Non-Lipschitz approach to quantum mechanics

An attempt to reconcile quantum mechanics with Newtonʹs laws represented by the non-Lipschitz formalism has been made. As a proof-of-concept, a line of equally spaced atoms was studied. It appeared that enforcement of atom incompressibility required relaxation of the Lipschitz condition at the points of contact. This, in turn, led to fractional powers and discreteness of values of the basic parameters including energy and action, and finally, to the uncertainty relationship between positions and velocities. In addition to that, the relaxation of the Lipschitz condition caused instability of velocity with respect to small changes of the atom position, and that introduced an element of randomness in the system behavior. It was shown that the only model for the probability evolution which incorporates all the new properties of the motions is the Schrödinger equation. This means that quantum mechanics can be derived from Newtonʹs laws if an unnecessary mathematical restriction—the Lipschitz condition—is removed from the mathematical formalism. Non-local properties of the model, as well as spin-effects and relativistic corrections are discussed.