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{D}\mathrm{i}\mathrm{f}\mathrm{f}$($\mathbb{R}$^{n})\ltimes C^\infty($\mathbb{R}$^{n})$, which is the configuration space of compressible fluids, on the space $\mathrm{\Psi }$ of wave functions. In particular, we show that the Madelung transform is a Poisson map taking the natural Poisson bracket on $\mathrm{\Psi }$ to the compressible fluid Poisson bracket, and observe that the Madelung transform provides an example of “Clebsch variables” for the hydrodynamical system.