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Geometry of the Madelung transform

The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an isometry) between the space of wave functions and the cotangent bundle to the density space equipped with the Fubini-Study metric and the Fisher-Rao information metric, respectively. We also show that Fusca's momentum map property of the Madelung transform is a manifestation of the general approach via reduction for semi-direct product groups. Furthermore, the Hasimoto transform for the binormal equation turns out to be the 1D case of the Madelung transform, while its higher-dimensional version is related to the problem of conservation of the Willmore energy in binormal flows.


Introduction
In 1927 E. Madelung [14] introduced a transformation, which now bears his name, in order to give an alternative formulation of the linear Schrödinger equation for a single particle moving in an electric field as a system of equations describing the motion of a compressible inviscid fluid. Since then other derivations have been proposed in the physics literature primarily in connection with various models in quantum hydrodynamics and optimal transport, cf. [16,20,15].
In this paper we focus on the geometric aspects of Madelung's construction and prove that the Madelung transform possesses a number of surprising properties. It turns out that in the right setting it can be viewed as a symplectomorphism, an isometry, a Kähler morphism or a generalized Hasimoto map. Furthermore, geometric properties of the Madelung transform are best understood not in the setting of the L 2 -Wasserstein geometry but (an infinite-dimensional analogue of) the Fisher-Rao information geometry-the canonical Riemannian geometry of the space of probability densities. These results can be summarized in the following theorem (a joint version of Theorems 2. 4

and 3.3 below).
Main Theorem. The Madelung transform is a Kähler morphism between the cotangent bundle of the space of smooth probability densities, equipped with the (Sasaki)-Fisher-Rao metric, and an open subset of the infinitedimensional complex projective space of smooth wave functions, equipped with the Fubini-Study metric.
The statement is valid in both the Sobolev topology of H s -smooth functions and Fréchet topology of C ∞ -smooth functions. In a sense the Madelung transform resembles the passage from Euclidean to polar coordinates in the infinite-dimensional space of wave functions, where the modulus is a probability density and the phase corresponds to fluid's vector field. The above theorem shows that, after projectivization, this transform relates not only equations of hydrodynamics and those of quantum physics, but the corresponding symplectic structures underlying them as well. Surprisingly, it also turns out to be an isometry between two well-known Riemannian metrics in geometry and statistics.
This result reveals tighter links between hydrodynamics, quantum information geometry and geometric quantum mechanics. Important in our constructions is a reformulation of Newton's equations on these spaces of diffeomorphisms and probability densities. This reformulation can be viewed as an extension of Arnold's formalism for the Euler equations of ideal hydrodynamics [1,2].
Our first motivation comes from hydrodynamics where groups of diffeomorphisms arise as configuration spaces for flows of compressible and incompressible fluids in a domain M (typically, a compact connected Riemannian manifold with a volume form µ). When equipped with a metric given at the identity diffeomorphism by the L 2 inner product (corresponding essentially to the kinetic energy) the geodesics of the group Diff(M ) of smooth diffeomorphisms of M describe motions of the gas of noninteracting particles in M whose velocity field v satisfies the inviscid Burgers equatioṅ v + ∇ v v = 0.
On the other hand, when restricted to the subgroup Diff µ (M ) of volumepreserving diffeomorphisms, the L 2 -metric becomes right-invariant, and its geodesics can be viewed as motions of an ideal (that is, incompressible and inviscid) fluid in M whose velocity field satisfies the incompressible Euler Here the pressure gradient ∇p is defined uniquely by the divergence-free condition on the velocity field v and can be viewed as a constraining force on the fluid. What we describe below can be regarded as an extension of this framework to various equations of compressible fluids, where the evolution of density becomes foremost important.
Our second motivation is to study the geometry of the space of densities. Namely, consider the projection π : Diff(M ) → Dens(M ) of the full diffeomorphism group Diff(M ) onto the space Dens(M ) of normalized smooth densities on M . The fiber over a density ν consists of all diffeomorphisms φ that push forward the Riemannian volume form µ to ν, that is, φ * µ = ν. It was shown by Otto [17] that π is a Riemannian submersion between Diff(M ) equipped with the L 2 -metric and Dens(M ) equipped with the (Kantorovich-Wasserstein) metric used in the optimal mass transport. More interesting for our purposes is that a Riemannian submersion arises also when Diff(M ) is equipped with a right-invariant homogeneous SobolevḢ 1 -metric and Dens(M ) with the Fisher-Rao metric which plays an important role in geometric statistics, see [9].
