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Theory: many interacting worlds

The many-interacting-worlds simulation implements the proposal of Hall, Deckert and Wiseman [2014]: quantum mechanics as the statistics of N classical worlds that repel each other. This document gives the equations with the notation of js/model.js, lists the exact results the simulator checks against, and separates published theory from the simulator's working choices. Usage and parameters: many-interacting-worlds-simulation-guide.en.md; Italian version: many-interacting-worlds-theory.it.md.

1. From the wave function to trajectories

Madelung [1927] rewrote the Schrödinger equation as a fluid: with ψ = √ρ · exp(iS/ħ), the density ρ and the velocity field v = ∇S/m obey classical hydrodynamics plus one extra term, the quantum potential

Q = -(ħ²/2m) · (∂²√ρ/∂x²) / √ρ

Bohm [1952] read the same equations as mechanics: point particles follow trajectories guided by ψ, and every quantum effect enters through the force -∂Q/∂x. In both readings the wave function remains a fundamental object.

The many interacting worlds (MIW) approach drops it. Take N worlds, each a classical particle with a definite position x[n] and velocity v[n], and let ρ be approximated by the empirical density of the worlds themselves. Q then becomes an inter-world potential: each world feels its neighbours, and the force that in Bohm's theory came from ψ now comes from a repulsion between nearby worlds. The wave function, if one still wants it, is an emergent description of the ensemble; nothing in the dynamics refers to it. Precursors with a continuum of trajectories are Poirier [2010] and Schiff and Poirier [2012]; the finite-N theory with its toy force is Hall, Deckert and Wiseman [2014]; the "mass density of worlds" reading is discussed by Sebens [2015] and Boström [2015].

2. The model in one dimension

Units ħ = m = 1. The N worlds keep a fixed order x[0] < x[1] < ... < x[N-1], with gaps g[n] = x[n+1] - x[n], inverse gaps s[n] = 1/g[n] (s[-1] = s[N-1] = 0) and differences D[n] = s[n] - s[n-1]. The inter-world potential used by HDW and by model.js is

U = (1/8) · Σ_n D[n]²

which is the discrete form of the average of Q over equally weighted worlds. Its gradient is exact and local (only nearest gaps enter), so forces cost O(N):

dU/dx[m] = (1/4) · ( D[m]·(s[m]² + s[m-1]²) - D[m+1]·s[m]² - D[m-1]·s[m-1]² )

with out-of-range D treated as zero. Each world then follows Newton's equation in the external potential V(x) = ω²x²/2 (ω = 0 for a free particle) plus this repulsion:

a[m] = -dU/dx[m] - ω²·x[m]

The repulsion diverges as 1/g³ when two worlds approach, so worlds never cross and the ordering is preserved for all times. With N = 1 the potential U vanishes identically: a single world is purely classical. Quantum behaviour is a collective property of the ensemble, not of any one world.

3. Exact results at finite N

These are theorems of the toy model, not large-N approximations, and the simulator displays each one next to the measured value.

Stationary oscillator. A configuration at rest is stationary when dU/dx[m] = -ω²x[m] for every m, which reduces to D[n] = -2ω·x[n] and hence to the recurrence s[n] = s[n-1] - 2ω·x[n] closed by s[N-1] = 0. groundState solves it by shooting on the leftmost world. The configuration satisfies the sum rule

Σ_n x[n]² = (N-1)/(2ω)

Zero-point energy. The energy of that configuration is exactly

E = (N-1)·ω/2

which tends to the quantum ground state energy ω/2 per world as N grows, and is zero for N = 1: the zero-point energy is the energy stored in the repulsion.

Uncertainty relation. Define the momentum spread of the ensemble as Δp² = Var(v) + 2U/N. The second term plays the role of the osmotic (quantum) part of the momentum in stochastic and Bohmian mechanics: it is the part stored in the inter-world interaction. For the stationary oscillator the product is exactly

Δx·Δp = (1 - 1/N)·ħ/2

approaching the Heisenberg bound ħ/2 from below as N grows, and vanishing for N = 1 [Hall, Deckert and Wiseman 2014, eq. 22].

Free spreading. Switching the trap off (ω = 0) at t = 0 from the stationary configuration gives the exact solution xn = xn·√(1+ω²t²), the same law as the width of a quantum Gaussian packet. The repulsion converts U into kinetic energy, exactly as the quantum packet converts Q into spreading.

Interference. Two packets launched against each other do not pass through one another (worlds cannot cross): the repulsion channels the worlds into bands of alternating high and low density with spacing close to πħ/p, the analogue of the fringes of the two-packet interference pattern. This result is numerical, not closed form, and reproduces the behaviour reported by HDW.

4. Measurement: collapse as conditioning

MIW has no collapse postulate. All N worlds are equally real; an observer lives in one of them and sees the definite position of that world. A position measurement with resolution δ does two things:

So the "collapse as intersection of realities" intuition holds in one precise sense: the post-measurement state is the selection of the sub-ensemble of worlds that agree with the record, and Δx collapses from the width of the whole ensemble to at most δ. It is never a fusion: no two worlds ever become one, and the discarded worlds go on existing. This mechanism is different from the branch convergence explored in the many-worlds convergence simulation, where distinct histories merge into one node and amplitudes add: here nothing merges and nothing interferes at the measurement, the ensemble is only split.

The squeezing has a cost: conditioning leaves the kept worlds packed at distances of order δ with a large interaction energy U, which immediately reconverts into spread. That is the Heisenberg kick of a sharp position measurement, visible in the simulator as the kept bin exploding outwards.

5. What the toy model leaves out

Bibliography

Hydrodynamic and pilot-wave formulations

Trajectory ensembles without a wave function

Many interacting worlds