The Madelung transform as a momentum map

The Madelung transform relates the non-linear Schrödinger 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 $\mathrm{Diff}(\mathbb{R}^{n}) \ltimes H^∞(\mathbb{R}^n; \mathbb{R})$, which is the configuration space of compressible fluids, on the space $Ψ = H^∞(\mathbb{R}^{n}; \mathbb{C})$ of wave functions. In particular, this implies that the Madelung transform is a Poisson map taking the natural Poisson bracket on $Ψ$ to the compressible fluid Poisson bracket. Moreover, the Madelung transform provides an example of "Clebsch variables" for the hydrodynamical system.