标题： 弱值的时间导数 显示英文标题

标题： Time Derivatives of Weak Values

摘要： 物理属性的时间导数常常会引出另一个有意义的属性。 由于弱值提供了从期望值无法推导出的经验性见解，本文探讨了可以从弱值的时间导数中获得哪些物理属性。 它表明，一般来说，规范不变弱值的时间导数既不是弱值，也不是规范不变量。 提出了两个条件以确保弱值的左时间导数或右时间导数也是规范不变弱值。 在这些条件下，可以为弱值推导出一个局部的类似厄伦费斯特定理，从而为弱值的时间导数提供自然的解释。 值得注意的是，系统位置的一个单独测量得到的弱值提供了关于两个额外未测量弱值的信息：系统的局部速度和加速度，分别通过初始弱值的一阶和二阶时间导数。 这些发现也为实验者提供了将弱值理论转化为实际实验室设置的指导方针，为创新量子技术铺平了道路。 一个例子展示了如何通过位置弱值的一阶和二阶时间导数，在特定的位置和时间确定电磁场。 显示英文摘要

摘要： The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a gauge-invariant weak value is neither a weak value nor a gauge-invariant quantity. Two conditions are presented to ensure that the left- or right-time derivative of a weak value is also a gauge-invariant weak value. Under these conditions, a local Ehrenfest-like theorem can be derived for weak values giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.