Hydrodynamic Quantum Mechanics: Wave Function as Superfluid Phase

This paper presents a hydrodynamic interpretation of nonrelativistic quantum mechanics within the Hydrodynamic Quantum Gravity (HQG) programme. The central result is that the standard Schrödinger formalism can be rewritten exactly as a continuity equation for a density field plus an Euler-like momentum equation containing an additional dispersive term (“quantum potential”). In this reading, quantum behaviour is modelled as effective fluid dynamics of a coherent phase field. Interpretation note: “superfluid” language is used here as an effective mathematical description of vacuum behaviour. The claim is not that space is a material fluid or that there exists a preferred rest frame. Rather, the point is that some aspects of vacuum dynamics can be modelled using the same mathematics that describes superfluids, while remaining consistent with operational Lorentz invariance and purely relational motion. Key elements discussed: Madelung decomposition and the emergence of a quantum-stress/quantum-potential term as an effective dispersive pressure. Phase single-valuedness and circulation quantisation as a natural route to discrete states in a phase-field description (treated as an effective/topological constraint). The de Broglie–Bohm / pilot-wave lineage as a closely related mathematical reformulation, with emphasis on what is re-expressed (not “replaced”) by the hydrodynamic variables. Walking-droplet experiments as an analogue system illustrating how interference-like and orbit-quantisation-like phenomena can arise from coupled wave–particle dynamics (as analogy, not identity). The aim is to clarify what is strictly equivalent (the hydrodynamic rewrite) versus what is interpretive or open (ontology of the phase field, measurement modelling, and extensions to relativistic/QFT domains). This paper is part of a series of preprints exploring effective-medium style descriptions of gravity, electromagnetism, and quantum phenomena within a single mathematical language. This paper is part of a series. Companion preprints develop the gravitational and electromagnetic components of the HQG framework and discuss additional open problems and testable predictions. Discussion and feedback: https://github.com/Gptham123456/Hydrodynamic-Quantum-Gravity/discussions