Lévy-Leblond equation and Eisenhart–Duval lift in Koopman–von Neumann mechanics

Abstract

The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.