In the present paper we prove the Kähler property of the Madelung transform thus establishing a close relation of the cotangent space of the space of densities and the projective space of wave functions on M . Furthermore, this transform also identifies many Newton-type equations on these spaces that are naturally related to equations of fluid dynamics.
As an additional perspective, the connection between equations of quantum mechanics and hydrodynamics described below might shed some light on the hydrodynamical quantum analogs studied in [5,4]: the motion of bouncing droplets in certain vibrating liquids manifests many properties of quantum mechanical particles. While bouncing droplets have a dynamical boundary condition with changing topology of the domain every period, apparently a more precise description of the phenomenon should involve a certain averaging procedure for the hydrodynamical system in a periodically changing domain. Then the droplet-quantum particle correspondence could be a combination of the averaging and Madelung transform.

Madelung transform as a symplectomorphism
In this section we show that the Madelung transform induces a symplectomorphism between the cotangent bundle of smooth probability densities and the projective space of smooth non-vanishing complex-valued wave functions.
Definition 2.1. Let µ be a (reference) volume form on M such that M µ = 1. The space of probability densities on a compact connected oriented nmanifold M is where H s (M ) denotes the space of real-valued functions on M of Sobolev class H s with s > n/2 (including the case s = ∞ corresponding to C ∞ functions). 1 The space Dens s (M ) can be equipped in the standard manner with the structure of a smooth infinite-dimensional manifold (Hilbert, if s < ∞ or Fréchet, if s = ∞). It is an open subset of an affine hyperplane in H s (M ). Its tangent bundle is trivial Likewise, the (regular part of the) co-tangent bundle is The pairing is given by 1 From a geometric point of view it is more natural to define densities as volume forms instead of functions. This way, they become independent of the reference volume form µ. However, since some of the equations studied in this paper depends on the reference volume form µ anyway, it is easier to define densities as functions to avoid notational overload. ρ θ Madelung ψ Figure 1. Illustration of the Madelung transform Φ on S 1 . For x ∈ S 1 , a probability density ρ(x) > 0 and a dual infinitesimal probability density θ(x) are mapped to a wave function ψ(x) = ρe iθ ∈ C, which is defined up to rigid rotations of the complex plane.
It is independent of the choice of θ in the coset [θ] since Mρ µ = 0.
Definition 2.2. The Madelung transform is a map Φ which to any pair of functions ρ : M → R >0 and θ : M → R associates a complex-valued function Remark 2.3. The latter expression defines a particular branch of the square root ρe iθ . The map Φ is unramified, since ρ is strictly positive. Note that this map is not injective because θ and θ+4πk have the same image. Despite this fact, there is, as we shall see next, a natural geometric setting in which the Madelung transform (2) becomes invertible.
which, up to scaling by 4, is a symplectomorphism 2 with respect to the canonical symplectic structure of T * Dens s (M ) and the symplectic form of the Kähler structure on PH s (M, C).
Proof. We need to establish the following three steps: (i) Φ is well-defined, (ii) Φ is smooth, surjective and injective and (iii) Φ is symplectic.
(i) Let ρ ∈ Dens s (M ). Recall that the elements of T * ρ Dens s (M ) are cosets of H s functions on M modulo constants and given any θ ∈ H s (M, R) and any τ ∈ R the Madelung transform maps (ρ, θ +τ ) to √ ρe i(θ+τ )/2 . If s > n/2 then standard results on products and compositions of Sobolev functions (cf. e.g., [18]) show that it is smooth as a map to H s (M, C). Furthermore, we have so that that cosets (ρ, [θ]) are mapped to cosets [ψ], i.e., the map is welldefined.
