This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincare integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint classical-quantum density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

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混合量子经典动力学中的马德隆变换和概率密度

本文扩展了量子力学的马德隆-玻姆公式，以描述经典系统和量子系统的时间可逆相互作用。马德隆变换的辛几何导致从标准量子理论中识别扩展波姆轨迹的混合经典-量子拉格朗日路径。由于不再保留经典辛形式，因此明确提出了庞加莱积分的非平凡演化。然而，混合波姆轨迹的经典相空间分量识别了由量子坐标参数化的哈密顿流，并且该流与经典子系统的运动相关。此外，还明确提出了联合经典量子密度的连续性方程。虽然量子子系统的冯诺依曼密度算子在构造上总是正定的，但混合密度通常允许是无符号的。然而，本文最后提出了一个无限的混合哈密顿族，其相应的演化保留了经典子系统的概率密度符号。