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 $\hbar = mσ$. 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.