Same maths, different names · Audit 02
Write a wavefunction as an amplitude and a phase and the Schrödinger equation splits into two fluid equations: one for how much is where, one for how it moves, with a single extra term. This audit records who uses that rewriting, what each field takes the fluid to be, which way the implication runs, and what has to be added before it will run backwards.
In plain terms
Quantum mechanics describes a particle with a wavefunction, a wave of complex numbers spread through space. Each complex number has a size and an angle. In 1926, months after Schrödinger published his equation, Erwin Madelung noticed that if you write the wavefunction as a size and an angle separately, the equation splits in two, and the two halves are the equations of a flowing fluid. The size, squared, is how much fluid is at each point. The angle, and how fast it changes from place to place, is the fluid’s velocity. There is one term that has no counterpart in an ordinary fluid: an energy that depends on how sharply the density varies from place to place, and that makes the fluid spread and disperse in ways an ordinary fluid does not. It is the only place Planck’s constant appears in the two equations. It is not, on its own, everything that makes the theory quantum; the rules about which angles are allowed matter as much.
The rewriting is used in fields that use different words for it. Clouds of ultracold atoms really are fluids that obey these equations, complete with whirlpools whose strength comes in fixed units. Light in certain crystals can be treated as a fluid of light, with the distance along the beam standing in for time. Electrons in plasmas and in microchips are modelled this way. So is the invisible dark matter in galaxies, in one leading theory. And a group of computer scientists used a cousin of the same trick backwards to make smoke in films look right, by simulating a wavefunction because it was easier than simulating the air.
There is a catch, and it is the interesting part. Going from wavefunction to fluid always works, wherever the wavefunction is not zero. Going back does not, in general. A fluid is described by a density and a velocity; to rebuild the wavefunction from them you also need the angle to come back to itself, up to whole turns, after any closed loop, which means the fluid’s circulation around any loop can only take certain values, and even then a fluid split into separate pieces has forgotten how the angles of the pieces line up. The fluid equations do not contain those rules. Takabayasi saw the first in 1952 and Wallstrom made it sharp in 1994. The consequence is narrow but firm: the rewriting by itself gives you no way to derive quantum mechanics from a fluid. Anyone proposing an underlying fluid has to supply, separately, the rule that fixes the circulation, the statistics, and the way many particles are correlated, and the last of these is the hardest, because for more than one particle the “fluid” does not flow through ordinary space at all.
The rest of the page is a ledger. It lists the fields that use the rewriting, what each calls it, when each first had it, what each assumes, and a verdict on how exact the match is: exact when it is the same equations on a stated domain, limited when it holds only after a named approximation, analogy when only some features line up. Every claim carries a note on how far the source was actually read.
What is being audited
Take the Schrödinger equation for one spinless particle of mass m in a real scalar potential V, with no magnetic vector potential, and write the scalar wavefunction in polar form with a real amplitude and a real phase. Then, wherever the amplitude is not zero and the solution is smooth enough, the single complex equation is equivalent to two real ones:
∂tρ + ∇·(ρv) = 0
∂tS + (∇S)²/2m + V + Q = 0, Q = −(ħ²/2m) · ∇²√ρ / √ρ
and, taking the gradient, m(∂t + v·∇)v = −∇(V + Q)
The first equation is continuity: ρ is conserved and carried by v. The second is a Hamilton–Jacobi equation for the phase; its gradient is an Euler equation for an inviscid fluid driven by the ordinary potential and by Q, the quantum potential. Q has the units of energy, not of pressure; its effect on the flow can be written as the divergence of a quantum stress, and the name “quantum pressure” is a convention rather than a dimensional fact. With the nonlinear term g|ψ|²ψ added, as in the Gross–Pitaevskii equation, the same steps give a fluid with an additional ordinary pressure gρ²/2 alongside Q. The Fisher-information identity that is sometimes offered as an interpretation of Q reads ∫ρQ d3x = (ħ²/8m)∫|∇ρ|²/ρ d3x, and holds when ρ is regular enough and boundary terms vanish; it is a statement about the average of Q, not about Q at a point.
Three conditions travel with the identity, and each row of the ledger is read against them.
First, the domain. Where ψ = 0 the polar variables are undefined; what happens to them as a zero is approached depends on the state. In the first excited state of the harmonic oscillator, ψ ∝ x e−x²/2, Q tends to a finite value from either side of the node even though its defining expression is undefined there. A travelling wave on a ring, ψ ∝ einθ, is an excited stationary state with no zeros at all, constant density and Q = 0. And a real wavefunction can have a nodal surface with no circulation around it. So “the fluid has holes at the nodes” is the right picture only for some states; the honest statement is that the rewriting is exact on the set where ρ > 0 and its continuation across zeros has to be examined case by case.
Second, the flow is irrotational wherever it is defined, since v is a gradient. Circulation can nonetheless be non-zero around a zero of ψ, where the phase winds, and also around a loop that the domain itself does not allow to be shrunk to a point, as on the ring, where there are no zeros. Quantised circulation is a fact about the topology of the region on which ψ is defined, not only about nodes.
