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On Hydrodynamic Formulations of Quantum Mechanics and the Problem of Sparse Ontology

Aric Hackebill, Bill Poirier

Abstract

Hydrodynamic reformulations of the Schrödinger equation suggest an interpretation of quantum mechanics in terms of a fluid flowing on configuration space. In the discrete hydrodynamic view, this fluid is not fundamental but emerges from many underlying microscopic fluid components whose collective behavior reproduces quantum phenomena. The most developed realization of this idea is the discrete many interacting worlds (MIW) framework, in which discrete particle-like worlds interact via inter-world forces and quantum probabilities are grounded in direct world counting. But there is also an older, continuous version of MIW. After reviewing the hydrodynamic and MIW formalisms, and emphasizing some of their interpretational advantages over the Everettian Many Worlds and Bohmian approaches, we argue that all discrete hydrodynamic models face a generic structural difficulty, which we call the problem of sparse ontology. Because wavefunctions typically branch under decoherence, the discrete components of the fluid are repeatedly partitioned into sub-ensembles, thereby thinning their density in configuration space and driving the dynamics away from the quantum regime once the components become sufficiently sparse. We conclude that successful hydrodynamic completions of quantum mechanics plausibly require an essentially continuous ontology.

On Hydrodynamic Formulations of Quantum Mechanics and the Problem of Sparse Ontology

Abstract

Hydrodynamic reformulations of the Schrödinger equation suggest an interpretation of quantum mechanics in terms of a fluid flowing on configuration space. In the discrete hydrodynamic view, this fluid is not fundamental but emerges from many underlying microscopic fluid components whose collective behavior reproduces quantum phenomena. The most developed realization of this idea is the discrete many interacting worlds (MIW) framework, in which discrete particle-like worlds interact via inter-world forces and quantum probabilities are grounded in direct world counting. But there is also an older, continuous version of MIW. After reviewing the hydrodynamic and MIW formalisms, and emphasizing some of their interpretational advantages over the Everettian Many Worlds and Bohmian approaches, we argue that all discrete hydrodynamic models face a generic structural difficulty, which we call the problem of sparse ontology. Because wavefunctions typically branch under decoherence, the discrete components of the fluid are repeatedly partitioned into sub-ensembles, thereby thinning their density in configuration space and driving the dynamics away from the quantum regime once the components become sufficiently sparse. We conclude that successful hydrodynamic completions of quantum mechanics plausibly require an essentially continuous ontology.
Paper Structure (37 sections, 53 equations, 4 figures)

This paper contains 37 sections, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Hydrodynamic View
  3. DMIW in Detail
  4. Dynamics and Locality
  5. Translating to the Standard Wavefunction Picture
  6. Measurement Statistics
  7. Connection with the Broader DHV
  8. A Simple System that Creates Branch Splitting
  9. The Wavefunction Approach
  10. The CHV Approach
  11. The DMIW Approach
  12. The Sparse World Argument
  13. Generalizing the Argument
  14. Why World Thinning Is Unavoidable
  15. Non-Splitting World Dynamics
  16. ...and 22 more sections

Figures (4)

  • Figure 1: Taxonomy of hydrodynamic interpretations with representative examples.
  • Figure 2: The hydrodynamic density, representing the Madelung fluid, is depicted by a blue spherical region to illustrate its localization around a small area (idealized here for clarity; in reality, the density is Gaussian). As the system passes through the analyzer, the density splits along the $z$-axis into two branches corresponding to the spin components. Overlaid are typical DMIW trajectories: the dashed pink line traces the path of our world, while the solid blue lines represent trajectories of other worlds in the ensemble. The worlds separate into two sub-ensembles displaced along $z$, mirroring the splitting of the hydrodynamic density.
  • Figure 3: Schematic of the concatenated SG sequence used to illustrate repeated branching. The dashed pink line indicates the trajectory of our world, while the solid blue lines trace the trajectories of other worlds in the discrete world ensemble. The blue tube depicts the high-density region of the hydrodynamic distribution $\rho(\vec{x},t)=\lvert\psi(\vec{x},t)\rvert^2$ from standard quantum theory, which branches over time, but remains continuous (in spacetime). As branching proceeds, discrete trajectories diverge and become sparse relative to this persistent density, illustrating the onset of sparse ontology.
  • Figure 4: Normalized histogram plots of the interference patterns resulting from $2\times10^4$ runs for (a) 1, (b) 2, (c) 3, (d) 5, and (e) 10 DMIW worlds passing through a double-slit barrier of slit width 0.01 mm (dotted red ) at a time. Panel (f) shows the standard pilot-wave prediction, consistent with quantum mechanics.