Quantum geodesics on quantum Minkowski spacetime

Abstract

We apply a recent formalism of quantum geodesics to the well-known quantum Minkowski spacetime [x i , t] = ıλ p x i with its flat quantum metric as a model of quantum gravity effects, with λ p the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order λ p frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity v of a Gaussian with width initially centred at the origin changes its shape but its centre of mass moves with , an order 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.