Quantum--Fluid Correspondence for Systems of Nonrelativistic Spin-$\frac{1}{2}$ Particles

Michio Yamada; Naoki Sato

arxiv: 2605.18247 · v1 · pith:W2LKREQ6new · submitted 2026-05-18 · 🪐 quant-ph · math-ph· math.MP

Quantum--Fluid Correspondence for Systems of Nonrelativistic Spin-frac{1}{2} Particles

Naoki Sato , Michio Yamada This is my paper

Pith reviewed 2026-05-20 10:13 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords quantum hydrodynamicsPauli equationMadelung transformationspin-1/2 particlesfluid dynamicsmany-particle systemsEuler flowquantum fluids

0 comments

The pith

A charged fluid with internal spin satisfies the Pauli equation for nonrelativistic spin-1/2 particles and maps collections of them to Euler flow in 3n dimensions.

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

The paper shows that endowing a charged fluid with an internal spin degree of freedom makes its continuity and Euler equations match the Pauli equation exactly. This supplies a hydrodynamic picture for the quantum mechanics of one spinning particle. For n interacting particles the whole system is recast as classical Euler flow in a 3n-dimensional space. The construction extends the Madelung fluid description from spinless to spin-1/2 particles and supplies a fluid representation for many-body quantum systems.

Core claim

A charged fluid endowed with an internal spin degree of freedom naturally satisfies the Pauli equation for a nonrelativistic spin-1/2 particle, and that a collection of n such interacting fluids can be reformulated as an Euler flow in 3n dimensions, thereby providing a natural representation of a system of n Pauli particles. These results provide a fluid-mechanical derivation of the Pauli equation and extend the Madelung, or quantum-hydrodynamic, picture to many-particle quantum systems.

What carries the argument

Charged fluid equipped with an internal spin degree of freedom whose coupling to density and velocity fields produces continuity and Euler equations identical to the Pauli equation.

If this is right

A system of n Pauli particles is equivalent to a single Euler flow in 3n dimensions.
An n-qubit quantum computer can in principle be realized as n interacting fluids.
The same system can be realized as a 3n-dimensional Euler flow.
The Pauli equation receives a direct fluid-mechanical derivation.
The Madelung quantum-hydrodynamic picture extends to many-particle systems that carry spin.

Where Pith is reading between the lines

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

The mapping may permit classical fluid experiments to simulate spin-dependent quantum dynamics.
It suggests testing whether laboratory fluids with controlled internal degrees of freedom can reproduce simple spin-precession or entanglement effects.
Extensions to external potentials or particle interactions could be checked by adding corresponding body forces to the fluid equations.

Load-bearing premise

The internal spin degree of freedom can be defined and coupled to the velocity and density fields so that the resulting continuity and Euler equations are exactly equivalent to the Pauli equation.

What would settle it

Derive the continuity and momentum equations from the fluid with the postulated internal spin and check whether every term matches the corresponding term in the Pauli equation; any unmatched term or extra force would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.18247 by Michio Yamada, Naoki Sato.

**Figure 2.** Figure 2: Schematic illustration of the Madelung correspondence. To [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

read the original abstract

We show that a charged fluid endowed with an internal spin degree of freedom naturally satisfies the Pauli equation for a nonrelativistic spin-1/2 particle, and that a collection of n such interacting fluids can be reformulated as an Euler flow in 3n dimensions, thereby providing a natural representation of a system of n Pauli particles. These results provide a fluid-mechanical derivation of the Pauli equation and extend the Madelung, or quantum-hydrodynamic, picture to many-particle quantum systems. In particular, they imply that an n-qubit quantum computer can, at least in principle, be realized as a suitable combination of n fluids, or equivalently as a 3n-dimensional Euler flow.

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.

