov's quantization.

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…Considering a de-quantization formalism [36], we construct certain quantum deformations of the classical Einstein configurations in the classical limit. Such a model defines a nonholonomic almost Kähler generalization of the Einstein gravity on cotangent bundle.…”

“…The aim of this work is to show how Karabegov's approach to Fedosov deformation quantization [19,35,36] can be naturally extended for almost Kähler manifolds endowed with canonical geometric structures generated by semi-Riemannian and/or Einstein metrics and Lagrange-Finsler and Hamilton-Cartan fundamental generating functions. This paper is motivated by the following results: In Refs.…”

On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization

“…In [11] we associated to each natural star product * on a Poisson manifold (M, {·, ·}) the mappings S, T :…”

“…In [11] we related to any natural deformation quantization on a symplectic manifold M a canonical formal symplectic groupoid isomorphic to the formal pair symplectic groupoid (M ×M , M diag ), where M diag is the diagonal of M ×M . The isomorphism was given via the source and target mappings of the formal symplectic groupoid of the deformation quantization.…”

Fedosov’s formal symplectic groupoids and contravariant connections

“…On the other hand, it was shown in [10] that the corresponding formal symplectic groupoids 'with separation of variables' can be naturally extended from Kähler manifolds to Kähler-Poisson manifolds, while it is impossible to extend the star-products with separation of variables to the Kähler-Poisson manifolds in a naive direct way (see [8]). In this paper we show that the construction of the formal symplectic groupoids of Fedosov's deformation quantizations from [9] can be naturally extended to the Poisson manifolds endowed with a torsion-free Poisson contravariant connection. We call the formal symplectic groupoids obtained via this construction Fedosov's formal symplectic groupoids.…”