Discrete-Time Path Distributions on Hilbert Space

TL;DR

This paper constructs a discrete-time path distribution on a Hilbert space to represent the kinetic part of the Feynman path integral, extending previous work to a more general setting and establishing its well-definedness for smooth potentials.

Contribution

It introduces a new construction of path distributions on Hilbert spaces for discrete-time Feynman integrals, broadening the mathematical framework beyond sequence spaces.

Findings

Path distribution on Hilbert space constructed

Discrete-time Feynman integral shown to be well-defined

Applicable to various boundary conditions and smooth potentials

Abstract

We construct a path distribution representing the kinetic part of the Feynman path integral at discrete times similar to that defined by Thomas [1], but on a Hilbert space of paths rather than a nuclear sequence space. We also consider different boundary conditions and show that the discrete-time Feynman path integral is well-defined for suitably smooth potentials.