Riemannian exponential and quantization

TL;DR

This paper introduces two quantization methods linked to a linear connection on manifolds, demonstrating their equivalence via a novel property of the Riemannian exponential, and extends the approach to broader functions using distribution fields.

Contribution

It presents a new equivalence result for quantization methods based on the Riemannian exponential and extends quantization to a wider class of functions using distribution fields.

Findings

Demonstrates the equivalence of two quantization methods.

Identifies a new property of the Riemannian exponential.

Extends quantization to functions of broad types using distribution fields.

Abstract

This article continues and completes our previous work [14] J. Phys. Commun. 2 (2018) 025007. 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 [14]. The two methods allow quantize 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. On the other hand, the extension of the previously mentioned quantization to functions of a very broad type can be carried out by generalizing the method of [14] in terms of fields of distributions.