Desk Editor's Note private letter to a colleague

The paper gives a fluid derivation of the Pauli equation by adding an internal spin field to the fluid, then maps n particles to 3n-dimensional Euler flow, but the spin-velocity coupling looks like it may need an extra relation not required by standard Pauli initial data.

read the letter

The core claim is that a charged fluid with its own spin degree of freedom reproduces the Pauli equation exactly, and that n interacting copies of this fluid can be rewritten as ordinary Euler flow in 3n dimensions. That is the main new piece: an extension of the Madelung picture that includes spin-1/2 and many-body systems at once, with a side suggestion about fluid-based n-qubit models. The motivation is clear and the target is well-defined. If the steps close without hidden assumptions, it supplies a concrete hydrodynamic route to the Pauli dynamics that was missing before. The many-particle reformulation is the part that could actually be useful for thinking about classical analogs of multi-qubit systems. The abstract is thin, but the idea itself is a straightforward next step from existing quantum-hydro work. The weak point is the coupling between the spin vector field and the velocity. The construction needs the spin current and torque terms to cancel everything that would otherwise deviate from Pauli evolution. The stress-test note flags that this cancellation may only hold once an auxiliary link is imposed between the spin direction and the phase gradient, a link that is not part of the usual Pauli initial-value problem. If the full derivation shows the cancellation is automatic for arbitrary spinor data, the concern disappears; otherwise the mapping is more restrictive than advertised. Either way, the paper should state the initial-data assumptions explicitly. This is the sort of thing that matters to people working on quantum hydrodynamics or classical representations of spin systems. A reader already familiar with Madelung-type derivations will see the extension quickly and can judge whether the extra constraint is acceptable. It is worth sending to referees who know both fluid mechanics and the Pauli equation so they can check the algebra directly. I would not desk-reject it.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that a charged fluid endowed with an internal spin degree of freedom satisfies the Pauli equation for a nonrelativistic spin-1/2 particle. It further asserts that a collection of n such interacting fluids can be reformulated as an Euler flow in 3n dimensions, thereby providing a fluid-mechanical derivation of the Pauli equation and a representation of n-particle quantum systems, with implications for realizing n-qubit quantum computers as fluids or higher-dimensional Euler flows.

Significance. If the derivation is free of auxiliary constraints and holds for arbitrary initial data, the result would extend the Madelung quantum-hydrodynamic picture to spin-1/2 particles and many-body systems. This offers a classical fluid analogy for quantum spin dynamics and a potential new representation for quantum information processing.

major comments (1)

The central construction defines an internal spin vector field s(x,t) coupled to the velocity v(x,t) such that continuity and Euler equations reproduce the Pauli dynamics. However, the torque and spin-current terms appear to cancel non-Pauli contributions only after imposing a specific relation between s and the phase gradient of the spinor; this relation is not part of the standard Pauli initial-value problem. The manuscript must demonstrate that the cancellation holds identically for arbitrary initial spinor data without this extra constraint (see the single-particle section and the derivation of the spin evolution equation).

minor comments (2)

The abstract and introduction should include at least one key equation or step from the single-particle derivation to make the central claim more concrete for readers.
Notation for the spin current and the coupling term should be defined explicitly with reference to the Pauli equation components to avoid ambiguity in the many-particle extension.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major concern below and have revised the single-particle section to provide the requested demonstration.

read point-by-point responses

Referee: The central construction defines an internal spin vector field s(x,t) coupled to the velocity v(x,t) such that continuity and Euler equations reproduce the Pauli dynamics. However, the torque and spin-current terms appear to cancel non-Pauli contributions only after imposing a specific relation between s and the phase gradient of the spinor; this relation is not part of the standard Pauli initial-value problem. The manuscript must demonstrate that the cancellation holds identically for arbitrary initial spinor data without this extra constraint (see the single-particle section and the derivation of the spin evolution equation).

Authors: We thank the referee for identifying this point. The spin vector s is obtained directly from the spinor via the standard definition s = (ħ/2) (ψ† σ ψ)/|ψ|², which is part of the Madelung-type transformation and does not constitute an external constraint. Starting from an arbitrary initial spinor ψ(x,0), the corresponding initial v and s are fixed by the transformation; the subsequent evolution under the fluid equations then reproduces the Pauli dynamics identically because the torque and spin-current contributions cancel by direct substitution of the spinor-derived expressions. We have added an explicit verification in the revised single-particle section, including the general initial-data case and the derivation of the spin evolution equation, confirming that no auxiliary relation is imposed beyond the definition of the fluid variables themselves. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained

full rationale

The paper constructs a charged fluid model with an internal spin degree of freedom whose continuity and Euler equations are shown to reproduce the Pauli equation, extending the Madelung picture to spin-1/2 particles and multi-particle systems. This is presented as a direct fluid-mechanical derivation rather than a reparameterization of quantum inputs. No load-bearing self-citations, self-definitional steps, or fitted quantities renamed as predictions are identifiable from the abstract or context; the central claim begins from classical fluid equations augmented by spin coupling and arrives at the quantum dynamics without reducing to its own assumptions by construction. Concerns about auxiliary constraints on initial data pertain to the validity or generality of the equivalence, not to circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full manuscript required to audit the ledger.

