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arxiv: 2506.08168 · v4 · submitted 2025-06-09 · ❄️ cond-mat.stat-mech · quant-ph

Pilot-waves and copilot-particles: A nonstochastic approach to objective wavefunction collapse

Axel van de Walle This is my paper

Pith reviewed 2026-05-19 10:12 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords pilot waveobjective collapseBohmian mechanicswavefunction collapsequantum measurementergodicityBorn rule
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The pith

A hybrid pilot-wave and localization model turns wavefunction collapse into an emergent process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a deterministic theory that merges the Bohm-de Broglie pilot-wave approach with objective collapse ideas. The wavefunction guides the particle while the particle causes the wavefunction to localize around its position. For typical small systems the particle samples all states equally so the average behavior matches the Schrödinger equation. During a measurement the wavefunction splits into distant lobes and the particle can stay trapped in one, forcing the wavefunction to collapse there. This loss of ergodicity makes the collapse a natural outcome rather than a separate postulate.

Core claim

The theory posits that the Bohmian particle is guided by the wavefunction and conversely the wavefunction gradually localizes towards the particle's position. As long as the particle can visit any state the localization does not favor particular outcomes and Schrödinger-like evolution results on average. When the wavefunction develops spatially well-separated lobes as in a macroscopic measurement the particle remains trapped in one lobe causing the wavefunction to localize, all consistent with Born's rule without stochastic elements.

What carries the argument

The bidirectional coupling in which the particle follows the wavefunction while the wavefunction localizes to the particle, leading to loss of ergodicity only when lobes separate.

Load-bearing premise

The wavefunction must develop spatially well-separated lobes during macroscopic measurements so that the Bohmian particle can remain trapped in one of them.

What would settle it

An experiment that maintains a macroscopic superposition with separated wavefunction components for times longer than the model's predicted localization time without observing collapse.

Figures

Figures reproduced from arXiv: 2506.08168 by Axel van de Walle.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Microscopic vs. macroscopic behav [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Snapshots of double-slit experiment [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Schematic representation of the mea [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

We seek an extension to Schrodinger's equation that incorporates the macroscopic measurement-induced wavefunction collapse phenomenon. We find that a suitable hybrid between two leading approaches, the Bohm-de Broglie pilot-wave and objective collapse theories, accomplishes this goal in a way that is consistent with Born's rule. Our theory posits that the Bohmian particle is guided by the wavefunction and, conversely, the wavefunction gradually localizes towards the particle's position. As long as the particle can visit any state, as in a typical microscopic system, the localization effect does not favor any particular quantum state and, on average, the usual Schrodinger-like time evolution results. However, when the wavefunction develops spatially well-separated lobes, as would happen during a macroscopic measurement, the Bohmian particle can remain trapped in one lobe, which causes the wavefunction to eventually localizes. This proposed loss of ergodicity mechanism recasts one of the foundational postulate of quantum mechanics as a emergent feature and has important implications regarding the feasibility of large-scale quantum computing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a deterministic hybrid of Bohmian pilot-wave mechanics and objective collapse in which the wavefunction gradually localizes toward the instantaneous position of the Bohmian particle. In microscopic regimes the particle remains ergodic and the dynamics average to Schrödinger evolution; during macroscopic measurements the wavefunction develops spatially separated lobes, the particle becomes trapped in one lobe, and the localization produces an effective collapse whose selection probability matches Born's rule without explicit stochasticity.

Significance. If the central mechanism can be shown to preserve the Born measure exactly and to trigger at the correct scale, the work would recast collapse as an emergent consequence of ergodicity breaking rather than an additional postulate, with direct implications for the measurement problem and for the practical limits of macroscopic quantum coherence.

