Invariant Kinematics on a One-Dimensional Lattice in Noncommutative Geometry

TL;DR

This paper addresses translation invariance issues in a one-dimensional noncommutative lattice by introducing a scale-dependent effective mass and a compensating potential, applicable to both classical and quantum particles.

Contribution

It proposes a novel method using an effective mass and potential to restore invariance in noncommutative lattice kinematics, applicable to classical and quantum particles.

Findings

Effective mass depends on lattice scale and local properties.

A compensating potential can counteract induced metric effects.

Formalism applies to both classical and quantum particles.

Abstract

In a one-dimensional lattice, the induced metric (from a noncommutative geometry calculation) breaks translation invariance. This leads to some inconsistencies among different spectator frames, in the observation of the hoppings of a test particle between lattice sites. To resolve the inconsistencies between the different spectator frames, we replace the test particle's bare mass by an effective locally dependent mass. This effective mass also depends on the lattice constant - i.e. it is a scale dependent variable (a "running" mass). We also develop an alternative approach based on a compensating potential. The induced potential between a spectator frame and the test particle is attractive on the average. We then show that the entire formalism holds for a quantum particle represented by a wave function, just as it applies to the mechanics of a classical point particle.