(ii) Surjectivity and smoothness of Φ are evident. To prove injectivity for the cosets recall that inverting the Madelung map amounts essentially to rewriting of a non-vanishing complex-valued function in polar coordinates. Since preimages for a given ψ differ by a constant polar argumentθ = θ + 2πk, they define the same coset [θ]. Similarly, changing ψ by a constant phase does not affect the argument coset [θ], which implies injectivity of the map between the cosets (ρ, [θ]) and [ψ]. 3 (iii) The canonical symplectic form on T * Dens s (M ) is given by Since Mρ k µ = 0 it follows that it is well-defined on the cosets [θ i ]. The symplectic form on PH s (M, C) is given by The tangent vectors can be described as cosets One can see that Ω PH s is well-defined on the coset vectors which follows from M Re ψψ k µ = 0 and a straightforward calculation. Finally, for ψ = Φ(ρ, is T (ρ,[θ]) Φ(ρ, [θ]) = 1/2(ρ/ρ + iθ)Φ(ρ, [θ]). Then (5) gives which completes the proof.
Remark 2.5. In § 4 the inverse Madelung transform is defined for any C 1 function with no restriction on strict positivity of |ψ| 2 . It can be defined similarly in a Sobolev setting. Furthermore, extending Fusca [7], we will also show that it can be understood as a momentum map for a natural action of a certain semi-direct product group. Thus the Madelung transform relates the standard symplectic structure on the space of wave functions and the linear Lie-Poisson structure on the corresponding dual Lie algebra.
Remark 2.6. The fact that the Madelung transform is a symplectic submersion between the cotangent bundle of the space of densities and the unit sphere S ∞ ⊂ H s (M, C\{0}) of non-vanishing wave functions was proved by von Renesse [20]. The stronger symplectomorphism property proved in Theorem 2.4 is achieved by considering projectivization PH s (M, C\{0}).  (6) is Hamiltonian with respect to the symplectic structure induced by the complex structure of L 2 (M, C). Indeed, recall that the real part of a Hermitian inner product defines a Riemannian structure and the imaginary part defines a symplectic structure, so that Ω(ψ 1 , ψ 2 ) := Im ψ 1 , ψ 2 L 2 = Re iψ 1 , ψ 2 L 2 defines a symplectic form Ω corresponding to the complex structure J(ψ) = iψ. The Hamiltonian function for the Schrödinger equation (6) is where F : (0, ∞) → R is a primitive function of f , namely F = f .
Observe that the L 2 -norm of any solution ψ of (6) is conserved in time. Furthermore, the Schrödinger equation is also equivariant with respect to a constant phase shift ψ(x) → e iτ ψ(x) and therefore descends to the projective space PH s (M, C). It can be viewed as an equation on the complex projective space, a point of view first suggested in [11].
Proposition 2.7 (cf. [14,20]). The Madelung transform (3) maps the family of Schrödinger equations (6) to the following system on T * Dens s (M ) Equation (8) has a hydrodynamic formulation as an equation for a barotropic- with potential velocity field v = ∇θ.
Remark 2.8. Note that (8) only makes sense for ρ > 0, whereas the NLS equation makes sense even when ρ ≥ 0. In particular, the properties of the Madelung transform imply that if one starts with a wave function such that |ψ| 2 > 0 everywhere, then it remains strictly positive for all t for which the solution to equation (8) is defined, since this holds for ρ = |ψ| 2 by the continuity equation. Thus, |ψ| 2 can become non-positive only if v = ∇θ stops being a C 1 vector field (so that the continuity equation breaks).
Proof. Since the transformation (ρ, [θ]) → ψ is symplectic, it is enough to work out the Hamiltonian (7) expressed in (ρ, [θ]). First, notice that so that Thus, the Hamiltonian on T * Dens s (M ) corresponding to the Schrödinger Hamiltonian (7) is Since the result now follows from Hamilton's equationṡ for the canonical symplectic form (4) scaled by 1/4. Corollary 2.9. The Hamiltonian system (8) on T * Dens s (M ) for potential solutions of the barotropic equation (9) is mapped symplectomorphically to the Schrödinger equation (6).
The choice e = 0 gives a Schrödinger formulation for potential solutions of Burgers' equation, which describe geodesics in the L 2 -type Wasserstein metric on Dens s (M ). Thus, the geometric framework links the optimal transport for cost functions with potentials with the compressible Euler equations and the NLS-type equations described above. The vortex filament equation is known to be Hamiltonian with respect to the Marsden-Weinstein symplectic structure on the space of curves in R 3 and with Hamiltonian given by the length functional, see, e.g., [2].
