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

TL;DR

This paper examines hydrodynamic reformulations of quantum mechanics, highlighting a structural problem called sparse ontology, and suggests that a continuous ontology may be necessary for these models to succeed.

Contribution

It reviews hydrodynamic and MIW formalisms, identifies the sparse ontology problem, and argues for the need of an essentially continuous ontology in quantum models.

Findings

Discrete hydrodynamic models face a structural difficulty called sparse ontology.

Wavefunction branching leads to thinning of fluid components in configuration space.

Continuous ontologies may be necessary for successful hydrodynamic quantum models.

Abstract

Hydrodynamic reformulations of the Schr\"odinger 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.