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arXiv:2510.27170

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

Partha Ghose

Published Oct 31, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2510.27170

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Nelson's stochastic mechanics links quantum mechanics to an underlying Brownian motion with the identification ℏ=mσ\hbar = mσℏ=mσ. Ghose's interpolating equation introduces a continuous parameter λλλ that suppresses the quantum potential Q[ψ]Q[ψ]Q[ψ] and yields a smooth transition between quantum (λ=0λ=0λ=0) and classical (λ=1λ=1λ=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→1λ\to 1λ→1 limit of this stochastic σσσ--λλλ hierarchy. The KvN phase-space amplitude provides an operator representation of the classical Liouville equation, while the λλλ parameter acts as a projection flow from the complex projective Hilbert manifold CPn\mathbb{C}P^nCPn to its classical quotient CP∗/U(1)\mathbb{C}P^*/U(1)CP∗/U(1), implementing phase superselection. This unified picture links quantum, stochastic, and classical dynamics within a single continuous framework.
In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This procedure was based on a Koopman-von Neumann approach where classical equations are reformulated into a quantumlike form. In this article, we develop a different derivation of quantum equations, based on purely classical stochastic arguments, taking some elements from the Bohm-Fenyes-Nelson approach. This study starts from a remark noticed by different authors [M. Born, Physics in My Generation (Pergamon Press, London, 1956); E. Prugovecki, Stochastic Quantum Mechanics and Quantum Spacetime (Reidel, Dordrecht, 1986)], suggesting that physical continuous observables are stochastic by nature. Following this idea, we study how intrinsic stochastic properties can be introduced into the framework of classical mechanics. Then we analyze how the quantum theory can emerge from this modified classical framework. This approach allows us to show that the transition from classical to quantum formalism (for a spinless particle) does not require real postulates, but rather soft generalizations.
While it is well-known that quantum mechanics can be reformulated in terms of a path integral representation, it will be shown that such a formulation is also possible in the case of classical mechanics. From Koopman-von Neumann theory, which recasts classical mechanics in terms of a Hilbert space wherein the Liouville operator acts as the generator of motion, we derive a path integral representation of the classical propagator and suggest an efficient numerical implementation using fast fourier transform techniques. We then include a first quantum correction to derive a revealing expression for the semi-classical path integral, which augments the classical picture of a single trajectory through phase space with additional wave-like spreading.
The so-called Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. We use the gauge freedom of the KvN equation to derive separable solutions of the KvN equation, and by extension the classical Liouville equation. The canonical ensemble solution of statistical mechanics is shown to arise from separable solutions. We discuss the Hermiticity of the Liouvillian operator, and treat a case where its spectrum is discrete. Finally, we analyze an uncertainty relation between the so-called dynamical time and the Liouvillian, and argue that such a relation is natural and can be understood entirely within a classical framework.
We develop a general `classicalization' procedure that links Hilbert-space and phase-space operators, using Weyl's operator. Then we transform the time-dependent Schrödinger equation into a phase-space picture using free parameters. They include position Q and momentum P . We expand the phase-space Hamiltonian in an ℏ-Taylor series and fix parameters with the condition that coefficients of ℏ 0 , -iℏ 1 ∂/∂ Q and iℏ 1 ∂/∂ P vanish. This condition results in generalized Hamilton equations and a natural link between classical and quantum dynamics, while the quantum motion-equation remains exact . In this picture, the Schrödinger equation reduces in the classical limit to a generalized Liouville equation for the quantum-mechanical system state. We modify Glauber's coherent states with a suitable phase factor S ( Q , P , t ) and use them to obtain phase-space representations of quantum dynamics and quantum-mechanical quantities.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
Nelson's stochastic mechanics may be understood as a stochastic underpinning, or reconstruction, of nonrelativistic quantum mechanics, once the diffusion scale is fixed by ℏ\hbarℏ and the admissible states are restricted by the usual single-valuedness condition on the wavefunction. In this note I briefly indicate what this route achieves and why it remains conceptually attractive. Four advantages are emphasized. First, it supplies a clear configuration-space stochastic picture of the underlying processes. Second, the Born rule is built in from the outset, with ∣ψ∣2|ψ|^2∣ψ∣2 arising as the probability density ρρρ of the underlying diffusion process rather than as an independent postulate. Third, it offers a markedly different perspective on measurement and nonlocality: in particular, collapse need not be treated as an extra axiom, and the nonlocality associated with entangled states is softened relative to the deterministic Bohmian guidance picture. Fourth, by tying quantumness to a diffusion scale, it naturally suggests a continuum of physical descriptions ranging from the strictly classical to the strictly quantum-mechanical regime. I conclude by proposing a natural distance scale in stochastic mechanics and examining its implications for testing possible limits of Bell correlations.