Third, and decisive for the direction of implication: reconstruction. On each connected region where ρ > 0, an irrotational velocity whose circulation around every loop is an integer multiple of h/m determines the phase factor eiS/ħ up to a single constant phase, and hence ψ up to an overall phase; a globally single-valued real S need not exist when the circulation is non-zero. The winding numbers themselves are not lost: each is the circulation around its loop divided by h/m, and is read off v directly. What the condition demands is that those values be integers. That is Takabayasi’s condition (1952), and Wallstrom (1994) showed that without it the Madelung equations are strictly weaker than the Schrödinger equation. But two further pieces of information are lost as well. Taking the gradient of S to get v discards a constant per region, and when the region where ρ > 0 has several disconnected pieces, the relative phases between pieces are not recorded in (ρ, v) at all. Two real packets f and g with disjoint supports give ψ+ = f + g and ψ− = f − g the same density and the same current, and different futures once the packets overlap. Markowich and Sierra (2019) turn this into a non-uniqueness theorem for the quantum-hydrodynamic initial-value problem. So: Schrödinger implies Madelung wherever ρ > 0; Madelung implies Schrödinger only with the quantisation condition added, and only on a connected region or with the relative phases supplied separately.
Two columns in the ledger do the real work. The dictionary column says what plays the role of what, so that a verdict can be checked rather than believed. The direction column says which way the implication runs: from a wave equation to a fluid form, or from a fluid to a wave equation. The verdict judges the rewriting within the equation each row starts from; whether that equation is a good description of the physical system is a separate question, and it is answered separately in the “model” line of each row. A shared equation is also not a shared problem: the notes say where the meaning of ρ, the boundary data and the observables differ even when the equations do not, and in this audit that difference is often the whole story.
The ledger
Each verdict applies to the correspondence named in that row’s dictionary column, not to the field as a whole. The physical adequacy of the starting equation is stated on the “model” line and does not enter the verdict.
| Field | Name used · earliest source located | Dictionary | Direction of implication | Assumptions added or dropped · model · where it breaks | Verdict |
|---|---|---|---|---|---|
| Quantum mechanics | Hydrodynamic form; Madelung equations; quantum hydrodynamicsMadelung 1926 (note), 1927 · Takabayasi 1952 (quantisation condition) · Wallstrom 1994 (inequivalence) | |ψ|² ↔ ρ, a probability density · ∇S/m ↔ v · Q ↔ quantum potential · single-valued ψ ↔ ∮ mv·dl = nh on every loop · relative phases of disconnected regions ↔ not represented | Wave equation → fluid form: exact where ρ > 0. Fluid form → wave equation: only with the quantisation condition, on a connected region or with relative phases supplied. | adds ρ > 0 and enough smoothness; one particle, or configuration space for N particles · model the Schrödinger equation itselfbreaks at zeros of ψ the polar variables are undefined and their continuation is state-dependent; the reverse direction fails without Takabayasi’s condition (Wallstrom) and loses relative phases across disconnected regions (Markowich–Sierra); for N particles the “fluid” flows in 3N-dimensional configuration space | exact |
| de Broglie–Bohm theory | Pilot wave; guidance equation; causal interpretation; Bohmian mechanicsde Broglie 1927 · Bohm 1952 · Holland 1993 (monograph) | The same two equations · plus particles with dx/dt = v(x,t) · plus the “quantum equilibrium” assumption that particles are distributed as |ψ|² | Same mathematics, added ontology. The wave remains fundamental; the particle rides the Madelung flow. | adds point particles; an initial distribution equal to |ψ|² · model standard quantum mechanics plus trajectoriesbreaks nothing in the equations; the additions are interpretive and make no new prediction where quantum equilibrium holds; the guiding velocity is singular where ψ = 0 and global existence of trajectories had to be proved (Berndl et al. 1995) | exact |
| Stochastic mechanics | Nelson’s derivation; conservative diffusionNelson 1966 · Wallstrom 1989, 1994 (objection) · Reddiger & Poirier 2023 (review) | Brownian motion with diffusion coefficient ħ/2m and no friction · Newton’s law for the mean acceleration ↔ the Madelung pair | Stochastic process → Madelung equations, offered as a derivation of quantum mechanics. Runs the transform backwards. | adds an underlying diffusion; a definition of mean acceleration · model a proposed foundation, not an established onebreaks at exactly the point the transform is not invertible: the original assumptions reach the Madelung equations, not Schrödinger’s, because the quantisation condition is not derived (Wallstrom). Later responses are assessed in Audit 35, not here | limited |
