Quantum geodesics on quantum Minkowski spacetime

We apply a recent formalism of quantum geodesics to the well-known quantum Minkowski spacetime [x(i), t] = t(lambda p)chi(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 root 2 beta initially centred at the origin changes its shape but its centre of mass moves with < x >/< t >= v(1 + 3 lambda(2)(p)/2 beta+ O(lambda(3)(p))), an order lambda(2)(p) 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.