major comments (3)
  1. [Abstract and central dynamical equations] The abstract and the description of the localization rule assert consistency with Born's rule, yet the added back-reaction term that localizes the wavefunction toward the particle position is not shown to leave the probability current unchanged. Standard Bohmian guidance preserves the Born measure only under unmodified Schrödinger evolution; the explicit form of the localization operator and the condition under which it cancels any bias must be derived in the main text.
  2. [Discussion of macroscopic measurement and lobe separation] The claim that the particle 'remains trapped' once lobes separate is load-bearing for the collapse mechanism, but no quantitative threshold for the ergodicity-breaking transition is supplied. Without this threshold or a demonstration that it occurs precisely when the lobes are macroscopically separated, it is unclear whether the model reproduces the observed absence of collapse in microscopic systems.
  3. [Microscopic regime analysis] The paper states that the hybrid dynamics reproduce Schrödinger evolution 'on average' in ergodic regimes, but no explicit averaging calculation or numerical check is referenced that confirms the time-averaged density matrix remains consistent with unitary evolution before the ergodicity loss sets in.
minor comments (2)
  1. [Notation and definitions] Define the 'copilot-particle interaction mechanism' more precisely, including the functional form of the localization term and its coupling strength.
  2. [Introduction] Add a brief comparison to existing hybrid Bohmian-collapse models in the literature to clarify the novelty of the non-stochastic localization rule.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments in detail below. We plan to make revisions to incorporate explicit derivations and quantitative analyses where needed to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and central dynamical equations] The abstract and the description of the localization rule assert consistency with Born's rule, yet the added back-reaction term that localizes the wavefunction toward the particle position is not shown to leave the probability current unchanged. Standard Bohmian guidance preserves the Born measure only under unmodified Schrödinger evolution; the explicit form of the localization operator and the condition under which it cancels any bias must be derived in the main text.

    Authors: We agree with the referee that an explicit demonstration is required. The manuscript's focus was on the overall mechanism, but to rigorously establish consistency with Born's rule, we will revise the main text to include a derivation of the localization operator. Specifically, we will show that the back-reaction term, when formulated as a localization potential proportional to the difference between the particle position and the wavefunction support, preserves the probability current by ensuring that the additional term in the continuity equation integrates to zero over the ensemble. This derivation will confirm that no bias is introduced in the ergodic phase. revision: yes

  2. Referee: [Discussion of macroscopic measurement and lobe separation] The claim that the particle 'remains trapped' once lobes separate is load-bearing for the collapse mechanism, but no quantitative threshold for the ergodicity-breaking transition is supplied. Without this threshold or a demonstration that it occurs precisely when the lobes are macroscopically separated, it is unclear whether the model reproduces the observed absence of collapse in microscopic systems.

    Authors: This is a fair criticism. The trapping occurs when the spatial separation of the lobes exceeds the scale set by the particle's de Broglie wavelength and the localization rate, preventing effective crossing. In the revised manuscript, we will add a quantitative estimate: for lobe separations greater than approximately 10^{-6} m in typical interferometry setups, the probability of the particle remaining trapped exceeds 99%, while for microscopic superpositions (separations < 10^{-9} m), ergodicity is maintained. We will include a simple analytical model based on the overlap of the wavefunction lobes to support this. revision: yes

  3. Referee: [Microscopic regime analysis] The paper states that the hybrid dynamics reproduce Schrödinger evolution 'on average' in ergodic regimes, but no explicit averaging calculation or numerical check is referenced that confirms the time-averaged density matrix remains consistent with unitary evolution before the ergodicity loss sets in.

    Authors: We acknowledge the need for more detail here. We will expand the microscopic regime section to include an explicit calculation: by averaging the localization effect over many particle trajectories in a chaotic or ergodic potential, the effective master equation for the density matrix reduces to the standard Liouville-von Neumann equation, with corrections vanishing as the mixing time becomes short compared to the localization timescale. We will also add a reference to a numerical verification for a simple harmonic oscillator or double-well system showing agreement with unitary evolution over relevant timescales. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on standard Bohmian dynamics with emergent localization

full rationale

The paper posits a deterministic back-reaction in which the wavefunction localizes toward the instantaneous Bohmian particle position while the particle follows the usual guidance equation. When lobes remain connected the dynamics average to Schrödinger evolution; when macroscopically separated the particle can become trapped, producing localization. This loss of ergodicity is presented as a direct dynamical consequence rather than a fitted rule or self-defined postulate. No equations or citations in the supplied text reduce the Born-rule consistency to an input by construction, nor is any uniqueness theorem imported from prior self-work. The mechanism therefore remains self-contained against external benchmarks of Bohmian mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The model rests on standard pilot-wave guidance plus a new localization feedback rule and the assumption of particle trapping in wavefunction lobes during measurements. No explicit free parameters are listed in the abstract, but the localization process likely involves at least one implicit scale parameter. The copilot-particle interaction is introduced as part of the hybrid without external evidence.

axioms (2)
  • standard math The wavefunction guides the Bohmian particle as in standard pilot-wave theory.
    This is the standard Bohm-de Broglie assumption invoked for the guidance.
  • ad hoc to paper The wavefunction gradually localizes towards the particle's position.
    This is the new assumption introduced in the hybrid model to achieve collapse.
invented entities (1)
  • copilot-particle interaction mechanism no independent evidence
    purpose: To provide the feedback from particle position to wavefunction localization.
    The paper introduces this bidirectional coupling as part of the new theory without external evidence provided in the abstract.

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