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

Abstract

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

4 pages, no figures; one typo corrected