It turns out that the vortex filament equation becomes the equation of the 1D barotropic-type fluid (9) with ρ = k 2 and v = 2τ , where k and τ denote curvature and torsion of the curve γ, respectively.
Definition 2.12. The Hasimoto transformation assigns to a curve γ, with curvature k and torsion τ , a wave function ψ according to the formula This map takes the vortex filament equation to the 1D NLS equation leads to a multiplication of ψ(x) by an irrelevant constant phase e iα ). In particular, the filament equation becomes a completely integrable system whose first integrals are obtained by pulling back those of the NLS equation. The first integrals for the filament equation can be written in terms of the total length dx, the torsion τ dx, the squared curvature k 2 dx, followed by τ k 2 dx etc. Langer and Perline [12] established symplectic properties of the Hasimoto transform. It turns out that the Marsden-Weinstein symplectic structure expressed in terms of the curvature and torsion is mapped by the Hasimoto transform to the constant symplectic structure on wave functions. (The original statement in [12] is more complicated, since the passage from the curve γ to its curvature and torsion requires taking two extra derivatives.) This symplectic property has the following heuristic explanation. The Marsden-Weinstein symplectic structure Ω M W on curves in R 3 is, essentially, averaging of the standard symplectic structures in all normal planes to the curve γ. Furthermore, one can regard the curvature magnitude |k(x)| as the radial coordinate in each normal plane, while (twice) the integral of torsion θ(x) = 2 τ (x) dx as the angular coordinate (since torsion is by definition the angular velocity of the rotation of the normal vector). This means that the passage from affine coordinates in normal planes to the polar ones (k 2 , θ/2) is a symplectic map: dx ∧ dy = (1/2)d(k 2 ) ∧ dθ. On the other hand, k and θ are (adjusted) polar coordinates of the wave function ψ(x) = k(x)e iθ(x) . So one arrives at the standard symplectic structure on the wave functions, regarded as complex-valued functions.
The following proposition relates the Hasimoto transform to the classical Madelung transform, see Section 2.
Proposition 2.14. The Hasimoto transformation is the Madelung transform in the 1D case.
This can be seen by comparing Definitions 2.2 and 2.12 which make the Hasimoto transform seem much less surprising. Alternatively, one may note that for ψ(x) = ρ(x)e iθ(x)/2 the pair (ρ, v) with v = ∇θ satisfies the compressible Euler equation, while in the one-dimensional case these variables are expressed via the curvature Remark 2.15. The filament equation has a higher-dimensional analog for membranes (i.e., compact oriented surfaces Σ of co-dimension 2 in R n ) as a skew-mean-curvature flowq = J(MC(q)), where q ∈ Σ is any point of the membrane, MC(q) is the mean curvature vector to Σ at the point q and J is the operator of rotation by π/2 in the positive direction in every normal space to Σ. This equation is again Hamiltonian with respect to the Marsden-Weinstein structure Ω M W on membranes of co-dimension 2 and with a Hamiltonian function given by the (n − 2)-dimensional volume of the membrane, see e.g. [19].
An intriguing problem in this area is the following.
Question 2.16. Find an analogue of the Hasimoto map, which sends a skew-mean-curvature flow to an NLS-type equation for any n.
The existence of the Madelung transform and its symplectic property in any dimension is a strong indication that such an analog should exist. Indeed, in any dimension by means of the Madelung transform one can pass from the wave function evolved according to an NLS-type equation to the polar form of ψ, i.e. to its magnitude √ ρ and the phase θ, so that the pair (ρ, v) with v = ∇θ will evolve according to the compressible Euler equation. Thus for a surface Σ of co-dimension 2 moving according to the skew-meancurvature flow, the problem boils down to interpreting the corresponding characteristics (ρ, ∇θ) similarly to the one-dimensional curvature and torsion. (Note that both the pair (ρ, θ) and the co-dimension 2 surface Σ in R n can be encoded by two functions of n − 2 variables).
