Classical dynamics on three dimensional fuzzy space: Connecting the short and long length scales

TL;DR

This paper derives the classical equations of motion for a particle in three-dimensional fuzzy space, revealing unconventional features like energy cut-offs, variable limiting speeds, and orbital precession similar to general relativity, with implications for observational tests.

Contribution

It introduces a novel path integral formulation for particles in fuzzy space and uncovers unique classical dynamics that connect short and long length scales.

Findings

Predicts a maximum energy cutoff.

Shows orbital precession akin to general relativity.

Identifies constraints on non-commutative parameters.

Abstract

We derive the path integral action for a particle moving in three dimensional fuzzy space. From this we extract the classical equations of motion. These equations have rather surprising and unconventional features: They predict a cut-off in energy, a generally spatial dependent limiting speed, orbital precession remarkably similar to the general relativistic result, flat velocity curves below a length scale determined by the limiting velocity and included mass, displaced planar motion and the existence of two dynamical branches of which only one reduces to Newtonian dynamics in the commutative limit. These features place strong constraints on the non-commutative parameter and coordinate algebra to avoid conflict with observation and may provide a stringent observational test for this scenario of non-commutativity.