Getzler Symbol Calculus and Deformation Quantization

TL;DR

This paper develops a Fedosov quantization method that integrates odd variables and extends Getzler's pseudodifferential calculus, leading to a deformation of the exterior algebra on Riemannian manifolds.

Contribution

It introduces a Fedosov-type construction incorporating odd variables and an analogous composition formula to Getzler's calculus, enabling new deformation quantizations.

Findings

Constructed Fedosov quantization with odd variables.

Derived an analogous formula to Getzler's calculus.

Defined a deformation of the exterior algebra on Riemannian manifolds.

Abstract

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.