L\'{e}vy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics

TL;DR

This paper explores how to incorporate spin into Koopman-von Neumann mechanics by deriving the Lévy-Leblond equation through Eisenhart-Duval lift, offering a non-relativistic approach to include spin in classical frameworks.

Contribution

It introduces a novel method to include spin in KvN mechanics via null reduction of the massless Dirac equation in Eisenhart-Duval lift, expanding the classical-quantum analogy.

Findings

Lévy-Leblond equation derived from Eisenhart-Duval lift in KvN formalism

Demonstration of spin inclusion in classical mechanics without relativistic assumptions

Application to a one-dimensional classical system without magnetic interactions

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. In Eur. Phys. J. Spec. Top. {\bf 227}, 2195 (2019) it was argued that the Spohn equation [Ann. Phys. {\bf 282}, 420 (2000)] is the correct classical framework for the Koopman-von Neumann theory corresponding to the Dirac equation. However, after L\'{e}vy-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 L\'{e}vy-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.