On the Operator Origins of Classical and Quantum Wave Functions

TL;DR

This paper introduces Operator Mechanics, a formalism that unifies classical and quantum wave functions as sections of a phase space bundle, revealing their common origin in a pre-quantum operator algebra without requiring a Hilbert space.

Contribution

It develops a pre-quantum algebraic framework called Operator Mechanics that derives classical and quantum wave functions and equations from a unified operator algebraic structure.

Findings

Both classical and quantum wave functions are sections of a phase space bundle.

The Schrödinger equation is derived from the Koopman-von Neumann equation.

Quantum and classical structures originate from a pre-quantum operator algebra.

Abstract

We investigate operator algebraic origins of the classical Koopman-von Neumann wave function $\psi_{KvN}$ as well as the quantum mechanical one $\psi_{QM}$. We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular, $\psi_{KvN}$ and $\psi_{QM}$ are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle ${\cal E}$ follows from this. $\psi_{KvN}$ and $\psi_{QM}$ are shown to be sections in ${\cal E}$. The difference between $\psi_{KvN}$ and $\psi_{QM}$ originates from a quantization map interpreted as "twisting" of sections over ${\cal E}$. We also show that the Schr\"{o}dinger equation is obtained from the Koopman-von Neumann equation. What this suggests is that neither the Schr\"{o}dinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.