Deformation quantization of symplectic vector fields

TL;DR

This paper explores the deformation quantization of symplectic vector fields using Fedosov's method, demonstrating how these fields can be quantized into derivations of the star algebra and forming a non-abelian 2-cocycle.

Contribution

It introduces a method to quantize symplectic vector fields into derivations and constructs a non-abelian 2-cocycle, enabling quantization of Lie algebra actions.

Findings

Symplectic vector fields can be quantized to derivations of the star algebra.

Quantization yields a non-abelian 2-cocycle on the Lie algebra of symplectic vector fields.

Any Lie algebra action by symplectic vector fields can be quantized.

Abstract

In this paper, we study deformation quantization of symplectic vector fields \`a la Fedosov. We show that each symplectic vector field can be quantized to a derivation of the deformed star algebra. Moreover, we show that this quantization yields a non-abelian $2$-cocycle on the Lie algebra of symplectic vector fields with values in the deformed star algebra. Therefore, we can quantize any Lie algebra action by symplectic vector fields.