Stein's method, many interacting worlds and quantum mechanics

TL;DR

This paper explores new connections between Stein's method and Many Interacting Worlds theory, demonstrating how quantum position densities for higher energy levels can be characterized as fixed points and establishing convergence rates to classical distributions.

Contribution

It introduces a novel generalization of Stein's method for quantum states beyond the ground level, linking it to MIW theory and providing convergence results.

Findings

Quantum position densities for excited states are fixed points of a generalized Stein's method.

Established a rate of convergence to Maxwell distribution for particle positions.

Developed new techniques for Stein's method with singular solutions.

Abstract

Hall, Deckert and Wiseman (2014) recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical "worlds." A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall, Deckert and Wiseman (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we propose new connections between Stein's method and Many Interacting Worlds (MIW) theory. In particular, we show that quantum position probability densities for higher energy levels beyond the ground state arise as distributional fixed points in a new generalization of Stein's method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual "density approach" Stein solution (see Chatterjee and Shao (2011)) has a singularity.