In any dimension the square of the mean curvature vector can be regarded as a natural analog of the density ρ = MC 2 . In this case an analog of the total mass of the fluid, i.e. Σ ρ dσ, is the Willmore energy W(Σ) = Σ MC 2 dσ. An intermediate step in finding a higherdimensional Hasimoto map is then the following Conjecture 2.17. For a co-dimension 2 surface Σ ∈ R n moving by the skew-mean curvature flowq = J(MC(q)) the following equivalent properties hold: i) its Willmore energy W(Σ) is invariant, ii) its square mean curvature ρ = MC 2 evolves according to the continuity equationρ + div(ρv) = 0 for some vector field v on Σ.
The equivalence of the two statements is a consequence of Moser's theorem: if the total mass on a surface is preserved, the corresponding evolution of density can be realized as a flow of a time-dependent vector field.
Proposition 2.18. The conjecture is true in dimension 1.
Proof. In 1D the conservation of the Willmore energy is the time invariance of the integral W(γ) = γ k 2 dx or, equivalently, in the arc-length parameterization, of the integral γ |γ | 2 dx. The latter invariance follows from the following straightforward computation d dt It would be very interesting to find a higher-dimensional analog of the torsion τ for co-dimension 2 membranes. Note that the integral of the torsion has to play the role of an angular coordinate in the tangent spaces to Σ. The torsion would be the gradient part of the field v transporting the density ρ = MC 2 . Essentially, the question is how to encode a codimension 2 surface by its mean curvature and torsion. Presumably ∇θ as an analog of τ can be regarded as an angle of rotation ("the phase") of the vector MC, i.e. it might play a role of an exact 1-form.
This metric is invariant under the action of the diffeomorphism group. It is, in fact, the only Riemannian metric on Dens s (M ) with this property, cf. e.g., [3]. Next We will refer to this metric as the Sasaki-Fisher-Rao metric.
Next, recall that the canonical (weak) Fubini-Study metric on the complex projective space PH s (M, C) ⊂ PL 2 (M, C) is given by Proof. We have which proves the theorem.
The metric property in Theorem 3.3 combined with the symplectic property in Theorem 2.4 yields the following.
This result can be compared with the result of Molitor [15] who described a similar construction using (the cotangent lift of) the L 2 Wasserstein metric in optimal transport but obtained an almost complex structure on T * Dens s (M ) which is not integrable. It appears that the Fisher-Rao metric is a more natural choice for such constructions: its lift to T * Dens s (M ) admits a compatible complex (and Kähler) structure. It would be interesting to write down Kähler potentials for all metrics compatible with (17) and identify which of these are invariant under the action of the diffeomorphism group.

3.2.
Geodesics of the Sasaki-Fisher-Rao metric. As an isometry the Madelung transform maps geodesics of the Sasaki metric to geodesics of the Fubini-Study metric. The latter are projective lines in the projective space of wave functions. To see which submanifolds are mapped to projective lines by the Madelung transform we need to describe geodesics of the Sasaki-Fisher-Rao metric.
where v = v(t, x) and σ = σ(t, x) are time-dependent periodic functions on the line and the prime stands for the x-derivative. It can be viewed as a high-frequency limit of the 2-component Camassa-Holm equation, cf. [21]. It turns out that this system is closely related to the Kähler geometry of the Madelung transform and the Sasaki-Fisher-Rao metric (15). Consider the semi-direct product G = Diff s+1 0 (S 1 ) H s (S 1 , S 1 ), where Diff s+1 0 (S 1 ) is the group of circle diffeomorphisms that fix a prescribed point and H s (S 1 , S 1 ) is the space of Sobolev S 1 -valued maps of the circle. The group multiplication is given by Define a right-invariant Riemannian metric on G at the identity element by If t → (ϕ(t), α(t)) is a geodesic in G then v =φ • ϕ −1 and σ =α • ϕ −1 satisfy equations (18). Lenells [13] showed that the map is an isometry from G to a subset of {ψ ∈ H s (S 1 , C) | ψ L 2 = 1}. Moreover, solutions to (18) satisfying S 1 σ dx = 0 correspond to geodesics in the complex projective space PH s (S 1 , C) equipped with the Fubini-Study metric.
