The dynamics of the Koopman-van Hove angular and linear momentum operators

TL;DR

This paper explores the Koopman-van Hove formalism, rigorously constructing angular and linear momentum operators within it, analyzing their dynamics, and applying the framework to classical systems like Kepler's problem and oscillators.

Contribution

It provides a rigorous exposition of the Koopman-van Hove framework and introduces operator analogues for classical momentum, along with their group actions and dynamics.

Findings

Constructed operator analogues for classical angular and linear momentum.

Analyzed the group actions and momentum maps for these operators.

Applied the formalism to classical systems such as Kepler's problem and oscillators.

Abstract

A recent formalism capturing the classical-quantum coupling in a Hamiltonian theory for probabilistic classical mechanics has been proposed: the Koopman-van Hove formulation. The aims of this report are twofolds. First, we rigourously expose this new framework for a non-expert audience and present a construction for the operatorial analogues to the classical angular and linear momentum operators. We then investigate the group actions generating their average as momentum maps as well as their associated dynamics. Finally, we illustrate the theory by deriving the Koopman-van Hove formulation to the Kepler problem, harmonic and anharmonic oscillators.