pith-pipeline@v0.9.0 · 5647 in / 1071 out tokens · 31258 ms · 2026-05-20T10:13:57.616643+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Derivation of the Pauli equation for a fluid with spin (Section 4); hydrodynamic form (31)–(34) with spin-stress Π
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Many-particle system as 3n-dimensional Euler flow (Section 6, Proposition 6.1)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Hamiltonian structure with Poisson operator J (Section 5)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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A brief history of liquid computers

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work page doi:10.1098/rstb.2018.0372 2019
[2]

Hamiltonian formalism of extended magneto- hydrodynamics

isbn: 978-3-030-74277-5. doi: 10.1007/978-3-030-74278-2 . [AKY15] H. M. Abdelhamid, Y. Kawazura, and Z. Yoshida. “Hamiltonian formalism of extended magneto- hydrodynamics”. In: Journal of Physics A: Mathematical and Theoretical 48.23 (2015), p. 235502. doi: 10.1088/1751-8113/48/23/235502. 27 [Boh52a] David Bohm. “A Suggested Interpretation of the Quantu m...

work page doi:10.1007/978-3-030-74278-2 2015
[3]

Simulation of Quantum Computers: Revie w and Acceleration Opportuni- ties

[Cic+25] Alessio Cicero et al. “Simulation of Quantum Computers: Revie w and Acceleration Opportuni- ties”. In: ACM Transactions on Quantum Computing 7.1 (Nov. 2025), Article

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Eine anschauliche Deutung der Gleichung vo n Schr¨ odinger

AIP Confer- ence Proceedings. New York: American Institute of Physics, 1982 , pp. 47–66. doi: 10.1063/1.33647. [Mad26] Erwin Madelung. “Eine anschauliche Deutung der Gleichung vo n Schr¨ odinger”. In:Die Natur- wissenschaften 14 (1926), p

work page doi:10.1063/1.33647 1982
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Quantentheorie in hydrodynamischer For m

doi: 10.1007/BF01504657. [Mad27] Erwin Madelung. “Quantentheorie in hydrodynamischer For m”. In: Zeitschrift f¨ ur Physik40.3–4 (1927), pp. 322–326. doi: 10.1007/BF01400372. [Men+24] Zhaoyuan Meng et al. “Simulating unsteady ﬂows on a super conducting quantum processor”. In: Communications Physics 7 (2024), p

work page doi:10.1007/bf01504657 1927
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doi: 10.1038/s42005-024-01845-w . [Moo90] Cristopher Moore. “Unpredictability and undecidability in dyn amical systems”. In: Phys. Rev. Lett. 64 (20 May 1990), pp. 2354–2357. doi: 10.1103/PhysRevLett.64.2354. [Mor82] Philip J. Morrison. “Poisson Brackets for Fluids and Plasmas” . In: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Sys...

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[11]

Hamiltonian description of the ideal ﬂuid

AIP Conference Proceedings. New York: American Institute of Physics, 1982, pp. 13–46. doi: 10.1063/1.33633. [Mor98] P. J. Morrison. “Hamiltonian description of the ideal ﬂuid”. I n: Reviews of Modern Physics 70.2 (1998), pp. 467–521. doi: 10.1103/RevModPhys.70.467. 28 [MTW17] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler . Gravitation. Repr...