Our results show that this isometry is a particular case of Theorem 3.3. Proof. First, observe that the mapping (20) can be rewritten as (ϕ, α) → Φ(π(ϕ), α), where Φ is the Madelung transform and π is the projection ϕ → ϕ * µ specialized to the case M = S 1 . Next, observe that the metric (19) in the case S 1 σdx = 0 is the pullback of the Sasaki metric (15) by the mapping It follows from the change of variables formula by the diffeomorphism ϕ that the condition S 1 σdx = 0 corresponds to S 1α ϕ dx = 0. Hence, the description of the 2-component Hunter-Saxton equation as a geodesic equation on the complex projective L 2 space is a special case of that on T * Dens s (M ) with respect to the Sasaki-Fisher-Rao metric (15).

Madelung transform as a momentum map
In § 2 we described the Madelung transform as a symplectomorphism from T * Dens(M ) to PH s (M, C\{0}) which associates a wave function ψ = √ ρe iθ/2 (modulo a phase factor e iτ ) to a pair (ρ, [θ]) consisting of a density ρ of unit mass and a function θ (modulo an additive constant). Here, we start by outlining (following [7]) another approach, which shows that it is In §4.7 below we present a unifying point of view which explains the origin of the Madelung transform as the momentum map in a semidirect product reduction.

4.1.
A group action on the space of wave functions. We start by defining a group action on the space of wave functions. First, observe that it is natural to think of H s (M, C) as a space of complex-valued half-densities on M . Indeed, ψ ∈ H s (M, C) is assumed to be square-integrable and |ψ| 2 is interpreted as a probability measure. Half-densities are characterized by how they are transformed under diffeomorphisms of the underlying space: the pushforward ϕ * ψ of a half-density ψ on M by a diffeomorphism ϕ of M is given by the formula This formula explains the following natural action of a semidirect product group on the vector space of half-densities.
This action descends to the space of cosets [ψ] ∈ PH s (M, C).
Thus, a wave function ψ is pushed forward under the diffeomorphism ϕ as a complex-valued half-density, followed by a pointwise phase adjustment by e −ia/2 . An easy computation gives the following Lie algebra action  These two components allow one to obtain ρ and ρ dθ and hence, by integration, to recover θ modulo an additive constant. (The ambiguity involving an additive constant in the definition of θ corresponds to recovering the wave function ψ modulo a constant phase factor).
For a positive function ρ satisfying M ρ µ = 1 the pair (ρ dθ, ρ) can be identified with (ρ, [θ]) in T * Dens(M ), where the momentum variable m = ρ dθ is naturally thought of as an element of X(M ) * . Note, however, that this definition of the inverse Madelung works in greater generality: the momentum variable m is defined even when ρ is allowed to be zero, although θ cannot be recovered there.
Remark 4.4. So far we viewed ψ as a function on an n-manifold M . One can also consistently regard ψ as a complex half-density = ψ µ 1/2 . The set of complex half-densities on M is denoted √ Ω n (M ) ⊗ C indicating that it is "the square root" of the space Ω n (M ) of n-forms. Then the map M in (23) can be understood as follows. For a half-density ∈ √ Ω n (M ) ⊗ C the second component¯ of the map M is understood as a tensor product (ψψ) µ = ρ µ of two half-densities on M , thus yielding the density ρ ∈ Dens s (M ). One can show that the first component Im (¯ d ) of M can be regarded as an element m ⊗ µ = ρ dθ ⊗ µ ∈ Ω 1 (M ) ⊗ H s (M ) Ω n (M ). Namely, given a reference density µ, for any half-density = f (x)µ 1/2 define its differential d := df (x) ⊗ µ 1/2 . While the differential d depends on the choice of the reference density, the momentum map does not. Proof. Given a different reference volume form ν = h(x)µ with a positive function h > 0 one has where we dropped the term withḡg √ h d( √ h) since it is purely real. Suppose that a Lie algebra g acts on a Poisson manifold P and denote its action by A : g → X(P ) where A(ξ) = ξ P . Let , denote the pairing of g and g * . Definition 4.7. A momentum map associated with a Lie algebra action A(ξ) = ξ P is a map M : P → g * such that for every ξ ∈ g the function H ξ : P → R defined by H ξ (p) := M(p), ξ for any p ∈ P is a Hamiltonian of the vector field ξ P on the Poisson manifold P , i.e., X H ξ (p) = ξ P (p).
