Brownian Motion and Finite Approximations of Quantum Systems over Local Fields

TL;DR

This paper establishes a stochastic approach to approximate quantum systems over local fields, demonstrating convergence of finite models to the full system and extending classical methods to a non-Archimedean context.

Contribution

It provides a novel stochastic proof of finite approximability for Schrödinger operators over local fields, including a Feynman-Kac formula and convergence results.

Findings

Brownian motion over local fields can be approximated by finite random walks

Finite propagators converge to the infinite system propagator

Feynman-Kac formula holds for finite quantum systems over local fields

Abstract

We give a stochastic proof of the finite approximability of a class of Schr\"odinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.