From quantum hydrodynamics to Koopman wavefunctions I

TL;DR

This paper introduces the Koopman-van Hove formulation of classical mechanics, linking classical wavefunctions to phase space distributions and demonstrating how von Neumann operators can represent singular point particles.

Contribution

It develops a new classical mechanics framework based on Koopman wavefunctions and von Neumann operators, extending quantum hydrodynamics concepts to classical phase space.

Findings

Classical Liouville density arises as a momentum map from contact transformations.

Koopman wavefunctions alone cannot reproduce arbitrary distributions.

Von Neumann operators enable representation of point particles in phase space.

Abstract

Based on Koopman's theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classical wavefunctions. Upon applying the Madelung transform from quantum hydrodynamics in the new context, we show how the Koopman wavefunction picture is insufficient to reproduce arbitrary classical distributions. However, this problem is entirely overcome by resorting to von Neumann operators. Indeed, we show that the latter also allow for singular $\delta-$like profiles of the Liouville density, thereby reproducing point particles in phase space.