Renormalization group in quantum mechanics

We establish the renormalization group (RG) equation for the running action in the context of a one-quantum-particle system. This equation is deduced by integrating one Fourier mode after another in the path integral formalism. It is free from the well known pathologies which appear in quantum field theory due to the sharp cutoff. We show that for an arbitrary background path the usual local form of the action is not preserved by the flow. To cure: this problem we consider a more general action than usual, which is stable in the RG flow. This allows us to obtain a new consistent RG equation for the action.