Thus, Lie algebra actions that admit momentum maps are Hamiltonian actions and the pairing of the momentum map at a point p ∈ P with an element ξ ∈ g defines a Hamiltonian function associated with the Hamiltonian vector field ξ P at that point p.
A momentum map M : P → g * of a Lie algebra g is infinitesimally equivariant if for all ξ, η ∈ g one has H [ξ,η] = {H ξ , H η }, which means that not only for any Lie algebra vector defines a Hamiltonian vector field on the manifold, but also the Lie algebra bracket of two such fields corresponds to the Poisson bracket of their Hamiltonians.

4.4.
Madelung transform is a momentum map. We now show (following Fusca [7]) that the transformation M is a momentum map associated with the action (22).  Proof. The Hamiltonian vector field for the function H ξ is X H ξ = −i dH ξ where the differential is obtained from To find the variational derivative let φ ∈ H s (M, C) be a test function. Then and comparing with (22) one obtains X H ξ (ψ) = 4V ξ (ψ).
Moreover, the Madelung transform turns out to be an infinitesimally equivariant momentum map, as was verified in [7].
In the case the entropy is constant or the pressure is independent of σ this system describes a barotropic flow, see equations (9). Note that, while the density evolves as an n-form, the entropy evolves as a function. However, according to the continuity equation, passing to the entropy density ς = σρ one can regard the corresponding group as the semidirect product S (2) = Diff(M ) (H s (M ) ⊕ H s (M )), which leads to a Hamiltonian picture on the dual s * (2) . By applying the multicomponent Madelung transform M (2) one can rewrite and interpret this system on the space of rank-1 spinors H s (M, C 2 ). Indeed, the evolution of the momentum m ⊗ µ = v ⊗ µ iṡ Observe that an invariant subset of solutions is given by those with momenta m = ρ dθ + ς dτ , where θ, τ ∈ H s (M )/R. They can be regarded as analogs of potential solutions of the barotropic fluid equations. We thereby obtain a canonical set of equations on T * H s (M ) 2 given by for a Hamiltonian of the form Using the multicomponent Madelung transform and Theorem 4.11 this gives (up to scaling by 4) a Hamiltonian system for the spinorψ = (ψ 1 , ψ 2 ) ∈ H s (M, C 2 ) with the Hamiltonian given by for the potential where the functional U is related to the pressure function P (ρ, σ) of the compressible Euler equation. The corresponding Schrödinger equation reads Conversely, one can work backwards to obtain a fluid formulation of various quantum-mechanical spin Hamiltonians, such as the Pauli equations for spin 1/2 particles of a given charge.

4.7.
Geometry of semi-direct product reduction. In this section we present the geometric structure behind the semi-direct product reduction which reveals the origin of the Madelung transform as the moment map above.
Let N be a Lie subgroup of a Lie group G. Assume that G acts from the left on a linear space V (a left representation of G). The quotient space of left cosets G/N is acted upon from the left by G. Assume now that G/N can be embedded as an orbit in V and let γ : G/N → V denote the embedding. Since the action of G on V induces a linear left dual action on V * we can construct the semi-direct product S = G V * .
Proposition 4.14. The quotient T * G/N is naturally embedded via a Poisson map in the Lie-Poisson space s * (the dual of the corresponding semidirect product algebra).
Proof. The Poisson embedding is given by where we use that T * G/N G/N × g * and s * g * × V . Now, the action of S on s * is where M : V * ×V → g * is the momentum map associated with the cotangent lifted action of G on V * . The corresponding infinitesimal action of s is Since the second component is only acted upon by g (or ξ), but not a (oṙ a), it follows from the embedding of G/N as an orbit in V that we have a natural Poisson action of S (or s) on T * G/N via the Poisson embedding (26).
Notice that the momentum map of S (or s) acting on s * is the identity: this follows since the Hamiltonian vector field on s * for H(m, b) = m, ξ + b,ȧ is given by (28).
