Differentiable-Path Integrals in Quantum Mechanics

TL;DR

This paper introduces a new approach to quantum mechanics using differentiable path integrals, which modifies standard theory at small time scales and ensures convergence of certain quantities.

Contribution

It proposes restricting path integrals to differentiable paths with at least b1 derivatives, introducing parameters that modify quantum behavior at short time scales.

Findings

The model reproduces standard quantum mechanics at large time scales.

It renders divergent quantities finite through path restrictions.

The approach maintains unitarity with appropriate parameter choices.

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.