Wick Quantization of Cotangent Bundles over Riemannian Manifolds

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.