Uncertainty and Wigner negativity in Hilbert-space classical mechanics

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.

💡 Research Summary

The paper revisits classical mechanics through the Koopman‑von Neumann (KvN) formalism, which embeds the Liouville probability density into a complex probability amplitude that lives in a Hilbert space. By treating the amplitude (\chi(q,p,t)) as a state vector (|\chi\rangle) and introducing a pair of auxiliary Hermitian operators—(\tilde q) (the generator of spatial translations) and (\tilde p) (the generator of momentum translations)—the author shows that classical canonical transformations are implemented as unitary operations generated by these “tilde‑variables.”

These tilde‑variables obey non‑commutative algebraic relations analogous to the canonical commutation relations of quantum mechanics: \