Geodesic multiplication and quantum kinematics in a Newtonian spacetime

TL;DR

This paper introduces a novel quantization method for a quantum particle in Newtonian spacetime using geodesic multiplication, providing detailed calculations in a 2D model.

Contribution

It proposes a new quantization approach based on geodesic multiplication and Riemann normal coordinates in nonrelativistic spacetimes.

Findings

New quantization scheme for Newtonian quantum mechanics

Explicit calculations in a 2D Newtonian spacetime model

Connection between geodesic multiplication and quantum kinematics

Abstract

We consider a quantum test particle in the background of a Newtonian gravitational field in the framework of Cartan's formulation of nonrelativistic spacetimes. We have proposed a novel quantization of a point particle which amounts to introducing its position operators as multiplication with the corresponding Riemann normal coordinates and momentum operators as infinitesimal right translation operators determined by geodesic multiplication of points of the spacetime. We present detailed calculations for the simplest model of a two-dimensional Newtonian spacetime.