Quantum geodesics on $\lambda$-Minkowski spacetime

TL;DR

This paper applies quantum geodesic formalism to $mbda$-Minkowski spacetime, revealing quantum gravity corrections to geodesic flow and particle modeling, with implications for quantum gravity effects at the Planck scale.

Contribution

It introduces quantum geodesic analysis on $mbda$-Minkowski spacetime and demonstrates quantum gravity corrections to classical geodesic behavior and particle models.

Findings

Quantum geodesic flow of a plane wave has order $mbda_p$ frequency correction.

Center of mass of a Gaussian moves with an order $mbda_p^2$ correction.

Point particles cannot be modeled as infinitely sharp Gaussians due to quantum gravity effects.

Abstract

We apply a recent formalism of quantum geodesics to the well-known bicrossproduct model $\lambda$-Minkowski quantum spacetime $[x^i,t]=\imath\lambda_p x^i$ with its flat quantum metric as a model of quantum gravity effects, with $\lambda_p$ the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order $\lambda_p$ frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity $v$ of a Gaussian with width $\sqrt{2\beta}$ initially centred at the origin changes its shape but its centre of mass moves with ${<x>\over<t>}=v(1+{\lambda_p^2\over 2\beta}+O(\lambda^3_p))$, an order $\lambda_p^2$ correction. This implies, at least within perturbation theory, that a `point particle' cannot be modelled as an infinitely sharp Gaussian due to quantum gravity corrections. For contrast, we also look at quantum geodesics on the noncommutative torus with a 2D curved weak quantum Levi-Civita connection.