Quantum, Stochastic, and Classical Dynamics Within A Single Geometric Framework

TL;DR

This paper presents a unified geometric framework that smoothly transitions between quantum, stochastic, and classical dynamics using a parameterized hierarchy, linking different regimes within a single mathematical structure.

Contribution

It demonstrates that the Koopman--von Neumann classical mechanics naturally arises as a limit of a stochastic hierarchy interpolating between quantum and classical regimes.

Findings

KvN formulation emerges as the $ ext{\lambda} o 1$ limit of the hierarchy

The $ ext{\lambda}$ parameter acts as a projection flow between quantum and classical phase spaces

Unified geometric framework links quantum, stochastic, and classical dynamics

Abstract

Nelson's stochastic mechanics links quantum mechanics to an underlying Brownian motion with the identification $\hbar = m\sigma$. Ghose's interpolating equation introduces a continuous parameter $\lambda$ that suppresses the quantum potential $Q[\psi]$ and yields a smooth transition between quantum ($\lambda=0$) and classical ($\lambda=1$) regimes. In this short note, we show that the Koopman--von Neumann (KvN) Hilbert-space formulation of classical mechanics emerges naturally as the $\lambda \to 1$ limit of this stochastic $\sigma$--$\lambda$ hierarchy. The KvN phase-space amplitude provides an operator representation of the classical Liouville equation, while the $\lambda$ parameter acts as a projection flow from the complex projective Hilbert manifold $\mathbb{C}P^n$ to its classical quotient $\mathbb{C}P^*/U(1)$, implementing phase superselection. This unified picture links quantum, stochastic, and classical dynamics within a single continuous framework.