Geodesic Curves on Quantized Manifolds

TL;DR

This paper defines and compares classical and quantum geodesics on quantized manifolds, demonstrating their compatibility and illustrating the concepts with the example of the quantum plane.

Contribution

It introduces a general framework for geodesics on quantized manifolds and shows their consistency with classical definitions, exemplified by the quantum plane.

Findings

Classical and quantum geodesics are compatible on quantized manifolds.

A general definition of geodesics in the quantum setting is provided.

The quantum plane serves as a concrete example illustrating the theory.

Abstract

A general definition of the curves and geodesics associated with a given connection on a quantized manifold is given. In the particular case of the functional quantization we define geodesics in the same way as in the classical case and we will show that the two definitions are compatible. As an example we examine our results for the quantum plane.