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arXiv:2512.05423

Nonlinear Classical Dynamics described by a Density Matrix in the Classical Limit

Gaspar Gonzalez, Angelo Plastino, Andrés Kowalski

quant-phnlin.CDSubmitted Dec 5, 2025• Announced Dec 8, 2025• 0 citations
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Abstract

We examine the classical limit of a fairly general nonlinear semiclassical hybrid system within a MaxEnt framework. The consistency of the hybrid dynamics requires algebraic constraints on quantum operators and smoothness conditions for the classical variables. Analytically, we demonstrate that the classical limit is characterized by a pure density matrix representing a single state, which reproduces the dynamics of its classical analogue. To illustrate the methodology, we revisit and synthesize two previously studied examples.

Comments
20 pages, 6 figures, 2 appendices

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We examine the classical limit of a fairly general nonlinear semiclassical hybrid system within a MaxEnt framework. The consistency of the hybrid dynamics requires algebraic constraints on quantum operators and smoothness conditions for the classical variables. Analytically, we demonstrate that the classical limit is characterized by a pure density matrix representing a single state, which reproduces the dynamics of its classical analogue. To illustrate the methodology, we revisit and synthesize two previously studied examples.
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We present a consistent framework of coupled classical and quantum dynamics. Our result allows us to overcome severe limitations of previous phenomenological approaches, such as evolutions that do not preserve the positivity of quantum states or that allow one to activate quantum nonlocality for superluminal signaling. A ``hybrid'' quantum-classical density is introduced, and its evolution equation derived. The implications and applications of our result are numerous: it incorporates the back-reaction of quantum on classical variables, and it resolves fundamental problems encountered in standard mean-field theories, and remarkably, also in the quantum measurement process; i.e., the most controversial example of quantum-classical interaction is consistently described within our approach, leading to a theory of dynamical collapse.
Abstract The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices ρ^(ξ){\hat{\rho }}(\xi )ρ^​(ξ) ρ ^ ( ξ ) , for ξ\xiξ ξ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.
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