Deterministic Many-Interacting-Worlds Method

• The topic is a formulation of quantum phenomena where a finite ensemble of classical-like worlds interact deterministically, replacing the traditional wavefunction.
• It uses Newton-like equations and tailored interworld forces to approximate quantum potentials, achieving convergence to canonical quantum distributions.
• Numerical and analytical studies validate MIW for ground states and extend to excited states, offering a robust framework for finite-particle quantum approximations.

The deterministic Many-Interacting-Worlds method, usually abbreviated MIW, is a formulation of quantum phenomena in which one replaces a fundamental wavefunction by a large but finite ensemble of classical-like worlds whose configurations evolve deterministically under ordinary forces supplemented by an interworld interaction. In the original proposal, a world is an entire universe with a well-defined particle-and-field configuration, quantum effects are supposed to arise from mutual interaction of nearby worlds in configuration space, and the wavefunction is relegated to an emergent or secondary description in an appropriate continuum limit (Hall et al., 2014). Subsequent work recast the method as a rigorous finite-particle approximation problem for specific stationary states, especially for the one-dimensional harmonic oscillator, and established quantitative convergence of empirical world distributions to the corresponding quantum position laws (McKeague et al., 2016).

1. Foundational conception and ontological commitments

In the foundational MIW proposal, the basic state is a set of $N$ worlds,

${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$

where each ${\bf x}_n(t)$ is a point in configuration space $\mathbb R^K$, with $K=DJ$ for $J$ distinguishable spinless particles in $D$ spatial dimensions (Hall et al., 2014). Each world is therefore not a branch of a universal wavefunction in Everett’s sense, but a sharply defined classical-like universe with definite configuration. The multiverse dynamics is deterministic: every world has a definite trajectory, and probabilities are interpreted epistemically, as ignorance about which world an observer occupies (Hall et al., 2014).

This ontology differs from standard quantum mechanics, Bohmian mechanics, and Everettian many-worlds in a precise way. Standard quantum mechanics takes the wavefunction and measurement probabilities as fundamental. Bohmian mechanics has one actual configuration guided by a physically real $\Psi_t({\bf q})$. Everettian approaches treat worlds as branches or quasi-classical sectors of a universal wavefunction. MIW instead posits many equally real worlds, no fundamental pilot wave, and direct interworld influence in configuration space (Hall et al., 2014).

A closely related continuum-limit reformulation identifies a world with a Bohmian trajectory and replaces the finite ensemble by an uncountable continuum of worlds. In that framework, the density of worlds is $\rho_t(q)=|\Psi_t(q)|^2$, the totality of worlds is a world continuum, and probability is interpreted as normalized world measure rather than primitive chance (Boström, 2014). This continuum theory is not itself a finite-world MIW algorithm, but it provides a conceptual limiting picture for discrete MIW constructions.

2. Dynamical structure and interworld interaction

The general MIW equations of motion are Newton-like: $$m^k \ddot{x}_n^k(t) = f^k({\bf x}_n(t)) + r_N^k({\bf x}_n(t);{\bf X}_t),$$ with ${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$0 and an interworld force ${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$1 chosen to approximate the Bohm force in the large-${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$2 limit (Hall et al., 2014). In the conservative subclass,

${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$3

so the method has Hamiltonian evolution with

${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$4

The interworld potential is motivated by matching the positive quantum kinetic-density term ${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$5, so that ${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$6 acts as a discrete analogue of Bohm’s quantum potential (Hall et al., 2014).

For the one-particle one-dimensional ordered-world approximation, the foundational paper takes

${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$7

and derives

${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$8

with ${\bf X}_t=\{ {\bf x}_1(t),\dots,{\bf x}_N(t)\},$9, ${\bf x}_n(t)$0 (Hall et al., 2014). The interaction is singular and repulsive when neighboring worlds approach one another, which preserves ordering and produces a quantum-like exclusion effect in configuration space. The corresponding force depends effectively on up to five worlds, even though the potential is written in nearest-neighbor form (Hall et al., 2014).

In the harmonic-oscillator ground-state analyses that followed, equivalent conventions often reverse the ordering and write the Hamiltonian as

${\bf x}_n(t)$1

with

${\bf x}_n(t)$2

where ${\bf x}_n(t)$3 and ${\bf x}_n(t)$4 (McKeague et al., 2016). The determinism is the same; only the indexing convention changes.

