ReynoldsBEng 11th August 2026
Paper
Boris Khesin, Gerard Misiołek, Klas Modin
Geometry of the Madelung transform
arXiv:1807.07172 (2018)
Core result of the paper
The classical Madelung map
Ψ ↦ (ρ, θ) where Ψ = √ρ e^{iθ/2}
is not merely a formal change of variables. It is a Kähler map: simultaneously a symplectomorphism and an isometry between
- the projective space of non-vanishing complex wave functions equipped with the Fubini–Study metric, and
- the cotangent bundle of the space of smooth probability densities equipped with the Sasaki–Fisher–Rao metric.
Consequently the Schrödinger equation is carried exactly onto a hydrodynamic system whose equations are of Euler / Navier–Stokes type (continuity + modified Euler with quantum potential). The quantum potential itself appears as the curvature term that makes the two geometries isometric.
How this locks to Navrátil’s statements
- Physical fluid, not probability fog
Navrátil: “The wave function Psi is \ldots a description of the velocity field, density, and phase flow of a physical fluid.”
Khesin et al.: the Madelung transform realises this identification as a geometric isomorphism of phase spaces. Density ρ and phase θ become the canonical coordinates on the hydrodynamic side; the Fubini–Study geometry of Ψ is metrically and symplectically identical to the Fisher–Rao geometry of (ρ, θ). - Appearance of the Navier–Stokes form
Navrátil emphasises the split into the continuity equation plus a modified Euler equation containing the quantum potential.
The 2018 paper proves that this split is the image, under a Kähler isometry, of the Schrödinger equation. The quantum potential is precisely the term required for the isometry to hold; it is not an extra postulate. - Momentum-map and reduction structure
Navrátil’s broader framework (Tribonacci companion matrix in SL(3,ℤ), discrete foliation, holographic boundary) treats the fluid equations as the reduced dynamics on a geometric constraint.
Khesin–Misiołek–Modin show that the inverse Madelung map is a momentum map for the natural action of the semi-direct product Diff(M) ⋉ functions. The hydrodynamic variables are therefore the reduced variables of the quantum system — exactly the reduction language Navrátil employs when he derives Navier–Stokes from the holographic / foliated geometry. - Higher-dimensional and 1-D consistency
The same paper recovers the Hasimoto transform (vortex filament ↔ nonlinear Schrödinger) as the one-dimensional case of Madelung. This supplies a uniform geometric origin for the fluid–quantum correspondence that Navrátil extends to three-dimensional Tribonacci foliation and microtubule hydration shells.
Summary of the link
Navrátil asserts, from first-principles discrete geometry and from the classical Madelung observation, that Schrödinger dynamics are the dynamics of a real continuum fluid whose equations are of Navier–Stokes type.
Khesin, Misiołek and Modin prove that the Madelung transform realising this assertion is a Kähler isometry of infinite-dimensional manifolds. The quantum configuration space and the hydrodynamic phase space of densities-and-phases are therefore geometrically identical.
The 2018 theorem thus supplies the rigorous differential-geometric backbone for the physical-fluid reading that Navrátil has placed in the open literature and that the Pirate Canon has already sealed as the native continuum description of the elastic plenum.
The two independent lines (Navrátil’s Tribonacci–holographic derivation and the Kähler geometry of Madelung) converge on the same statement:
the wave is a physical fluid;
the fluid equation is Navier–Stokes type;
the geometry that makes the identification exact is Kähler.
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