Feynman path integrals for discrete-variable systems: Walks on Hamiltonian graphs

TL;DR

This paper introduces a discrete-variable formulation of Feynman path integrals as walks on Hamiltonian graphs, bridging discrete and continuous quantum systems and enabling new computational approaches.

Contribution

It presents a novel, parameter-free discrete formulation of Feynman path integrals using graph walks, connecting discrete models to continuous quantum systems.

Findings

Derived expressions for partition functions and transition amplitudes

Recovered continuous Feynman path integrals in the continuum limit

Established a graph-based framework for discrete quantum systems

Abstract

We propose a natural, parameter-free, discrete-variable formulation of Feynman path integrals. We show that for discrete-variable quantum systems, Feynman path integrals take the form of walks on the graph whose weighted adjacency matrix is the Hamiltonian. By working out expressions for the partition function and transition amplitudes of discretized versions of continuous-variable quantum systems, and then taking the continuum limit, we explicitly recover Feynman's continuous-variable path integrals. We also discuss the implications of our result.