3. Harmonic-oscillator ground state and rigorous validation

The benchmark success of deterministic MIW is the one-dimensional harmonic-oscillator ground state. In the original proposal, minimizing the finite-${\bf x}_n(t)$5 Hamiltonian gives a static equilibrium with vanishing momenta and a recursion for the ordered world positions. In dimensionless form, the stationary configuration satisfies

${\bf x}_n(t)$6

and the finite-${\bf x}_n(t)$7 ground-state energy is

${\bf x}_n(t)$8

(Hall et al., 2014). This already showed that a deterministic equilibrium of interacting worlds can reproduce the Gaussian quantum ground-state profile qualitatively and, for the energy, asymptotically.

Later work turned this into a precise convergence theorem. Writing

${\bf x}_n(t)$9

the unique monotonic zero-mean solution has symmetry $\mathbb R^K$0 and empirical distribution

$\mathbb R^K$1

that converges to the standard Gaussian (Chen et al., 2020). The strongest quantitative result gives optimal bounds in both Kolmogorov and Wasserstein distance: $\mathbb R^K$2 and

$\mathbb R^K$3

Thus the deterministic MIW ground-state approximation satisfies

$\mathbb R^K$4

and the $\mathbb R^K$5 factor is driven by the tail geometry, specifically by the extreme world position $\mathbb R^K$6 (Chen et al., 2020).

The proof architecture uses Stein’s method and zero-bias coupling. For the empirical law of the deterministic world positions, the zero-biased distribution is uniform on each interval between adjacent worlds, which makes the finite configuration analytically tractable (Chen et al., 2020). This line of work established one of the clearest rigorous confirmations of MIW: a static finite classical-world system whose empirical measure reproduces the harmonic-oscillator Born distribution with sharp finite-$\mathbb R^K$7 error control.

4. Higher-energy states, generalized interworld potentials, and Stein theory

The main obstacle beyond the ground state is nodal structure. Harmonic-oscillator eigenstate densities are

$\mathbb R^K$8

so for $\mathbb R^K$9 the density has internal zeros because Hermite polynomials vanish inside $K=DJ$0 (McKeague et al., 2016). Rather than trying to realize all higher states from one universal interworld potential, one line of work introduces a target-dependent interworld potential. For densities of the form

$K=DJ$1

the generalized MIW ansatz is

$K=DJ$2

For $K=DJ$3, this reduces to the ground-state potential; for the first excited state, $K=DJ$4, $K=DJ$5, and

$K=DJ$6

(McKeague et al., 2016).

The equilibrium recursion for the first excited state becomes

$K=DJ$7

and, for even $K=DJ$8, every zero-median solution satisfies zero mean, symmetry, and “Maxwell variance”

$K=DJ$9

There is a unique strictly decreasing zero-mean solution, and its empirical distribution converges to

$J$0

the two-sided Maxwell distribution, at rate

$J$1

(McKeague et al., 2016).

The conceptual novelty of that work is a $J$2-generalized zero-bias transformation. If $J$3, then $J$4 is a fixed point of the corresponding generalized biasing map; in particular, the excited-state oscillator laws arise as distributional fixed points (McKeague et al., 2016). The technical difficulty is that the standard Stein equation becomes singular at interior zeros such as $J$5 for $J$6. The remedy is a modified Stein framework based on a Stein kernel and a transformed differential equation, which allows one to control bounded quantities despite the singularity (McKeague et al., 2016).

A later extension constructs deterministic MIW sequences for all fixed higher harmonic-oscillator levels by distributing world particles across the nodal intervals determined by the zeros of the Hermite polynomial. For the $J$7-th state,

$J$8

there exists a unique MIW sequence with prescribed point counts in each positivity region, and its empirical measure converges in Wasserstein-1 distance at rate

$J$9

when the point counts match the nodal masses (Loomis et al., 2024). These constructions are not, for $D$0, exact critical points of the full discrete Hamiltonian; they are approximate excited states that become asymptotically stationary away from the nodes (Loomis et al., 2024). This distinction is central to the present status of deterministic MIW beyond the ground state.

5. Numerical implementations, non-Gaussian models, and higher-dimensional extensions

The original MIW paper already proposed a deterministic relaxation algorithm for stationary states: start from an arbitrary configuration, set velocities to zero, integrate the MIW equations over a short interval, replace the configuration by the updated one, reset velocities to zero again, and iterate until convergence (Hall et al., 2014). Later numerical work generalized this idea by reconstructing the world density with smooth kernels instead of nearest-neighbor finite differences. In the kernel-based formulation,

$D$1

with Gaussian $D$2, and the approximate quantum potential is

$D$3

The deterministic eigenstate solver integrates

$D$4

for short time steps while resetting velocities to zero after each iteration. This produces 1D ground states, 1D excited states with imposed nodes, and 2D ground states, with Voronoi-cell density estimates used in higher dimensions (Herrmann et al., 2017).

