不确定性原理、不确定性关系与底层轨迹：费曼、纳尔逊、玻姆与持续卡茨-狄拉克动力学
The Uncertainty Principle, Uncertainty Relations, and Underlying Trajectories: Feynman, Nelson, Bohm, and Persistent Kac-Dirac Dynamics
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- Tagore Centre for Natural Sciences and Philosophy（泰戈尔自然科学与哲学中心）
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中文总结 AI 辅助
本文区分量子不确定性关系与广义不确定性原理，通过费曼路径、纳尔逊随机力学、玻姆力学及卡茨动力学证明不确定性关系不排除底层轨迹描述，从而驳斥其否定微观轨迹的本体论解读。
中文摘要 AI 辅助
应当区分量子不确定性关系与更广义的不确定性原理。前者是量子力学的精确统计推论；后者若被解释为排除共轭变量与粒子轨迹的同时确定性，则是一种额外的本体论主张，并非由不确定性关系本身所蕴含。我们利用费曼路径、纳尔逊的随机力学、玻姆力学以及有限速度的持续卡茨动力学来考察这一区分。费曼路径表现出类布朗的短时间标度，而纳尔逊则从维纳扩散并辅以动力学假设推导出薛定谔动力学。相比之下，玻姆力学在保留确定粒子轨迹的同时复现了量子预言。卡茨动力学提供连续、分段可微的有限速度轨迹，其扩散极限趋近于维纳运动学。此外，经过适当的威克旋转，其耦合的双扇区输运方程可映射为狄拉克方程。这些实例表明，实验确立的不确定性关系与显著不同的底层路径结构相容。因此，它们本身并不排除对量子动力学的微观轨迹描述。
英文摘要
A distinction should be made between quantum uncertainty relations and the broader uncertainty principle. The former are precise statistical consequences of quantum mechanics; the latter, when interpreted as excluding simultaneously definite conjugate variables and particle trajectories, is an additional ontological claim not implied by the uncertainty relations themselves. We examine this distinction using Feynman paths, Nelson's stochastic mechanics, Bohmian mechanics, and finite-speed persistent Kac dynamics. Feynman paths exhibit Brownian-like short-time scaling, while Nelson derives Schrödinger dynamics from Wiener diffusion supplemented by dynamical assumptions. Bohmian mechanics, by contrast, reproduces quantum predictions while retaining definite particle trajectories. Kac dynamics provides continuous, piecewise differentiable finite-speed trajectories whose diffusive limit approaches Wiener kinematics. Moreover, after an appropriate Wick rotation, its coupled two-sector transport equations can be mapped to the Dirac equation. These examples demonstrate that the experimentally established uncertainty relations are compatible with markedly different underlying path structures. They therefore do not, by themselves, rule out a microscopic trajectory description of quantum dynamics.
