Uncertainty and Wigner negativity in Hilbert-space classical mechanics

TL;DR

This work demonstrates that key quantum-like features, specifically uncertainty relations and Wigner negativity, can emerge from classical mechanics when formulated in the Koopman-von Neumann Hilbert-space framework. By introducing tilde-variables as generators of canonical transformations and constructing a doubled phase-space Wigner representation, the authors show that classical Liouville dynamics and its Hilbert-space extension yield a rich, noncommutative structure. The approach unifies classical and quantum formalisms in a common operator language and clarifies how generators of transformations differ between the theories, with implications for interpreting quantum momentum as a tilde-momentum. The results provide a principled, epistemic-free route to quantum-like phenomena within classical mechanics and offer a new lens for comparing Wigner functions across the classical-quantum boundary.

Abstract

Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.