Differentiable-path integrals in quantum mechanics

Abstract

A method is presented which restricts the space of paths entering the path integral of quantum mechanics to subspaces of Cα, by only allowing paths which possess at least α derivatives. The method introduces two external parameters, and induces the appearance of a particular time scale ϵD such that for time intervals longer than ϵD the model behaves as usual quantum mechanics. However, for time scales smaller than ϵD, modifications to standard formulation of quantum theory occur. This restriction renders convergent some quantities which are usually divergent in the time-continuum limit ϵ → 0. We illustrate the model by computing several meaningful physical quantities such as the mean square velocity 〈v2〉, the canonical commutator, the Schrödinger equation and the energy levels of the harmonic oscillator. It is shown that an adequate choice of the parameters introduced makes the evolution unitary.