Quantum geodesics on quantum Minkowski spacetime

We apply a recent formalism of quantum geodesics to the well-known quantum Minkowski spacetime [x , t] = ıλ x with its flat quantum metric as a model of quantum gravity effects, with λ the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order λ frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity v of a Gaussian with width 2 β initially centred at the origin changes its shape but its centre of mass moves with ⟨ x ⟩ ⟨ t ⟩ = v ( 1 + 3 λ p 2 2 β + O ( λ p 3 ) ) , an order λ 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.