Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles

TL;DR

This paper develops a method to transform Lagrange, Finsler, Hamilton, and Cartan spaces into almost Kaehler structures on tangent and cotangent bundles, enabling deformation quantization of gravity and mechanical models.

Contribution

It introduces a novel approach to apply Fedosov quantization to complex geometric structures related to gravity and mechanics.

Findings

Constructed almost Kaehler structures from Lagrange and Finsler spaces.

Developed Fedosov operators for nonlinear mechanical and gravity models.

Enabled deformation quantization on (co) tangent bundles with Einstein metrics.

Abstract

We provide a method of converting Lagrange and Finsler spaces and their Legendre transforms to Hamilton and Cartan spaces into almost Kaehler structures on tangent and cotangent bundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on effective phase spaces. This allows us to define the corresponding Fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on Lagrange- and Hamilton-Fedosov manifolds.