The Madelung transform as a momentum map

TL;DR

This paper demonstrates that the Madelung transform acts as a momentum map connecting the nonlinear Schrödinger equation with quantum hydrodynamics, revealing its geometric structure and role as a Clebsch variable.

Contribution

It proves that the Madelung transform is a momentum map for a specific group action, establishing its Poisson map property and geometric significance in fluid dynamics.

Findings

Madelung transform is a momentum map for the fluid group action.

The transform is a Poisson map between wave functions and fluid variables.

It provides a Clebsch variable representation for quantum hydrodynamics.

Abstract

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.