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 $C^\alpha$, by only allowing paths which possess at least $\alpha$ derivatives. The method introduces two external parameters, and induces the appearance of a particular time scale $\epsilon_D$ such that for time intervals longer than $\epsilon_D$ the model behaves as usual quantum mechanics. However, for time scales smaller than $\epsilon_D$, modifications to standard formulation of quantum theory occur. This restriction renders convergent some quantities which are usually divergent in the time-continuum limit $\epsilon\rightarrow 0$. We illustrate the model by computing several meaningful physical quantities such as the mean square velocity $\langle v^2 \rangle $, the canonical commutator, the Schrodinger equation and the energy levels of the harmonic oscillator. It is shown that an adequate choice of the parameters introduced makes the evolution unitary.