Is quantum mechanics based on an invariance principle?

Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through the requirement of an invariance principle under transformations that preserve the Heisenberg position-momentum inequality. These transformations acting on the position and momentum uncertainties are induced by isotropic space dilatations. This invariance imposes a change in the laws of classical mechanics that exactly corresponds to the transition to quantum mechanics. Space-time geometry is affected with possible consequences for quantum gravity