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arxiv: 2607.01024 · v1 · pith:7FWIMWV3new · submitted 2026-07-01 · 🧮 math.DG · math-ph· math.MP· math.SG

Madelung hydrodynamics and Poisson geometry of wave functions

Boris Khesin , Klas Modin This is my paper

Pith reviewed 2026-07-02 05:59 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPmath.SG
keywords Madelung transformmomentum mapprequantum bundlesdiffeomorphism groupsFubini-Study geometryFisher-Rao geometrycoadjoint orbitsPoisson geometry
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The pith

The Madelung transform, regarded as a momentum map, defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups on arbitrary oriented manifolds.

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

The paper shows that the Madelung transform between wave functions and hydrodynamic densities functions as a momentum map in the Poisson geometry of diffeomorphism groups. This construction yields prequantum bundles over the relevant coadjoint orbits for any oriented manifold. When wave functions have no zeros the map becomes a Kähler isomorphism between the infinite-dimensional Fubini-Study and Fisher-Rao geometries. The same framework supplies an infinite-dimensional convexity statement for torus actions and a momentum-map interpretation of the Wallstrom quantization condition via Morse-Bott densities. A reader would care because the result supplies a geometric dictionary that converts questions about quantum wave functions into questions about coadjoint orbits of fluid groups, extending known finite-dimensional pictures to manifolds of arbitrary topology.

Core claim

For arbitrary oriented manifolds the Madelung transform, regarded as a momentum map, naturally defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups. For wave functions without zeros the transform supplies a Kähler map between the infinite-dimensional Fubini-Study and Fisher-Rao geometries. For wave functions with noncritical zeros the transform is a symplectomorphism onto coadjoint orbits carrying Morse-Bott densities, thereby furnishing a momentum-map account of the Wallstrom quantization condition.

What carries the argument

The Madelung transform viewed as a momentum map in the Poisson geometry of the diffeomorphism group and its semidirect product extensions.

If this is right

  • The construction supplies an infinite-dimensional version of convexity results for Hamiltonian torus actions, giving a partial answer to Atiyah's question.
  • It extends the Kähler isomorphism between Fubini-Study and Fisher-Rao geometries from simply-connected to non-simply-connected manifolds.
  • It yields a momentum-map perspective on the Wallstrom quantization condition for the hydrodynamical form of quantum mechanics.
  • It relates the Madelung setting to the Marsden-Weinstein symplectic structures on knots and membranes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The prequantum-bundle construction could be used to import finite-dimensional geometric quantization techniques directly into the hydrodynamical formulation of quantum mechanics.
  • The result suggests that topological features of the base manifold (fundamental group, orientability) control which coadjoint orbits can be prequantized via wave functions.
  • One could test whether the same momentum-map property persists when the underlying manifold is allowed to have boundary or is taken to be non-orientable.

Load-bearing premise

Wave functions are generic (no zeros or only noncritical zeros) on an oriented manifold and the Madelung map is a momentum map in the infinite-dimensional Poisson geometry.

What would settle it

A concrete wave function with a critical zero on the circle whose image under the Madelung map fails to be a symplectomorphism onto the corresponding coadjoint orbit with Morse-Bott density.

Figures

Figures reproduced from arXiv: 2607.01024 by Boris Khesin, Klas Modin.

Figure 1
Figure 1. Figure 1: The M-projection maps connected components Ψcc ̸=0 of normal￾ized nonvanishing wave functions in Ψ = C∞(M, C) to coadjoint orbits in dc∗ . The latter are enumerated by elements of H1 (M, 2πZ). These pre￾quantum S 1 -bundles over orbits can be thought of as infinite-dimensional analogues of the Hopf fibration. Proof. It remains to prove the transitivity part of the theorem, i.e., to show that any path ψ(t) … view at source ↗
Figure 2
Figure 2. Figure 2: The orientation of any connected component C ⊂ γ is inherited from the orientation of C since the wave-function ψ restricted to a transver￾sal N is a local diffeomorphism on C near the zero-set γ. Now note that the 1-form λψ is linear in ψ. Thus, its differential is the 2-form (ψ˙ 1, ψ˙ 2)ψ 7→ 2 Im Z M ψ˙ 1ψ˙ 2 , which is the symplectic structure descending to the Fubini–Study one on the projectivization. … view at source ↗
Figure 3
Figure 3. Figure 3: For a wave function ψt satisfying the Schr¨odinger equation and developing zeros during the evolution, its M-projection “jumps” from one coadjoint orbit O(ν0,ϱ0) ∈ dc∗ to another orbit O(ν1,ϱ1) by switching them at the time t = t∗. extends from M\γ to M. The statement (f) emphasizes that the L 2 metric on normalized wave functions in S∞ 1 is S 1 -invariant and descends to the corresponding coadjoint orbits… view at source ↗
read the original abstract

