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

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→ 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}Pⁿ 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.