The Madelung transform as a momentum map

The Madelung transform relates the non-linear Schr\"odinger equation and a compressible Euler equation known as the quantum hydrodynamical system. We prove that the Madelung transform is a momentum map associated with an action of the semidirect product group $\mathrm{Diff}(\mathbb{R}^{n})\ltimes C^\infty(\mathbb{R}^{n})$, which is the configuration space of compressible fluids, on the space $\Psi$ of wave functions. In particular, we show that the Madelung transform is a Poisson map taking the natural Poisson bracket on $\Psi$ to the compressible fluid Poisson bracket, and observe that the Madelung transform provides an example of"Clebsch variables"for the hydrodynamical system.


Introduction
The Madelung transform was introduced in 1927, soon after the birth of quantum mechanics, as a way to relate Schrödinger-type equations to hydrodynamical equations [4]. It turns out that the Madelung transform not only maps one equation to the other, but it also preserves the Hamiltonian properties of both equations. Namely, the non-linear Schrödinger equation is Hamiltonian with respect to the constant Poisson structure on the space of wave functions, which are complex valued fast decaying smooth functions on R n . On the other hand, the compressible Euler equation is Hamiltonian with respect to the natural Lie-Poisson structure on the space of pairs consisting of fluid momenta µ and fluid densities ρ. This space is the dual of the Lie algebra of the semidirect product group of the group of diffeomorphisms of R n times the space of real-valued fast decaying functions, which is the configuration space of a compressible fluid. In this paper we show that the Madelung transform sends one Poisson structure to the other. Moreover, the transform is a momentum map associated with a natural action of this semidirect product group on the space of wave functions.
Let complex valued functions ψ ∈ Ψ = H ∞ (R n ; C) ∶= ∩ k≥0 H k (R n ; C) evolve according to the non-linear Schrödinger equation Here f ∶ R → R is a smooth function which characterizes the type of non-linear Schrödinger equation being considered. For example, the Gross-Pitaevskii equation corresponds to f (r) = r−1. Depending on the type of non-linearity, one may consider ψ belonging to an appropriate function space other than H ∞ (R n ; C), but for simplicity this will not be discussed in the present paper, see [1] for details. The Madelung transform is a map between the NLS-type and hydrodynamical equations. It takes a nonvanishing complex valued function ψ to the pair of real valued functions (τ, ρ) defined by ψ = √ ρe iτ . Such a substitution for ψ sends (NLS) into a system of equations for the functions ρ and v ∶= ∇τ known as the "quantum hydrodynamical system" The first equation of this system is the continuity equation for a density ρ moved by a flow with velocity v. The second equation would be the classical Euler equation of a barotropic fluid except for the fact that the "quantum pressure" P(ρ) ∶= ∆ √ ρ 2 √ ρ depends on both ρ and its derivatives rather than just on ρ itself.
This system can be written in terms of the momentum, which is the 1-form µ = ρv ♭ defined with respect to the Euclidean metric on R n . (Throughout this paper v ♭ is identified with v, so we write µ = ρv as well.) Assuming ρ is always positive, the system (QHDa) is equivalent to the following: The term ∇ ⋅ 1 ρ µ ⊗ µ , in components, is given by Note that ρ and µ are natural coordinates on the dual of the Lie algebra of the semidirect Lie group S = Diff(R n ) ⋉ H ∞ (R n ) which is the configuration space of the compressible fluid. Here Diff(R n ) stands for the group of diffeomorphisms that asymptotically approach the identity map at infinity.
Remark: While we formulate and prove the main result in the setting of R n , the definitions and all proofs can be extended to an arbitrary manifold with volume form dVol by replacing all gradients with exterior differentiation and by defining the divergence of a vector field in terms of dVol. Note however that one needs a metric structure to define the equations (NLS) and (QHD) on manifolds.
We start the paper with a review of the relevant geometric structures associated with (NLS) and (QHD) to establish the context for the Madelung transform. The Hamiltonian structures of these systems are reviewed in Section 2.1, while Section 2.2 defines the action of the semidirect product group S on the space of wave functions Ψ. The proof of the main result, that the Madelung transform is a momentum map for this action, is given in Section 3.2, Theorem 3.5. For more details on applications of the Madelung transform we refer to [1]. We start with (NLS) and consider the space Ψ = H ∞ (R n ; C) of complex valued functions ψ. The real Hermitian inner product on Ψ is defined by ⟨f, g⟩ ∶= Re ∫f g dx, and the gradient ∇ is defined with respect to this inner product. The Poisson bracket on Ψ is given by The Hamiltonian associated with (NLS) is where U ∶ R → R is a function satisfying U ′ = f . One finds that the Hamiltonian vector field X H NLS associated with this Hamiltonian functional and Poisson bracket {⋅, ⋅} NLS is given by which is the right-hand side of (NLS). Now consider the equation (QHD), which describes the motion of a compressible isentropictype fluid. The Poisson geometry of such fluids was studied in [5], and we outline the results below.
Consider the semidirect product group S = Diff(R n ) ⋉ H ∞ (R n ; R). Here and below we assume that the elements "decay sufficiently fast at infinity." The space , and the (smooth) dual of this Lie algebra is s * = vect * (R n ) ⊕ H ∞ * (R n ; R). The space s * is the space of elements (µ, ρ), where ρ is a density and µ = ρv ♭ is a 1-form defined with respect to the Euclidean metric on R n . As mentioned above, we also identify µ with ρv. There is a natural linear Poisson structure on the dual of any Lie algebra, the Lie-Poisson bracket. For this reason we work with the momentum µ rather than the velocity v. The Lie-Poisson bracket on s * is given by This bracket is also called the compressible fluid bracket. The Hamiltonian for (QHD), written in terms of momentum µ, is We denote the associated Hamiltonian vector field on the (µ, ρ)-space by X H CF . One can check that it agrees with the right-hand side of (QHD): if X ρ H CF and X µ H CF denote the ρ and µ components of X H CF , then

