Non-lipschitz Approach to Quantum Mechanics

An attempt to reconcile quantum mechanics with Newton’s 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, leads 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 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 motion, is the Schrodinger equation. This means that quantum mechanics can be derived from the Newton’s laws if an unnecessary mathematical restriction — the Lipschitz condition — is removed from the mathematical formalism.