| Superfluids and condensates | Gross–Pitaevskii hydrodynamics; superfluid velocity; quantised vorticesLandau 1941 (curl vs = 0) · Onsager 1949, Feynman 1955 (quantised circulation) · Pitaevskii 1961, Gross 1961 · Vinen 1961 (measured) | |ψ|² ↔ condensate number density, a matter density · ∇S/m ↔ superfluid velocity · Q ↔ quantum pressure · healing length ξ = ħ/√(2mgn) ↔ scale of Q · quantised vortex ↔ Takabayasi’s condition made physical, ∮ v·dl = nh/m | Wave equation (mean field) → fluid, and the fluid is real. The quantisation condition that must be assumed in row 1 is here observed. | adds nothing beyond ρ > 0 · model Gross–Pitaevskii mean field: dilute, weakly interacting, near zero temperature; the order parameter is not the many-body wavefunctionbreaks nothing within the equation: the GP equation and its Madelung form are “completely equivalent” (Barceló, Liberati & Visser 2001, eqs 12, 14, 17); the model is quantitative for dilute gases and qualitative for liquid helium | exact |
| Quantum plasmas and semiconductors | Quantum hydrodynamic model (QHD); Bohm potential; quantum pressureGardner 1994 (semiconductors) · Manfredi & Haas 2001 (electron gas) · Bonitz, Moldabekov & Ramazanov 2019 (critical review) | Velocity moments of the Wigner function ↔ fluid equations · each stream written in Madelung form · the ħ² term ↔ “quantum pressure” (Manfredi–Haas) or “Bohm potential” (Manfredi 2005) · closure ↔ an assumed equation of state | Many-body quantum kinetics → fluid, by moment truncation with a closure borrowed from the Madelung form. | adds a closure (Manfredi–Haas: equal amplitudes for all states, pressure a function of density); collisionless; often one dimension · model Wigner kinetics of a many-electron systembreaks the closure is an approximation, not a transformation; its range of validity is contested (Bonitz et al. 2019); the coefficient of the ħ² term depends on the closure | limited |
| Nonlinear optics | Fluids of light; photon fluid; dispersive shock wavesSpiegel 1980 (NLS in fluid form) · Wan, Jia & Fleischer 2007 (shocks) · Carusotto & Ciuti 2013 (review) · Glorieux et al. 2025 (review) | Paraxial envelope ↔ ψ · propagation distance z ↔ evolution variable (“time”) · intensity ↔ ρ · transverse phase gradient ↔ v · optical nonlinearity ↔ interaction g · diffraction ↔ kinetic term · absorption ↔ a sink term in the continuity equation | Optical wave equation → Schrödinger form → fluid form. The evolution variable is a space coordinate, with the field on the entrance plane as initial data. | adds nothing within the envelope equation · model paraxial and slowly-varying-envelope approximations; monochromatic light; a material model (Kerr in the idealisation; a saturable photorefractive nonlinearity in Wan et al.’s experiment)breaks outside the paraxial regime; with loss or gain the continuity equation acquires a source term (∂zρ + ∇⊥·(ρu) = −αρ) and the fluid is no longer conservative, but remains a fluid | exact |
| Cosmology (i) | Fuzzy (ultralight) dark matter; wave dark matter; Schrödinger–PoissonHui, Ostriker, Tremaine & Witten 2017 (Madelung form, Sec. II.B) | m|ψ|² ↔ dark-matter mass density (their convention, eq. 22) · ∇S/m ↔ velocity field · Q ↔ quantum pressure at the de Broglie scale · Poisson equation ↔ self-gravity · standard hydrodynamics codes ↔ solvers for the Madelung form | Wave equation → fluid form, exact within the Schrödinger–Poisson system where ρ > 0. | adds nothing within the system · model a non-relativistic self-gravitating scalar field, and the hypothesis that dark matter is onebreaks at zeros of the density, as in row 1. Interference is not a failure: a superposition of counter-propagating waves with unequal amplitudes has ρ > 0 everywhere and well-defined Madelung variables. What fails is a classical single-velocity fluid closure, which cannot represent collisionless multi-streaming. The exact Madelung equations also carry a single current velocity at each point of positive density; they describe the interference through density, current and quantum stress rather than through several local velocities (Hui et al., Appendix E) | exact |
| Cosmology (ii) | Schrödinger method for collisionless matterWidrow & Kaiser 1993 | Schrödinger–Poisson with a free numerical ħ ↔ an approximation to the Vlasov–Poisson system, via the Wigner or Husimi function · no Madelung variables used | Particles → wave equation as a numerical device. A different correspondence (Wigner–Vlasov), listed because it is often confused with row (i). | adds ħ chosen as small as practical · model cold collisionless dark matterbreaks the correspondence is with collisionless kinetics, controlled by the smallness of the numerical ħ; it is not the Madelung rewriting and is not exact | limited |
| Computational fluid dynamics | Incompressible Schrödinger flow; “Schrödinger’s smoke”Chern, Knöppel, Pinkall, Schröder & Weißmann 2016 | Classical velocity field ↔ the geometry of a two-component (ℂ²) wavefunction normalised to |ψ1|² + |ψ2|² = 1, which therefore never vanishes · vorticity ↔ carried by the spinor’s geometry, not by zeros · incompressibility ↔ a constraint imposed on the evolution · ħ ↔ a numerical parameter setting the circulation quantum and entering the modified dynamics; vortex-core thickness is treated separately | Classical fluid → a constrained Schrödinger-type system. A generalisation of the transform, run backwards, for a fluid with no quantum content. | adds a second component; pointwise normalisation; a pressure projection enforcing ∇·v = 0; a Landau–Lifshitz-type energy term (per the paper’s abstract) · model incompressible Euler flowbreaks this is not the scalar Madelung pair displayed above; it is a related construction whose relation to Euler dynamics carries its own qualifications; the wavefunction is a computational device, not a state | limited |
