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

TL;DR

This work introduces a geometric $\sigma$–$\lambda$ hierarchy that unifies quantum, stochastic, and classical dynamics by continuously suppressing the quantum potential $Q$ with the parameter $0\le\lambda\le1$ and identifying $\hbar=m\sigma$. It shows that the quantum regime ($\lambda=0$) reduces to Nelson's stochastic Schrödinger equation, while the classical limit ($\lambda=1$) yields Koopman–von Neumann (KvN) phase-space dynamics on a Lagrangian manifold with a phase-superselection quotient $\mathrm{CP}^*/U(1)$ that eliminates residual coherence. A phase-space density functional $F_{\sigma,\lambda}$ bridges configuration-space wave dynamics and phase-space Liouville flow, clarifying how classical mixtures of sheets emerge and how classical Liouville behavior is recovered in the appropriate limit. Overall, the framework provides a natural bridge between quantum, stochastic, and classical mechanics, offering a geometric perspective on controlled classicalization and the role of phase superselection in removing quantum coherence.

Abstract

Nelson's stochastic mechanics links quantum mechanics to an underlying Brownian motion with the identification $\hbar = mσ$. Ghose's interpolating equation introduces a continuous parameter $λ$ that suppresses the quantum potential $Q[ψ]$ and yields a smooth transition between quantum ($λ=0$) and classical ($λ=1$) regimes. In this short note, we show that the Koopman--von Neumann (KvN) Hilbert-space formulation of classical mechanics emerges naturally as the $λ\to 1$ limit of this stochastic $σ$--$λ$ hierarchy. The KvN phase-space amplitude provides an operator representation of the classical Liouville equation, while the $λ$ 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.