Geometry of the Madelung transform

TL;DR

This paper explores the geometric structure of the Madelung transform, revealing it as a Kähler map linking quantum wave functions and fluid densities, and connects it to other transforms and conservation laws in geometric analysis.

Contribution

It proves the Madelung transform is a Kähler map between wave functions and density space, and relates it to reduction theory and other geometric transforms.

Findings

Madelung transform is a Kähler map between wave functions and density space.

Fusca's momentum map property is explained via reduction for semi-direct product groups.

Higher-dimensional Hasimoto transform relates to Willmore energy conservation.

Abstract

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.