| Mathematics | Madelung transform as a symplectomorphism; geometric hydrodynamicsKhesin, Misiołek & Modin 2018 · Khesin & Modin 2026 (preprint) · Reddiger & Poirier 2023 | T*Dens(M), the cotangent bundle of positive densities with the Sasaki–Fisher–Rao metric ↔ the projective space of nowhere-vanishing wavefunctions with the Fubini–Study metric · Newton’s equations on densities ↔ Schrödinger-type equations · winding sectors ↔ classes in H¹(M, 2πℤ) | Both directions, as a theorem, on the domain ρ > 0 and within a fixed winding sector. | adds smoothness; nowhere-vanishing ψ; a fixed manifold M · model the Schrödinger equation on Mbreaks at the excluded set: zeros of ψ, and the winding that a non-simply-connected M allows without zeros; the 2026 preprint extends the picture to such manifolds and to a specified class of zeros, and gives a momentum-map reading of the Wallstrom condition | exact |
| Fluid mechanics (droplets) | Walking droplets; hydrodynamic pilot wavesCouder et al. 2005 · Couder & Fort 2006 · Bush 2015 (review) · Andersen et al. 2015 (contrary result) | Bouncing droplet ↔ particle · its self-generated surface wave ↔ pilot wave · no Madelung pair; the guiding wave obeys a Faraday-wave equation, not Schrödinger’s | Neither. A classical system that shares some behaviour with pilot-wave quantum mechanics, without the equations. | adds a vibrated bath; a real wave field with a wavelength fixed by the forcing · model a classical fluid experimentbreaks Bush (Sec. 4.2): “closer to de Broglie’s double-solution theory than to Bohmian mechanics”, with “numerous limitations as a quantum analog”; the double-slit interference claim of 2006 was not reproduced by Andersen et al. 2015 | analogy |
Related hydrodynamic formulations exist for relativistic fields (Klein–Gordon and Dirac; Takabayasi 1953) and in quantum chemistry’s trajectory methods (Wyatt 2005). Not separately audited.
Historical relationships
This structure has a single, dated origin. Schrödinger’s equation appeared in 1926; Madelung’s one-page note followed in November of the same year and the full paper in early 1927. What happened next was not rediscovery but rereading. De Broglie’s guidance rule, that a particle moves with velocity ∇S/m, was presented at the 1927 Solvay conference within months of Madelung’s paper; the two men were reading the same equation, one as a fluid and one as a particle guided by a wave, and I have not established whether de Broglie knew Madelung’s work, so the independence of the two readings is recorded as uncertain. Bohm’s 1952 papers revived the particle reading and named the quantum potential; Takabayasi’s response in the same year is the earliest source located for the circulation condition that the fluid reading needs and the wave reading supplies for free.
The superfluid line is separate. Landau’s 1941 theory already had a superfluid velocity that is irrotational (his eq. 7,1); Onsager (1949) and Feynman (1955, p. 35) argued that circulation must be quantised; Pitaevskii’s paper was submitted in September 1960 and Gross’s received in January 1961, and Vinen’s detection of single quanta of circulation was published in February 1961. Prediction preceded both the equation and the measurement, and the equation and the measurement arrived within months of each other; an earlier draft of this page claimed the measurement came first, which the dates do not support. Everything after that is deliberate transplant, and says so: Spiegel (1980) introduced the form to fluid dynamicists by name, Manfredi and Haas (2001) write “Madelung” in their derivation, Bush (2015, Sec. 4.1) cites Spiegel, Chern and colleagues (2016) cite Madelung, and the fuzzy-dark-matter literature calls its equations “the Madelung equations”. This is the opposite pattern from Audit 01: one origin, many named borrowings, and the interesting history is not who found it but what each borrower had to add.
Notes by row
Quantum mechanics exact
The transformation is exact as a change of variables wherever ψ ≠ 0 and the solution is smooth, and that qualification carries most of the weight on this page. What happens at zeros depends on the state, as the examples in the previous section show: the first excited oscillator state has a node at which Q has a finite limit; the ring state has quantised circulation and no zeros; a real standing wave has nodal surfaces with no circulation. Where the phase does wind around a nodal line, the circulation of mv is an integer multiple of h because ψ must return to itself. That is Takabayasi’s condition, and it is one of the places the fluid picture remembers that it came from a single-valued complex function; the other is the relative phase between disconnected regions of positive density, which (ρ, v) does not record and which Markowich and Sierra (2019) show makes the hydrodynamic initial-value problem non-unique when nodal domains merge. Drop the quantisation condition and the Madelung equations admit solutions with arbitrary circulation that correspond to no wavefunction at all; that is Wallstrom’s inequivalence.
The second qualification is the one most often passed over. For one particle the fluid flows through ordinary space. For N particles the wavefunction lives on the 3N-dimensional configuration space, and so does its Madelung fluid. Whatever a hydrodynamic reading of quantum mechanics means, it does not mean a single fluid in the room. Norsen, Marian and Oriols (2015) show what it would take to rewrite the N-particle theory in terms of fields in physical space: a countably infinite hierarchy of coupled equations for conditional wavefunctions, exact as an infinite hierarchy and without a generally exact finite truncation. This is the sharpest instance in the series of a shared equation that is not a shared problem.