We describe the Poisson geometry of the Madelung transform between quantum mechanics and hydrodynamics for generic wave functions. We prove that for arbitrary oriented manifolds this transform, being regarded as a momentum map, naturally defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups. Furthermore, we show that the Madelung framework provides a natural infinite-dimensional version of the convexity results for Hamiltonian torus actions, thus giving a partial answer to Atiyah's question. In particular, for wave functions without zeros our results provide a K\"ahler map between the infinite-dimensional Fubini--Study and Fisher--Rao geometries, thus extending previous results to non-simply-connected manifolds. Furthermore, for wave functions with noncritical zeros, the Madelung transform is shown to be a symplectomorphism to the coadjoint orbits with Morse--Bott densities. The latter, in turn, furnishes a novel momentum map point of view on the Wallstrom quantization condition for the hydrodynamical form of quantum mechanics. We also comment on the relation between the Madelung setting and the Marsden--Weinstein symplectic structures on knots and membranes.

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 claims that the Madelung transform, regarded as a momentum map, defines prequantum bundles for coadjoint orbits of semidirect extensions of diffeomorphism groups on arbitrary oriented manifolds. For wave functions without zeros it induces a Kähler map between the infinite-dimensional Fubini–Study and Fisher–Rao geometries; for wave functions with noncritical zeros it is a symplectomorphism onto coadjoint orbits carrying Morse–Bott densities, thereby furnishing a momentum-map interpretation of the Wallstrom quantization condition. The work also supplies an infinite-dimensional version of convexity results for Hamiltonian torus actions, partially answering Atiyah’s question, and comments on relations to Marsden–Weinstein structures on knots and membranes.

Significance. If the central identifications hold, the paper would provide a coherent Poisson-geometric bridge between quantum mechanics and hydrodynamics on general oriented manifolds, extending known finite-dimensional convexity and Kähler correspondences to the infinite-dimensional setting. The momentum-map treatment of the Wallstrom condition and the explicit construction of prequantum bundles constitute concrete technical contributions that could be useful for geometric quantization of fluid systems.

major comments (3)
  1. [Main theorem on prequantum bundles] The assertion that the Madelung map ρ: Ψ ↦ (density, velocity) is a momentum map for the coadjoint action of Diff(M) ⋉ C^∞(M) is load-bearing for every subsequent theorem on prequantum bundles and Kähler structures. The manuscript does not specify the precise infinite-dimensional manifold structure (Fréchet, Sobolev H^s for s > dim(M)/2 + 1, etc.) placed on the space of sections or on the diffeomorphism group. Without this choice the smoothness of the coadjoint action, the embeddedness of the orbit, and the identification of the pulled-back symplectic form with the Fubini–Study form cannot be verified. (Main theorem on prequantum bundles and the paragraph immediately following the definition of ρ.)
  2. [Section on the Kähler correspondence] The claim that the Madelung transform supplies a Kähler map between Fubini–Study and Fisher–Rao geometries for zero-free wave functions on non-simply-connected manifolds rests on the pull-back of the complex structure being well-defined and integrable. The argument must explicitly check that the almost-complex structure induced by the momentum map commutes with the coadjoint action; the current exposition leaves this verification implicit. (Section on the Kähler correspondence and the statement of the relevant theorem.)
  3. [Paragraph discussing the infinite-dimensional convexity result] The partial answer to Atiyah’s convexity question for infinite-dimensional torus actions is presented as a consequence of the Madelung momentum map. The convexity statement requires a precise definition of the moment map image and a compactness or properness hypothesis on the torus action; neither is stated explicitly, making it impossible to assess whether the infinite-dimensional case reduces to the finite-dimensional convexity theorem or requires additional hypotheses. (Paragraph discussing the infinite-dimensional convexity result.)
minor comments (2)
  1. [Introduction] The notation for the semidirect product group and its Lie algebra should be introduced once in the introduction and used consistently thereafter.
  2. [Discussion of related work] Several references to infinite-dimensional symplectic geometry (e.g., works on ideal fluids) are cited but the precise relation of the present construction to those results is not spelled out.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to incorporate the requested clarifications on the infinite-dimensional setting.

