Deconjugation of canonical variables and the Koopman-von Neumann theory

TL;DR

This paper explores the Koopman-von Neumann theory's deconjugation of variables, revealing its quantum origins and the necessity of auxiliary variables for canonical structure, with implications for quantum-classical interactions.

Contribution

It demonstrates that KvN theory derives from quantum mechanics and introduces auxiliary variables to preserve canonical structure, impacting quantum-classical interaction studies.

Findings

KvN theory arises from quantum mechanics.

Auxiliary variables are necessary for canonical structure.

KvN theory may have implications beyond classical physics.

Abstract

The Koopman-von Neumann (KvN) theory is one where the dynamical momentum is not canonically conjugate to position, i.e., position and momentum are deconjugated. From this point of view, we show that the KvN theory arises from quantum mechanics, extracting classical equations of motion from quantum ones. However, preserving the canonical structure of the theory requires introducing ``auxiliary'' canonical conjugates to position and momentum. We show that using the KvN formulation to study the interaction between quantum and classical systems forces the auxiliary variables to take on a physical role. While giving rise to classical behaviour, the KvN theory might be more than classical.