De Broglie–Bohm theory exact
Bohm’s theory uses the Madelung pair unchanged and adds a particle that follows the flow lines. Nothing in the equations changes, so the row is exact; what is added is a claim about what exists. Where the added particles are distributed as |ψ|², the theory’s predictions coincide with standard quantum mechanics by construction. The mathematical difficulty it inherits is the nodal one: the guiding velocity is singular where ψ vanishes, and showing that, for a large class of potentials and initial wavefunctions, trajectories nonetheless exist for all time for typical initial configurations took until Berndl and colleagues in 1995. Bush’s remark, that the walking-droplet system is closer to de Broglie’s double-solution idea than to Bohm, marks the difference between a pilot wave that is real and local and one that is the wavefunction itself.
Stochastic mechanics limited
Nelson’s 1966 derivation starts from particles undergoing frictionless Brownian motion with diffusion coefficient ħ/2m, imposes a form of Newton’s law on their mean acceleration, and arrives at the Madelung equations. It is the cleanest example in the series of running the transform backwards as a proposed foundation, and Wallstrom’s objection (1989, 1994) is that the original assumptions reach the Madelung system and not the Schrödinger equation, because nothing in the process forces the circulation to be quantised. Whether later work has answered that objection is the subject of Audit 35; Reddiger and Poirier’s 2023 review surveys the responses and this page records the review without assessing it.
Superfluids and condensates exact
Here the fluid is a fluid. The Gross–Pitaevskii equation is a nonlinear Schrödinger equation for the condensate order parameter, and its Madelung form, continuity plus an Euler equation with an ordinary pressure and a quantum pressure, is, in Barceló, Liberati and Visser’s words (2001, after their eq. 17), “completely equivalent” to it. The density is a number density of atoms, not a probability. The healing length ξ = ħ/√(2mgn) (Pitaevskii and Stringari 2016, eq. 4.39) is the scale below which the quantum pressure matters, and above which the condensate is an ordinary irrotational compressible fluid; that limit is what Audit 01 relied on to obtain an acoustic metric. Bogoliubov’s 1947 excitation spectrum already has this shape, linear at long wavelength and approaching the free-particle form at short, though the crossover was given its name by later authors.
The row’s importance for this series is that the quantisation condition, which row 1 has to postulate, is here observed. Landau’s 1941 superfluid velocity is irrotational by construction; Onsager and Feynman argued that circulation must come in units of h/m; Vinen detected single quanta of circulation in helium in 1961, and Yarmchuk, Gordon and Packard photographed vortex arrays in 1979. Whether this counts as evidence that the condition is “physical” rather than “assumed” depends on which way you read the implication. The order parameter is single-valued because it is a quantum object; the vortices are quantised because the order parameter is single-valued. The fluid inherits the rule from the wave; it does not explain it.
The verdict is exact because the rewriting is exact within the Gross–Pitaevskii equation. The model line is where the physics is qualified: the equation describes a dilute, weakly interacting gas near zero temperature, its order parameter is not the many-body wavefunction, and for liquid helium it is qualitative rather than quantitative. The same separation between the exactness of the rewriting and the adequacy of the starting equation is applied to every row.
Quantum plasmas and semiconductors limited
Gardner’s 1994 quantum hydrodynamic model for semiconductor devices and Manfredi and Haas’s 2001 fluid model for a quantum electron gas both arrive at Euler-type equations with an ħ² term by taking velocity moments of a kinetic (Wigner) equation and closing the hierarchy. Manfredi and Haas write each electron stream in Madelung form and assume all streams share the same amplitude; the result (their eq. 26) contains the term (ħ²/2m²)∂x[(∂x²√n)/√n], which they call the quantum pressure and which Manfredi (2005, eq. 4.20) notes is “sometimes called the Bohm potential”. This is not a change of variables. It is a truncation with a closure, and the closure is where the physics is either captured or lost. Bonitz, Moldabekov and Ramazanov’s 2019 review is titled with a question, “quo vadis?”, and the question is about exactly this. Limited, because the correspondence is with the closure and not with the kinetic equation.
Nonlinear optics exact
For a monochromatic beam in a medium with an intensity-dependent refractive index, the paraxial approximation turns Maxwell’s equations into a two-dimensional nonlinear Schrödinger equation in which the distance along the beam is the evolution variable and the field on the entrance face is the initial data; Glorieux and colleagues’ 2025 review states the mapping z → τ = z/c directly (their Sec. II.C). Within that envelope equation the Madelung rewriting is as exact as in row 1, which is why the verdict is exact; the approximations that produce the envelope equation, paraxiality, a slowly varying envelope and a material model, belong on the model line. Its fluid form is a “fluid of light”: intensity for density, transverse phase gradient for velocity, nonlinearity for interaction. Wan, Jia and Fleischer used it to observe dispersive shock waves in 2007 in a photorefractive crystal whose nonlinearity is saturable rather than Kerr, a difference they discuss and find small in the defocusing regime; Carusotto and Ciuti’s review states the correspondence and cites the vortex and superfluid-flow work built on it. Absorption does not remove the fluid picture: a loss term gives the continuity equation a sink, ∂zρ + ∇⊥·(ρu) = −αρ, so the fluid is non-conservative but still a fluid. An earlier version of this page had the problem as a boundary-value problem in z and loss as having no fluid counterpart; both were wrong. Spiegel’s 1980 note is the earliest source located in which the fluid form of the nonlinear Schrödinger equation was presented to fluid dynamicists as such.
