Uncertainty and Wigner negativity in Hilbert-space classical mechanics

TL;DR

This paper demonstrates that classical mechanics formulated in Hilbert space exhibits quantum-like features such as uncertainty relations and Wigner negativity, blurring the line between classical and quantum descriptions.

Contribution

It reveals that classical mechanics in Hilbert space naturally includes noncommutative operators, leading to uncertainty relations and Wigner negativity, traditionally associated with quantum mechanics.

Findings

Classical canonical transformations are generated by noncommutative Hermitian operators.

Uncertainty relations naturally arise in the Hilbert-space formulation of classical mechanics.

Wigner representation of classical states can exhibit negative quasi-probabilities.

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.