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Quantum-ﬂuid correspondence in relativistic ﬂuids with spin: from Madelung form to gravitational coupling

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Temperature Equilib rium in a Static Gravitational Field

[TE30] Richard C. Tolman and Paul Ehrenfest. “Temperature Equilib rium in a Static Gravitational Field”. In: Physical Review 36 (1930), pp. 1791–1798. doi: 10.1103/PhysRev.36.1791. [Tol30] Richard C. Tolman. “On the Weight of Heat and Thermal Equilibr ium in General Relativity”. In: Physical Review 35 (1930), pp. 904–924. doi: 10.1103/PhysRev.35.904. [Vid...

work page doi:10.1103/physrev.36.1791 1930
[15]

A Herculean task: classical simulation of qua ntum computers

doi: 10.1007/0-387-28145-2 . [Xu+25] Xiaosi Xu et al. “A Herculean task: classical simulation of qua ntum computers”. In: Science Bulletin 70 (2025), pp. 4104–4112. doi: 10.1016/j.scib.2025.10.016. 29 [YM16] Z. Yoshida and S. M. Mahajan. “Quantum spirals”. In: Journal of Physics A: Mathematical and Theoretical 49.5 (2016), p. 055501. doi: 10.1088/1751-811...

work page doi:10.1007/0-387-28145-2 2025

[1] [1]

A brief history of liquid computers

[Ada19] Andrew Adamatzky. “A brief history of liquid computers”. I n: Philosophical Transactions of the Royal Society B: Biological Sciences 374.1774 (2019), p. 20180372. doi: 10.1098/rstb.2018.0372. [AK21] V. I. Arnold and B. A. Khesin. Topological Methods in Hydrodynamics. 2nd ed. Vol

work page doi:10.1098/rstb.2018.0372 2019

[2] [2]

Hamiltonian formalism of extended magneto- hydrodynamics

isbn: 978-3-030-74277-5. doi: 10.1007/978-3-030-74278-2 . [AKY15] H. M. Abdelhamid, Y. Kawazura, and Z. Yoshida. “Hamiltonian formalism of extended magneto- hydrodynamics”. In: Journal of Physics A: Mathematical and Theoretical 48.23 (2015), p. 235502. doi: 10.1088/1751-8113/48/23/235502. 27 [Boh52a] David Bohm. “A Suggested Interpretation of the Quantu m...

work page doi:10.1007/978-3-030-74278-2 2015

[3] [3]

Simulation of Quantum Computers: Revie w and Acceleration Opportuni- ties

[Cic+25] Alessio Cicero et al. “Simulation of Quantum Computers: Revie w and Acceleration Opportuni- ties”. In: ACM Transactions on Quantum Computing 7.1 (Nov. 2025), Article

work page 2025

[4] [4]

Towards a Fluid Computer

doi: 10.1145/3762672. [CMP25] Robert Cardona, Eva Miranda, and Daniel Peralta-Salas. “ Towards a Fluid Computer”. In: Foun- dations of Computational Mathematics 25 (2025), pp. 1921–1937. doi: 10.1007/s10208-025-09699-6 . [EPR35] Albert Einstein, Boris Podolsky, and Nathan Rosen. “Can Qu antum-Mechanical Description of Physical Reality Be Considered Comple...

work page doi:10.1145/3762672 2025

[5] [5]

On Darboux theorems for geometr ic structures induced by closed forms

doi: 10.1017/CBO9780511795046 [Gr` a+24] Xavier Gr` acia et al. “On Darboux theorems for geometr ic structures induced by closed forms”. In: Revista de la Real Academia de Ciencias Exactas, F ´ ısicas y Naturales. Serie A. Matem´ aticas 118 (2024), p

work page doi:10.1017/cbo9780511795046 2024

[6] [6]

The zitterbewegung interpretation of q uantum mechanics

doi: 10.1007/s13398-024-01632-w . [Hes90] David Hestenes. “The zitterbewegung interpretation of q uantum mechanics”. In: Foundations of Physics 20.10 (1990), pp. 1213–1232. doi: 10.1007/BF01889466. [Hol93] Peter R. Holland. The Quantum Theory of Motion: An Account of the de Broglie–Bo hm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge Uni...

work page doi:10.1007/s13398-024-01632-w 1990

[7] [7]

Singular Poisson tensors

doi: 10.1017/CBO978051162 [Lit82] Robert G. Littlejohn. “Singular Poisson tensors”. In: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems . Ed. by M. Tabor and Y. M. Treve. Vol

work page doi:10.1017/cbo978051162

[8] [8]

Eine anschauliche Deutung der Gleichung vo n Schr¨ odinger

AIP Confer- ence Proceedings. New York: American Institute of Physics, 1982 , pp. 47–66. doi: 10.1063/1.33647. [Mad26] Erwin Madelung. “Eine anschauliche Deutung der Gleichung vo n Schr¨ odinger”. In:Die Natur- wissenschaften 14 (1926), p

work page doi:10.1063/1.33647 1982

[9] [9]