Cosmology exact limited
Two different uses share a name, and they are now two rows. Widrow and Kaiser in 1993 proposed solving the Schrödinger–Poisson system as a numerical stand-in for collisionless dark matter, with ħ a free parameter chosen as small as practical; their justification runs through the Wigner function and the Vlasov equation, not through Madelung, so the row is a different correspondence and is marked limited. The fuzzy-dark-matter programme, set out in full by Hui, Ostriker, Tremaine and Witten in 2017, takes the wave equation as physical, with a particle mass so small that the de Broglie wavelength is galactic, defines ρ = m|ψ|² (their eq. 22), and uses the Madelung form because, as they note after their eq. 24, standard hydrodynamics codes can be modified to include the quantum pressure. Within the Schrödinger–Poisson system that rewriting is exact where ρ > 0.
An earlier version of this page said the fluid form fails wherever streams cross and interfere, and that was wrong. Interference is not a failure of the transform: two counter-propagating waves of unequal amplitude give a density 1 + a² + 2a cos 2kx that is positive everywhere, with a perfectly good Madelung velocity. What cannot represent multi-streaming is a classical single-velocity fluid closure that treats the streams as ordinary matter. The exact Madelung fluid also has one velocity at each point of positive density; it carries the interference in its density, its current and its quantum stress, not in several local velocities. A classical kinetic distribution, conversely, can hold several velocities at one position without representing phase interference at all; the two failures are different. Hui and colleagues’ Appendix E treats the collision of streams and opens by separating single-valued fluid variables from wave superposition. The genuine failure points are the zeros of the density, as in row 1, and the practical difficulty of resolving fine interference structure in a hydrodynamic code, which is a numerical matter and not a mathematical one.
Computational fluid dynamics limited
“Schrödinger’s smoke” is the transform’s idea run backwards for a purpose with no quantum content at all, and it is a generalisation rather than the scalar pair displayed above. Chern and colleagues wanted to simulate incompressible Euler flow for computer graphics and found it easier to evolve a two-component wavefunction, normalised pointwise so that it never vanishes, under a Schrödinger-type equation with a constraint enforcing incompressibility and, per their abstract, a Landau–Lifshitz-type energy that shapes the vortical structure. The velocity is read from the geometry of the two-component field, and vorticity lives in that geometry rather than at zeros; a single Madelung phase would give only irrotational flow, and smoke is all vortex. The constant ħ becomes a numerical parameter: it sets the quantum of circulation and enters the modified dynamics, and the thesis derives an effective vortex radius from it separately, under stated assumptions (Chern 2017, Secs 10.2 and 10.4). Because the total density is constant, the scalar Q displayed above is identically zero for this field; whatever extra dynamics the construction has comes from the spinor structure, not from Q. The construction is exact for the constrained wave system it defines and its relation to Euler dynamics carries qualifications the paper sets out. It belongs in the ledger because it shows what the transform is when stripped of interpretation: a convenient parametrisation of certain velocity fields, no more. An earlier version of this page described the vortex filaments as nodal lines of the wavefunction; they are not, since the normalised field has none.
Mathematics exact
Khesin, Misiołek and Modin proved in 2018 that the Madelung transform is a symplectomorphism between the cotangent bundle of the space of positive densities and the projective space of nowhere-vanishing wavefunctions (PNAS, Theorem 14), and moreover a Kähler map between the Sasaki–Fisher–Rao metric on the fluid side and the Fubini–Study metric on the wave side (Theorem 2). That is the precise statement of “exact”: on the domain where ρ > 0, and within a fixed winding sector, the two systems are the same Hamiltonian system in different coordinates. Positive density alone does not remove global winding on a manifold with non-contractible loops; Khesin and Modin’s 2026 preprint enumerates the sectors by H¹(M, 2πℤ), extends the correspondence to a specified class of zeros of ψ (smooth codimension-two zero sets with quadratic degeneration of the density), and reads the Wallstrom condition as a momentum-map statement. It is a preprint, and its treatment of generic strong solutions should not be taken as covering every nodal evolution, nor as deriving quantum mechanics from an independently specified fluid.
Walking droplets analogy
A droplet bouncing on a vibrated bath generates a surface wave and is then steered by it. Couder, Fort and colleagues showed walking, orbiting and, in 2006, what they reported as single-slit and double-slit interference statistics. The system is a real pilot wave, and Bush’s 2015 review is careful about what that does and does not buy (Sec. 4.2): the guiding wave is a Faraday wave whose wavelength is set by the forcing, not by the particle’s momentum; the system is closer to de Broglie’s double solution than to Bohm’s theory; and it has “numerous limitations as a quantum analog”. Andersen and colleagues could not reproduce the double-slit result in 2015. No Madelung equation appears anywhere in the droplet system, so the row is an analogy and stays one.
