Convergence of empirical distributions in an interpretation of quantum mechanics

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 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.

Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising

“…More generally, however, for and the interworld potential leads to forces on each world that act to reproduce quantum phenomena such as Ehrenfest’s theorem, spreading of wave packets, tunneling through a barrier, and interference effects [ 29 ]. For the case of a 1D oscillator, , it has further been shown, in the limit , that the average energy per world of the MIW ground state converges to the quantum groundstate energy [ 29 ], and that the corresponding stationary distribution of worlds samples the usual quantum Gaussian probability distribution [ 29 , 39 ].…”

Stein’s method and approximating the quantum harmonic oscillator

“…In the ground state, there is no movement because all the momenta p n have to vanish for the total energy to be minimized. In this case, as mentioned above, Hall et al (2014) showed that the particle locations x n satisfy (1) and McKeague and Levin (2016) showed that the empirical distribution tends to a standard Gaussian distribution.…”

“… 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. …”

“…Hall et al conjectured that the empirical distribution of these locations converges to Gaussian as the total number of worlds N increases. McKeague and Levin (2016) recently proved such a result and provided a rate of convergence. More specifically, McKeague and Levin showed that if x 1 , … x N is a decreasing, zero-mean sequence of real numbers satisfying the recursion relation xn+1=xn−1x1+⋯+xn, then the empirical distribution of the x n tends to standard Gaussian when N → ∞.…”

A general parametric Stein characterization

“…in network analysis (Franceschetti and Meester, 2006), random matrices (Mackey et al, 2014) or even quantum physics (McKeague and Levin, 2015))…”