Liouville dynamics for optimal stochastic phase-space representations of quantum mechanics

Eduard Prugovečki

Eduard Prugovečki

J. Bertrand, Guy Rideau

J. Bertrand, Guy Rideau

Eduard Prugovečki

Partha Ghose

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.