The dictionary
Terms that name the same object across rows, on the domain ρ > 0, except where an entry says otherwise.
- Density
- |ψ|² (or m|ψ|²) · probability density (quantum mechanics, Bohm) · condensate number density (superfluids) · intensity (optics) · dark-matter mass density (cosmology) · a bookkeeping field (CFD). The same symbol, five meanings; only in superfluids is it a fluid density in the ordinary sense
- Velocity
- ∇S/m · Madelung velocity · Bohmian guidance velocity · superfluid velocity vs · transverse phase gradient (optics)
- Quantum potential
- Q = −(ħ²/2m)∇²√ρ/√ρ, an energy · “quantum pressure” (superfluids, optics, cosmology) and “Bohm potential” (plasmas) by convention · enters the momentum equation as the divergence of a quantum stress · its density-weighted integral is (ħ²/8m)∫|∇ρ|²/ρ, a Fisher information (Heifetz & Cohen 2015, eq. 23), under regularity and boundary conditions
- Quantisation condition
- ∮ mv·dl = nh · Takabayasi’s condition · single-valuedness of ψ · Onsager–Feynman quantised circulation · a momentum-map condition (Khesin–Modin 2026) · the step Nelson’s original assumptions do not supply (Wallstrom)
- Relative phases
- The constant of integration lost in passing from S to v, one per connected region of ρ > 0 · not recorded in (ρ, v) · source of the non-uniqueness in Markowich–Sierra
- Winding sector
- The integers nγ = (m/h)∮γ v·dl over the non-contractible loops of the region, a class in H¹(M, 2πℤ) · recorded in v, as its circulations (Khesin–Modin 2026, Lemmas 2.5–2.6) · what the quantisation condition requires is that these values be integers, not that they be supplied
- Zeros of ψ
- Where the polar variables are undefined · vortex cores when the phase winds (superfluids) · nodal surfaces without circulation for real wavefunctions · a finite limit of Q in some states, a divergence in others · absent in the ring state and in the normalised CFD field
- Irrotational flow
- curl v = 0 away from zeros · Landau’s curl vs = 0 · the reason a single-component wavefunction cannot simulate smoke
- Characteristic scales
- healing length ξ = ħ/√(2mgn), the scale below which Q competes with the interaction pressure in a condensate · de Broglie wavelength, the scale of interference structure and of Q’s influence in fuzzy dark matter · numerical ħ in the computational construction, which sets the circulation quantum and, separately, an effective vortex radius, and for which the scalar Q vanishes · related points of comparison, not one threshold under three names
- Evolution variable
- t in every row except optics, where it is the propagation distance z, with the entrance plane supplying the initial data
- Configuration space
- For N particles the Madelung fluid lives on ℝ3N; only the one-particle and mean-field rows have a fluid in ordinary space; Norsen, Marian and Oriols give the physical-space rewriting and its infinite hierarchy
Verdict
Three claims are usually run together. The ledger supports the first, supports the second with a change of meaning, and neither supports nor excludes the third.
First: for sufficiently smooth solutions of the scalar Schrödinger equation, the Madelung substitution gives an exact local hydrodynamic representation wherever the wavefunction is non-zero. Reconstruction of the wavefunction from density and velocity requires the circulation condition, a connected domain or separately supplied relative phases, and appropriate regularity; evolution through zeros needs case-by-case care. On the domain ρ > 0 and within a fixed winding sector, with the relative phases of disconnected regions supplied, the correspondence is a theorem about symplectic manifolds.
Second: the same substitution applies exactly within several approximate physical models: the Gross–Pitaevskii equation for condensates, the paraxial envelope equation for light, the Schrödinger–Poisson system for fuzzy dark matter. In each the meaning of ρ changes, from a probability to a number density, an intensity or a mass density. The assumptions of those models must be distinguished from the exactness of the substitution, and the ledger now does that on a separate line. Condensates are the case where the fluid is physically a fluid and the quantisation condition is visible as quantised vortices; the fluid inherits its single-valuedness from the quantum object it describes rather than explaining it. The plasma and semiconductor models, the Widrow–Kaiser method and the computational construction are correspondences of a different kind, each limited by a closure, an approximation or a generalisation named in its row.
Third: that quantum mechanics is, or can be derived from, an underlying fluid. These correspondences establish shared mathematical structure. They do not, by themselves, establish or exclude an underlying fluid theory. What the ledger does show is where such a theory’s obligations lie: it must supply the circulation scale h/m and the relative phases between disconnected regions that (ρ, v) does not carry, the statistical interpretation, and the many-particle correlations that put the Madelung fluid in configuration space. A classical complex field is not itself a quantum assumption, as the optics and computational rows show; reproducing quantum theory from one is the substantive task, and the rewriting on this page does none of it.
