Geometrical origin of the *-product in the Fedosov formalism

TL;DR

This paper explores the geometric foundations of Fedosov's *-product in deformation quantization, linking Weyl algebra structures with symplectic geometry and illustrating applications in quantum mechanics.

Contribution

It reveals the geometric origin of the *-product within fibre bundle theory and establishes relations between Weyl algebra connections and symplectic structures.

Findings

Proved properties of the Weyl algebra product

Established relations between Weyl and symplectic connections

Provided examples of Fedosov formalism in quantum mechanics

Abstract

The construction of the *-product proposed by Fedosov is implemented in terms of the theory of fibre bundles. The geometrical origin of the Weyl algebra and the Weyl bundle is shown. Several properties of the product in the Weyl algebra are proved. Symplectic and abelian connections in the Weyl algebra bundle are introduced. Relations between them and the symplectic connection on a phase space M are established. Elements of differential symplectic geometry are included. Examples of the Fedosov formalism in quantum mechanics are given.