Deconjugation of canonical variables and the Koopman-von Neumann theory

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.