From Koopman-von Neumann Theory to Quantum Theory

TL;DR

This paper introduces a new classical action-based approach within Koopman-von Neumann theory that naturally derives quantum operators and equations from classical phase space functions, offering a novel perspective on quantization.

Contribution

It presents a new phase space function with clear physical meaning that leads to a derivation of quantum operators and evolution equations from classical entities.

Findings

Defines a classical action in phase space with physical meaning.

Derives Schrödinger's equation from classical phase space functions.

Provides a natural explanation for quantization rules.

Abstract

Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function $S^{(K)}(q,p,t)$ whose physical meaning is unknown. We show that a different $S(q,p,t)$, with well-defined physical meaning, may be introduced without destroying the attractive "quantum-like" mathematical features of the KvN theory. This new $S(q,p,t)$ is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schr\"odinger's equation if functions on phase space are reduced to functions on configuration space. This new kind of "quantization" does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing $\frac{\partial}{\partial p}$ by $0$ and $p$ by $\frac{\hbar}{\imath} \frac{\partial}{\partial q}$, thus providing an explanation for the common quantization rules.