Convergence of empirical distributions in an interpretation of quantum mechanics

TL;DR

This paper proves that in a many-interacting-worlds model of quantum mechanics, the particle configurations become Gaussian as the number of worlds increases, and the resampling process converges to an Ornstein-Uhlenbeck process, aligning with quantum solutions.

Contribution

It provides a rigorous proof that the particle distribution converges to the quantum ground state and the resampling process approaches a quantum-like stochastic process.

Findings

Particle configurations become asymptotically Gaussian.

Resampling process converges to Ornstein-Uhlenbeck process.

Model aligns with stationary and time-dependent quantum solutions.

Abstract

From its beginning, there have been attempts by physicists to formulate quantum mechanics without requiring the use of wave functions. An interesting recent approach takes the point of view that quantum effects arise solely from the interaction of finitely many classical "worlds." The wave function is then recovered (as a secondary object) from observations of particles in these worlds, without knowing the world from which any particular observation originates. Hall, Deckert and Wiseman [Physical Review X 4 (2014) 041013] have introduced an explicit many-interacting-worlds harmonic oscillator model to provide support for this approach. In this note we provide a proof of their claim that the particle configuration is asymptotically Gaussian, thus matching the stationary ground-state solution of Schroedinger's equation when the number of worlds goes to infinity. We also construct a Markov chain based on resampling from the particle configuration and show that it converges to an Ornstein-Uhlenbeck process, matching the time-dependent solution as well.