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arXiv:2407.11231 · Physical Review Research

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

Amir Kalev, Itay Hen

Published Feb 27, 2025• 3 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevresearch.7.013220

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In this paper, we construct a ppp-adic path integral via ppp-adic multiple integrals. This integral describes the evolution of a wave function Ψ(x)Ψ(x)Ψ(x), defined as a map from a domain in Cp\mathbb{C}_pCp​ to Cp\mathbb{C}_pCp​. Unlike the standard ppp-adic quantum mechanics formulated for functions Qp→C\mathbb{Q}_p \to \mathbb{C}Qp​→C, our construction works directly on Cp\mathbb{C}_pCp​-valued wave functions. Using ppp-adic Dirac delta measures and ppp-adic additive characters, we define the propagator as a finite-partition multiple integral, which allows explicit iterative integration. As an application, we compute the Feynman propagator for free particles and obtain a closed-form expression analogous to the classical counterpart. Our method provides a discrete, constructive alternative that complements the existing distribution-theoretic frameworks for ppp-adic functional integration.