The same kernel-based direction was later applied to systems with singular points or non-smooth boundaries. A 2026 study replaces problematic potentials by asymptotically smooth surrogates, uses adaptive kernel density estimation inside the Bohm-like quantum potential, and again applies deterministic relaxation with the velocity Verlet algorithm. For the Coulomb singularity $D$5, the paper employs smooth models such as

$D$6

while a finite-depth well is smoothed by

$D$7

The reported simulations cover a 1D finite trap and 2D Coulomb ground and excited states, with results said to be consistent with matrix Numerov calculations (Chen et al., 28 May 2026).

A distinct non-Gaussian extension treats the one-dimensional Coulomb potential in the first excited state, with target density

$D$8

There the worlds are placed on the half-line $D$9, the empirical density is modified to

$\Psi_t({\bf q})$0

and the deterministic world positions satisfy

$\Psi_t({\bf q})$1

The paper proves convergence of the empirical density to $\Psi_t({\bf q})$2 on the half-line and reports numerical agreement with the exact quantum density for $\Psi_t({\bf q})$3 and $\Psi_t({\bf q})$4 (Chen et al., 2022). This does not develop full time-dependent MIW dynamics, but it shows that deterministic stationary-state constructions are not confined to Gaussian oscillator models.

6. Probability, continuum limits, and open problems

Probability in deterministic MIW is typically introduced as self-locating uncertainty: all worlds are equally real, but an observer does not know which world they occupy (Hall et al., 2014). In the continuum-of-worlds reformulation, this becomes measure-theoretic: the amount of worlds in a region $\Psi_t({\bf q})$5 is

$\Psi_t({\bf q})$6

and Born probabilities are normalized world measure rather than primitive probabilities (Boström, 2014). This shift from counting to measure is important because it avoids identifying probability with naïve discrete world-counting.

Foundational work on deterministic many-world theories strengthens that point. In one analysis, subjective probabilities are determined by a norm induced by the dynamics: the $\Psi_t({\bf q})$7-norm in Everettian quantum theory and the $\Psi_t({\bf q})$8-norm in Kent’s classical many-worlds model. The same work argues that objective probability should be identified with the proportion of worlds relative to an appropriate measure, and that frequency theorems require a product structure for that measure (Araújo, 2018). A related axiomatic treatment shows that, for a class of deterministic many-world theories, the natural world-weighting is $\Psi_t({\bf q})$9 in the quantum case, $\rho_t(q)=|\Psi_t(q)|^2$0 in a stochastic many-worlds case, and that no natural probability rule satisfying the axioms exists in a purely discrete copy-counting theory (Short, 2021). These papers do not analyze MIW directly, but they suggest that deterministic MIW needs a dynamically natural measure over worlds rather than bare finite counting.

Several limitations remain explicit across the MIW literature. Exact equivalence to quantum mechanics is not claimed for finite $\rho_t(q)=|\Psi_t(q)|^2$1; the target is the continuum limit (Hall et al., 2014). Rigorous convergence theory is strongest for the harmonic-oscillator ground state and, in modified or approximate forms, for fixed excited states of the oscillator (McKeague et al., 2016). The original universal-potential program for higher oscillator levels appeared analytically unworkable, which is why later papers adopt target-dependent interworld potentials or approximate excited-state recursions (McKeague et al., 2016). Numerical higher-dimensional implementations still require kernel smoothing, boundary devices, or prescribed nodes, and they remain sensitive to bandwidth choice, time step, and finite-region truncation (Herrmann et al., 2017). The singular-potential extension itself states no proof of a limit of the form

$\rho_t(q)=|\Psi_t(q)|^2$2

and therefore remains empirical rather than rigorous (Chen et al., 28 May 2026).

The deterministic Many-Interacting-Worlds method is therefore best characterized as both a realist ontology and a family of finite-particle approximation schemes. Its most secure achievements are the construction of deterministic equilibrium configurations whose empirical measures converge to quantum stationary distributions in a set of benchmark problems. Its unresolved questions concern general time-dependent accuracy, higher-dimensional many-body systems, phase-sensitive phenomena, nodal structures in excited states, spin, entanglement, and the status of a universal interworld interaction that would reproduce a broad sector of quantum theory without target-specific modification (Hall et al., 2014).