read point-by-point responses

  1. Referee: The assertion that the Madelung map ρ: Ψ ↦ (density, velocity) is a momentum map for the coadjoint action of Diff(M) ⋉ C^∞(M) is load-bearing for every subsequent theorem on prequantum bundles and Kähler structures. The manuscript does not specify the precise infinite-dimensional manifold structure (Fréchet, Sobolev H^s for s > dim(M)/2 + 1, etc.) placed on the space of sections or on the diffeomorphism group. Without this choice the smoothness of the coadjoint action, the embeddedness of the orbit, and the identification of the pulled-back symplectic form with the Fubini–Study form cannot be verified. (Main theorem on prequantum bundles and the paragraph immediately following the definition of ρ.)

    Authors: We agree that an explicit choice of manifold structure is required for rigor. In the revised manuscript we will specify that the space of wave functions is an open subset of the Sobolev Hilbert manifold H^s(M,ℂ) with s > dim(M)/2 + 1 and that Diff(M) ⋉ C^∞(M) is equipped with the corresponding Sobolev topology. With this structure the coadjoint action is smooth, orbits are embedded submanifolds, and the pull-back of the symplectic form coincides with the Fubini–Study form; we will add the corresponding verification to the proof of the main theorem. revision: yes

  2. Referee: The claim that the Madelung transform supplies a Kähler map between Fubini–Study and Fisher–Rao geometries for zero-free wave functions on non-simply-connected manifolds rests on the pull-back of the complex structure being well-defined and integrable. The argument must explicitly check that the almost-complex structure induced by the momentum map commutes with the coadjoint action; the current exposition leaves this verification implicit. (Section on the Kähler correspondence and the statement of the relevant theorem.)

    Authors: We will make the verification explicit. We will insert a short lemma showing that the almost-complex structure J pulled back by the momentum map is invariant under the coadjoint action: for any fundamental vector field X_M generated by the group action one has ℒ_{X_M} J = 0, which follows directly from the equivariance of the momentum map. This establishes that J is integrable and that the Madelung map is Kähler on the zero-free locus. revision: yes

  3. Referee: The partial answer to Atiyah’s convexity question for infinite-dimensional torus actions is presented as a consequence of the Madelung momentum map. The convexity statement requires a precise definition of the moment map image and a compactness or properness hypothesis on the torus action; neither is stated explicitly, making it impossible to assess whether the infinite-dimensional case reduces to the finite-dimensional convexity theorem or requires additional hypotheses. (Paragraph discussing the infinite-dimensional convexity result.)

    Authors: We will revise the paragraph to state explicitly that the image of the moment map consists of those coadjoint orbits whose densities have fixed total mass and whose velocities satisfy the appropriate integrability conditions, and that the torus action is assumed proper in the H^s topology. Under these hypotheses the convexity statement follows from the classical Atiyah–Guillemin–Sternberg theorem applied after reduction or by a direct argument using the momentum-map property; the revised text will make this reduction clear. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation self-contained in standard Poisson geometry

full rationale

The paper proves that the Madelung transform is a momentum map for semidirect extensions of Diff(M), yielding prequantum bundles and a Kähler correspondence between Fubini-Study and Fisher-Rao geometries on generic wave functions. No quoted step reduces by the paper's own equations to a fitted input, self-definition, or load-bearing self-citation chain; the claims extend prior results via explicit constructions on oriented manifolds without invoking uniqueness theorems or ansatzes from the authors' own prior work as the sole justification. The derivation therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is inferred from stated claims. No free parameters or invented entities are mentioned. The work relies on standard background in infinite-dimensional symplectic and Poisson geometry.

axioms (2)
  • standard math Standard results on momentum maps for diffeomorphism groups and their semidirect extensions.
    Invoked when the Madelung transform is regarded as a momentum map.
  • standard math Existence of prequantum bundles over the relevant coadjoint orbits.
    Used to conclude that the transform defines such bundles.

pith-pipeline@v0.9.1-grok · 5741 in / 1346 out tokens · 25378 ms · 2026-07-02T05:59:07.820573+00:00 · methodology

discussion (0)

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Reference graph

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