Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.1563 · The Annals of Applied Probability

Convergence of empirical distributions in an interpretation of quantum mechanics

Ian W. McKeague, Bruce Levin

Published Aug 1, 2016• 11 citations
arXivHTMLPDF

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 [Phys. Rev. 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 Schrödinger’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.

DOI
10.1214/15-aap1154

Related papers

Stein's method, many interacting worlds and quantum mechanics

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

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.

arXiv:1606.06618•Jun 21, 2016•2 citations

Stein's method, many interacting worlds and quantum mechanics

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

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.

Bernoulli•Jan 1, 2017•2 citations

An extension of many-interacting-worlds method on non-Guassian model

Wen Chen, An Min Wang

Discussions about whether quantum theory is determinism or indeterminism has lasted for a century. A new approach to standard quantum mechanics called many-interacting-worlds method based on many-worlds interpretation and de Broglie–Bohm mechanics provided the possibility to demonstrate probability from deterministic universe. The many-interacting-worlds method has been proved successful in the ground state of harmonic oscillator. In this article we extend this method to one dimensional Coulomb potential and construct a corresponding empirical density function. We also provide a theoretical proof of the convergence of density function. Our numerical simulation of one dimensional Coulomb potential in the first excited state obtains the consistent result with standard quantum mechanics and shows the applicability of many-interacting-worlds method. This research provides the possibility to extend many-interacting-worlds method to non-Gaussian quantum systems.

arXiv:2212.09020•Results in Physics•Jan 30, 2023•1 citation

Stein’s method and approximating the quantum harmonic oscillator

Ian W. McKeague, Erol A. Peköz, Yvik C. Swan

Hall et al. (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 et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may 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.

Bernoulli•Dec 12, 2018•8 citations

Many-Worlds and Schrodinger’s First Quantum Theory

Valia Allori, Sheldon Goldstein, Roderich Tumulka +1 more

Schrodinger’s first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schrodinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully this theory is seen to be an empirically adequate many-worlds theory and not an empirically inadequate theory describing a single world. Moreover, this formulation—Schrodinger’s first quantum theory—can be regarded as a formulation of the many-worlds view of quantum mechanics that is ontologically clearer than Everett’s.

•Jan 1, 2009•122 citations

A Bird’s-Eye View on a New Stochastic Interpretation of Quantum Mechanics

Olavo Leopoldino da Silva Filho, Marcello Caio Ramon e Barros Ferreira

Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.

Mathematics•Nov 6, 2025•3 citations
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 [Phys. Rev. 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 Schrödinger’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.
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.
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.
Discussions about whether quantum theory is determinism or indeterminism has lasted for a century. A new approach to standard quantum mechanics called many-interacting-worlds method based on many-worlds interpretation and de Broglie–Bohm mechanics provided the possibility to demonstrate probability from deterministic universe. The many-interacting-worlds method has been proved successful in the ground state of harmonic oscillator. In this article we extend this method to one dimensional Coulomb potential and construct a corresponding empirical density function. We also provide a theoretical proof of the convergence of density function. Our numerical simulation of one dimensional Coulomb potential in the first excited state obtains the consistent result with standard quantum mechanics and shows the applicability of many-interacting-worlds method. This research provides the possibility to extend many-interacting-worlds method to non-Gaussian quantum systems.
Hall et al. (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 et al. (2014) and proven by McKeague and Levin (2016) using Stein's method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may 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.
Schrodinger’s first proposal for the interpretation of quantum mechanics was based on a postulate relating the wave function on configuration space to charge density in physical space. Schrodinger apparently later thought that his proposal was empirically wrong. We argue here that this is not the case, at least for a very similar proposal with charge density replaced by mass density. We argue that when analyzed carefully this theory is seen to be an empirically adequate many-worlds theory and not an empirically inadequate theory describing a single world. Moreover, this formulation—Schrodinger’s first quantum theory—can be regarded as a formulation of the many-worlds view of quantum mechanics that is ontologically clearer than Everett’s.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.