Riemannian exponential and quantization

Abstract

This article continues and completes the previous one [18]. First of all, we present two methods of quantization associated with a linear connection given on a differentiable manifold, one of them being the one presented in [18]. The two methods allow quantization of functions that come from covariant tensor fields. The equivalence of both is demonstrated as a consequence of a remarkable property of the Riemannian exponential (Theorem 5.1) that, as far as we know, is new to the literature. In addition, we provide a characterization of the Schrödinger operators as the only ones that by quantization correspond to classical mechanical systems. Finally, it is shown that the extension of the above quantization to functions of a very broad type can be carried out by generalizing the method of [18] in terms of fields of distributions.