标题： 关于薛定谔动力学的对偶结构 显示英文标题

标题： On the dual structure of the Schrödinger dynamics

摘要： 本文阐明了薛定谔动力学在两个相关阶段中的双重结构：(1) 我们首先从零开始推导出实值薛定谔方程，而不参考经典力学、波动力学或光学，从而获得对薛定谔（波）函数的具体而清晰的解释。 从粒子密度分布函数在配置空间中分解为两个分量向量开始，我们对其施加非常简单的条件，如时空的平移不变性和在给定势函数下的通量守恒。 一个实值路径积分被构造成实值薛定谔方程的格林函数。 (2) 然后以与薛定谔方程兼容的方式研究量子随机路径动力学。 它们之间的关系类似于朗之万动力学与扩散方程的关系。 每条量子路径描述了配置空间中的\textquotedblleft 轨迹\textquotedblright ，例如双缝实验中单独发射的电子在测量板上依次留下一个点，而累积的点则产生由薛定谔函数绝对值平方预测的条纹图案。 我们从伊藤随机微分方程、费曼-卡茨公式和相关的抛物型偏微分方程之间的关系出发，其中一个是薛定谔方程的转换对象。 讨论了量子内在随机性以及量子路径之间的间接相关性等物理意义。 薛定谔函数（作为整体）与量子路径（作为其部分）之间的自指非线性相互关系被识别为量子动力学中的终极奥秘。 显示英文摘要

摘要： This paper elucidates the dual structure of the Schr\"{o}dinger dynamics in two correlated stages: (1) We first derive the real-valued Schr\"{o}dinger equation from scratch without referring to classical mechanics, wave mechanics, nor optics, and thereby attain a concrete and clear interpretation of the Schr\"{o}dinger (wave) function. Beginning with a factorization of the density distribution function of the particles to two component vectors in configuration space, we impose very simple conditions on them such as translational invariance of space-time and the conservation of flux under a given potential function. A real-valued path-integral is formulated as a Green function for the real-valued Schr\"{o}dinger equation. (2) We then study a quantum stochastic path dynamics in a manner compatible with the Schr\"{o}dinger equation. The relation between them is like the Langevin dynamics with the diffusion equation. Each quantum path describes a \textquotedblleft trajectory\textquotedblright\ in configuration space representing, for instance, a singly launched electron in the double-slit experiment that leaves a spot one by one at the measurement board, while accumulated spots give rise to the fringe pattern as predicted by the absolute square of the Schr\"{o}dinger function. We start from the relationship between the Ito stochastic differential equation, the Feynman-Kac formula, and the associated parabolic partial differential equations, to one of which\ the Schr\"{o}dinger equation is transformed. The physical significance of the quantum intrinsic stochasticity and the indirect correlation among the quantum paths and so on are discussed. The self-referential nonlinear interrelationship between the Schr\"{o}dinger functions (regarded as a whole) and the quantum paths (as its parts) is identified as the ultimate mystery in quantum dynamics.