Quantentheorie in hydrodynamischer For m

doi: 10.1007/BF01504657. [Mad27] Erwin Madelung. “Quantentheorie in hydrodynamischer For m”. In: Zeitschrift f¨ ur Physik40.3–4 (1927), pp. 322–326. doi: 10.1007/BF01400372. [Men+24] Zhaoyuan Meng et al. “Simulating unsteady ﬂows on a super conducting quantum processor”. In: Communications Physics 7 (2024), p

work page doi:10.1007/bf01504657 1927

[10] [10]

Unpredictability and undecidability in dyn amical systems

doi: 10.1038/s42005-024-01845-w . [Moo90] Cristopher Moore. “Unpredictability and undecidability in dyn amical systems”. In: Phys. Rev. Lett. 64 (20 May 1990), pp. 2354–2357. doi: 10.1103/PhysRevLett.64.2354. [Mor82] Philip J. Morrison. “Poisson Brackets for Fluids and Plasmas” . In: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Sys...

work page doi:10.1038/s42005-024-01845-w 1990

[11] [11]

Hamiltonian description of the ideal ﬂuid

AIP Conference Proceedings. New York: American Institute of Physics, 1982, pp. 13–46. doi: 10.1063/1.33633. [Mor98] P. J. Morrison. “Hamiltonian description of the ideal ﬂuid”. I n: Reviews of Modern Physics 70.2 (1998), pp. 467–521. doi: 10.1103/RevModPhys.70.467. 28 [MTW17] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler . Gravitation. Repr...

work page doi:10.1063/1.33633 1982

[12] [12]

Quantum computing of ﬂuid dy namics using the hydrodynamic Schr¨ odinger equation

isbn: 9780691177793. [MY23] Zhaoyuan Meng and Yue Yang. “Quantum computing of ﬂuid dy namics using the hydrodynamic Schr¨ odinger equation”. In:Physical Review Research 5.3 (2023), p. 033182. doi: 10.1103/PhysRevResearch.5.0331 [MY24] Zhaoyuan Meng and Yue Yang. “Quantum spin representatio n for the Navier–Stokes equation”. In: Physical Review Research 6....

work page doi:10.1103/physrevresearch.5.0331 2023

[13] [13]

Quantum-ﬂuid correspondence in relativistic ﬂuids with spin: from Madelung form to gravitational coupling

doi: 10.1142/S0217732396001806. [Sat25] Naoki Sato. “Quantum-ﬂuid correspondence in relativistic ﬂuids with spin: from Madelung form to gravitational coupling”. In: Classical and Quantum Gravity 42.2 (2025), p. 025017. doi: 10.1088/1361-6382/ad9fcd. [Sch30] Erwin Schr¨ odinger. “ ¨Uber die kr¨ aftefreie Bewegung in der relativistischen Quantenmec hanik”. ...

work page doi:10.1142/s0217732396001806 2025

[14] [14]

Temperature Equilib rium in a Static Gravitational Field

[TE30] Richard C. Tolman and Paul Ehrenfest. “Temperature Equilib rium in a Static Gravitational Field”. In: Physical Review 36 (1930), pp. 1791–1798. doi: 10.1103/PhysRev.36.1791. [Tol30] Richard C. Tolman. “On the Weight of Heat and Thermal Equilibr ium in General Relativity”. In: Physical Review 35 (1930), pp. 904–924. doi: 10.1103/PhysRev.35.904. [Vid...

work page doi:10.1103/physrev.36.1791 1930

[15] [15]

A Herculean task: classical simulation of qua ntum computers

doi: 10.1007/0-387-28145-2 . [Xu+25] Xiaosi Xu et al. “A Herculean task: classical simulation of qua ntum computers”. In: Science Bulletin 70 (2025), pp. 4104–4112. doi: 10.1016/j.scib.2025.10.016. 29 [YM16] Z. Yoshida and S. M. Mahajan. “Quantum spirals”. In: Journal of Physics A: Mathematical and Theoretical 49.5 (2016), p. 055501. doi: 10.1088/1751-811...

work page doi:10.1007/0-387-28145-2 2025