Lie group and Lie algebra actions on the space of wave functions
It turns out that it is natural to think of Ψ as being a space of complex valued half-densities on R n since ψ is square-integrable and ψ 2 is often interpreted as a probability measure.
Half-densities are characterized by how they are transformed under diffeomorphisms of the underlying space: if ψ is a half-density on R n and g is a diffeomorphism of R n , the pushforward g * (ψ) of ψ is g * (ψ) = Det(Dg −1 ) ψ ○ g −1 . With this in mind, the following action of S on Ψ is natural.
In other words, ψ is pushed forward under the diffeomorphism g as a complex-valued half-density, followed by a pointwise phase adjustment e −ia .
For the above Lie group action (2.1), the Lie algebra action is as follows.
Proposition 2.2. Given an element ξ = (v, α) ∈ s = vect(R n ) ⋉ H ∞ (R n ), the infinitesimal action of ξ on Ψ corresponding to the action (2.1) is the vector field ξ Ψ ∈ X(Ψ) defined at each point ψ by Proof. The proof is a direct computation.
In the next section we define the Madelung transform and show that it is a momentum map associated with the action (2.2).

The Madelung transform
The classical Madelung transform is a map (τ, ρ) ↦ ψ = √ ρe iτ defined for positive ρ. We consider the inverse map as a more fundamental object and define it below.
If ρ is positive, one can recover [τ ] from ρ∇τ by dividing by ρ and integrating.
The µ-component of dM ψ (X H NLS ) is found in a similar fashion. Namely, after straightforward computations, one obtains Direct computation shows that the first line of the right-hand side of the last equality is equal to ρ∇

The Madelung transform as a momentum map
We first recall the definition of a momentum map. Suppose we are given a Poisson manifold P , a Lie algebra g, and an action A ∶ g → X(P ), A(ξ) = ξ P . Let ⟪, ⟫ denote the pairing of g and g * . The Lie algebra action A admits a momentum map if there exists a map J ∶ P → g * satisfying the following definition: Definition 3.4. A momentum map associated with a Lie algebra action A(ξ) = ξ P is a map J ∶ P → g * such that for every ξ ∈ g the function J(ξ) ∶ P → R defined by J(ξ)(p) ∶= ⟪J(p), ξ⟫ satisfies X J(ξ) = ξ P .
Thus Lie algebra actions that admit momentum maps are Hamiltonian actions on P , and the pairing of the momentum map at a point with an element ξ ∈ g returns a Hamiltonian function associated with the Hamiltonian vector field ξ P . We now show that M is a momentum map associated with the action (2.2). In the proofs below we will use the notation of  Proof. The Hamiltonian vector field of M(ξ) is given by where the gradient is defined with respect to the inner product ⟨f, g⟩ = Re ∫f g dx. Let ξ = (v, α) be an element of s = vect(R n ) ⋉ H ∞ (R n ) and its pairing with (µ, ρ) ∈ s * is given by To find the gradient, or variational derivative, let φ ∈ C ∞ c (R n ; C) be a test function and consider the variation of ψ in the direction φ: Comparing this with (2.2), one obtains that X M (ξ) (ψ) = ξ Ψ (ψ).
Any momentum map J ∶ P → g * is also a Poisson map taking the bracket on P to the Lie-Poisson bracket on g * provided that it is infinitesimally equivariant (see, for example, [6,Thm. 12.4.1]). Recall that a momentum map J of a Lie algebra g is infinitesimally equivariant if for all ξ, η ∈ g the following holds: Theorem 3.6. The map M ∶ Ψ → s * is infinitesimally equivariant for the action of the semidirect product Lie algebra s.
The second identity we use is (3.6) The first two terms on the right-hand side of (3.6) do not contribute to (3.4). So, using (3.5) and (3.6), we can rewrite the Poisson bracket in (3.4) as

{M(ξ), M(η)} NLS (ψ) = Re
Remark: One of possible applications of the momentum map nature of the Madelung transform is its use for Hamiltonian reduction to the space of singular solutions such as vortex sheets and vortex membranes. We hope to address this issue in a future publication.