Wick Quantization of Cotangent Bundles over Riemannian Manifolds

TL;DR

This paper introduces a geometric method for constructing Wick symbols on cotangent bundles of Riemannian manifolds by endowing them with a formal Kähler structure, facilitating phase space quantization.

Contribution

It proposes a novel geometric procedure to define Wick symbols using a formal Kähler structure on cotangent bundles, extending quantization techniques to Riemannian manifolds.

Findings

Constructed Wick symbols on cotangent bundles using formal Kähler structures.

Demonstrated the method on constant curvature spaces and sigma models.

Showed the canonical symplectic form is preserved in the construction.

Abstract

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal K\"ahler structure. The formality means that the metric is lifted from the Riemannian manifold $Q$ to its phase space $T^\ast Q$ in the form of formal power series in momenta with the coefficients being tensor fields on the base. The corresponding K\"ahler two-form on the total space of $T^\ast Q$ coincides with the canonical symplectic form, while the canonical projection of the K\"ahler metric on the base manifold reproduces the original metric. Some examples are considered, including constant curvature space and nonlinear sigma models, illustrating the general construction.