We now return to the standard symplectic reduction (without semi-direct products). The dual n * of the subalgebra n ⊂ g is naturally identified with affine cosets of g * such that The momentum map of the subgroup N acting on g * by Ad * is then given by m → [m], since the momentum map of G acting on g * is the identity. If m, n = 0, i.e., m ∈ (g/n) * , then m ∈ [0] is in the zero momentum coset. Since we also have T * (G/N ) G/N × (g/n) * this gives us an embedding as a symplectic leaf in T * G/N G/N × g * . The restriction to this leaf is called zero-momentum symplectic reduction.
Turning to the semi-direct product reduction, we now have Poisson embeddings of T * (G/N ) in T * G/N and of T * G/N in s * . The combined embedding of T * (G/N ) as a symplectic leaf in s * is given by the map This implies that we have a Hamiltonian action of S (or s) on the zeromomentum symplectic leaf T * (G/N ) sitting inside T * G/N , which in turn sits inside s * . Since S has a natural symplectic action on s * and since G/N is an orbit in V V * * , we have, by restriction, a natural action of S on T * G/N . Furthermore, since the momentum map associated with S acting on s * is the identity, the Poisson embedding map (26) is the momentum map for S acting on T * G/N . Thus, the momentum map of S acting on T * (G/N ) is given by (30).
The above considerations are summarized in the following theorem. Appendix A. The functional-analytic setting The infinite-dimensional geometric constructions in this paper can be rigorously carried out in any reasonable function space setting in which the topology is at least as strong as C 1 , satisfies the functorial axioms of Palais [18] and admits a Hodge decomposition. The choice of the Sobolev spaces is very convenient for the purposes of this paper because many of the technical details which were used (explicitly or implicitly) in the proofs can be readily traced in the literature. We briefly review the main points below.
As introduced in the main body of the paper the notation Diff s (M ) stands for the completion of the group of smooth C ∞ diffeomorphisms of an ndimensional compact Riemannian manifold M with respect to the H s topology where s > n/2 + 1. This puts the Sobolev lemma at our disposal and thus equipped Diff s (M ) becomes a smooth Hilbert manifold whose tangent space at the identity T e Diff s (M ) consists of all H s vector fields on M , see e.g., [6], Section 2.
Using the implicit function theorem the subgroup Diff s µ (M ) = {η ∈ Diff s (M ) : η * µ = µ} consisting of those diffeomorphisms that preserve the Riemannian volume form µ can then be shown to inherit the structure of a smooth Hilbert submanifold with T e Diff s µ (M ) = {v ∈ T e Diff s : div v = 0}, cf. e.g., [6], Sections 4 and 8.
Standard results on compositions and products of Sobolev functions ensure that both Diff s and Diff s µ are topological groups with right translations ξ → ξ • η (resp., left translations ξ → η • ξ and inversions ξ → ξ −1 ) being smooth (resp., continuous) as maps in the H s topology, cf. [18]; Chapters 4 and 9. Furthermore, the natural projection given by η → π(η) = η * µ extends to a smooth submersion between Diff s+1 (M ) and the space of right cosets which can be identified with the space of probability densities on M of Sobolev class H s (cf. Section 2 above). More technical details, as well as proofs of all these facts, can be found in [6,18] and their bibliographies. Next, we turn to the Madelung transform which, for a fixed constant γ = 0, is and whose derivative isψ Given β > 0 the (scaled) Fubini-Study Hermitian structure on PC ∞ (M, C) is given by where φ, ψ L 2 = M φψµ. Ifψ 1 ,ψ 2 are of the form (32), then it follows from M ρ µ = 1 and Mρ i µ = Mθ i ρ µ = 0 that ψ L 2 = 1 and ψ i , ψ L 2 = 0. In this case and the associated symplectic structure iŝ Ω ψ (ψ 1 ,ψ 2 ) = Re FS ψ (iψ 1 ,ψ 2 ) = −Im FS ψ (ψ 1 ,ψ 2 ) Thus, the Madelung transform as defined by (31)  Thus, to make the Madelung transform defined by (31) an isometry (up to rescaling by β/4γ) we require that α = γ. If, in addition, it is to be a Kähler morphism, then we also require β = 4γ. Note that in this paper we set γ = 1 while in his original work Madelung used γ = /2 (as did von Renesse [20]).