Solutions of Koopman-von Neumann equations, their superpositions, orthogonality and uncertainties

TL;DR

This paper explores the Koopman-von Neumann formulation of classical mechanics, demonstrating how to solve eigenvalue problems, construct orthonormal superpositions, and derive classical uncertainty relations within a Hilbert space framework.

Contribution

It introduces methods to solve the KvN equations, construct superpositions, and establish orthogonality, advancing the application of Hilbert space techniques to classical mechanics.

Findings

Superpositions of differently-gauged Liouvillian eigenstates form orthonormal sets.

Separable solutions describe the canonical ensemble with temperature linked to separation constants.

Classical uncertainty relations naturally emerge in the KvN formalism.

Abstract

The Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. One would hope to solve eigenvalue problems, obtain orthonormal eigenstates of Hermitian operators and ascribe meaning to a coherent superposition of states, among other things. Here we consider the general KvN equation for a classical probability amplitude and show that its so-called gauge freedom allows the separation of variables. The amenability to Hilbert-space methods of the resulting KvN solutions is investigated. We construct superpositions from differently-gauged Liouvillian eigenstates, and find an orthonormal set among them. We find that some separable solutions describe the canonical ensemble with temperature related to the separation constant. Classical uncertainty relations arise naturally in the KvN formalism. We discuss one between the dynamical time and the Liouvillian in terms of the statistical description of classical systems.