What this means for a hydrodynamic theory of quantum mechanics or gravity. The Madelung form is the natural language for such a theory and it is exact on its domain, which is why the programme is not idle. The debt is specific. Why is circulation quantised in units of h/m, and what fixes the phases that density and velocity forget? What is the fluid for two or more particles, given that the equations put it in 3N dimensions, and given that the physical-space construction considered here requires an infinite hierarchy? And what is Q: an energy with a mechanism, or a term that only makes sense as the shadow of the wave equation? Audit 35 takes the first question up directly. A theory that does not answer them is a rewriting, and should say so.
What this page does not claim
It does not claim that the Madelung equations are equivalent to quantum mechanics; they are equivalent only with the quantisation condition, on a connected domain or with relative phases supplied, and only where ρ > 0. It does not claim that every stationary state has zeros, that Q always diverges at a zero, or that every nodal surface carries circulation; counterexamples are given above. It does not claim that interference defeats the fluid form; it defeats a classical single-velocity fluid closure, and the Madelung fluid, which also has one velocity per point, carries the interference in its density, current and quantum stress instead. It does not claim that Q is a pressure; it is an energy whose effect can be written as a stress. It does not claim that superfluids explain the quantisation condition; they exhibit it. It does not claim that the quantum hydrodynamic models of plasma and semiconductor physics are transformations of anything; they are closures. It does not claim that walking droplets obey the equations on this page; they do not. It does not adjudicate the responses to Wallstrom’s objection or the status of stochastic mechanics; that is Audit 35. And it does not claim completeness: the relativistic and spinor versions, quantum-chemical trajectory methods, and the many-body extensions of the transform are noted but not audited.
Extensions of the transform, and what is known about each
Every entry below was searched before being written.
Next in the series
Audit 03 is the Hamilton–Jacobi equation and the eikonal: the second Madelung equation with Q removed, formally the classical limit of this page and the common ancestor of optics, mechanics and semiclassical quantum theory. It is a calibration case with well-established connections, used to test the method before the contested ones.
References
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Change log
0.3 · 12 Sep 2026 Final pass after a second external review. Winding numbers separated from relative phases: the circulations of v record the winding sector, and only the relative phase constants between disconnected regions are absent from (ρ, v); reconstruction now stated for the circle-valued phase factor. Cosmology row corrected once more: the Madelung fluid has one velocity per point and carries interference in density, current and quantum stress; what fails is a classical single-velocity closure. Computational row: the numerical ħ sets the circulation quantum and enters the modified dynamics, vortex thickness is treated separately, and the scalar Q vanishes for the normalised field; “Coherence scale” replaced by “Characteristic scales”. “Only known exact rewriting” withdrawn for Norsen, Marian and Oriols. Opening equations scoped to a spinless scalar wavefunction with a real scalar potential; relativistic cases described as related formulations; Berndl et al. qualified; caustics statement narrowed; plain-language description of Q corrected. Chern 2017 thesis added as a reference with pinpoints. Published.
0.2 · 12 Sep 2026 Revised after external review. Reconstruction claim narrowed: circulation quantisation recovers the phase only on a connected region, and relative phases between disconnected regions are not carried by (ρ, v) (Markowich & Sierra 2019 added). Three statements about zeros corrected: not every excited stationary state has zeros (ring state), Q need not diverge at a zero (first excited oscillator state), and circulation can arise from the domain’s topology without zeros; “everything quantum sits in Q” removed. Cosmology split into two rows; the claim that interference defeats the fluid form withdrawn, since a superposition with unequal amplitudes has positive density and well-defined Madelung variables. Optics corrected: propagation from the entrance plane is an initial-value problem in z, loss appears as a sink term in the continuity equation, and Wan et al.’s medium was a saturable photorefractive crystal, not Kerr. Computational-fluid-dynamics row rebuilt around the normalised two-component construction, which has no zeros. Verdict standard made uniform: the verdict judges the rewriting within the row’s stated equation, and the physical adequacy of that equation is stated on a separate model line; optics and fuzzy dark matter accordingly moved from limited to exact. Q described as an energy, with the Fisher-information identity stated with its conditions. Khesin–Misiołek–Modin theorem numbers and metrics stated correctly; Khesin & Modin 2026 added as a preprint. Foundational conclusion brought within the evidence: the rewriting supplies no derivation of quantum mechanics, and a classical complex field is not itself a quantum assumption; Norsen, Marian & Oriols 2015 added on physical-space rewritings. Historical claim that circulation was measured before the Gross–Pitaevskii equation was written withdrawn (Pitaevskii submitted September 1960; Vinen published February 1961). Pinpoint references added in the notes where a claim rests on a specific passage.
0.1 · 11 Sep 2026 First draft on the series template. Sources verified before drafting; one row corrected during verification (Widrow & Kaiser 1993 justify their method through the Wigner–Vlasov correspondence, not through Madelung) and one attribution narrowed (Heifetz & Cohen give a scalar pressure-like term, not a stress tensor).
Method note. This page was drafted with AI assistance (Anthropic’s Claude) working from my brief and revised with me through external review. Bibliographic details were verified against publisher records for every reference before the text was written; the flag beside each says how far the content itself was consulted, and where a claim rests on a particular passage the section, equation or page is given in the text. Claims that a topic was unexplored were searched before being kept. The verdicts are mine and I expect some of them to be argued with; corrections are welcome and will be logged. Content on this site is licensed CC BY-SA 4.0.