单向电流作为首到达时间POVM：MS-Kijowski恒等式、物理解释及数学应用
The Unidirectional Current as First Arrival-Time POVM: An MS-Kijowski Identity, Physical Interpretation, and Mathematical Applications
AI总结：
该研究将MS单向首到达电流与Kijowski到达时间分布关联，构建到达时间POVM，揭示两种量子到达时间方法的等价性，结合探测器动力学与正算子测量结构。
AI中文摘要：
基于探测器的首到达模型与基于算符的到达时间可观测量是量子到达时间问题的两种概念上截然不同的方法。本文通过使用Marchewka和Schuss（MS）的正单向首到达电流作为基础，构建一族正到达时间算符，归一化后得到到达时间POVM，将这两种方法关联起来。首到达特性继承自具有Dirichlet边界的动力学，探测器响应通过非负谱函数λ(k)在振幅层面引入。对于一维自由粒子，本征值方程的解唯一选取λ(k)=π/(4k)，得到到达时间POVM。因此，归一化的单向电流可解释为广义玻恩定则概率密度，而非原始MS公式中的风险率。本文进一步证明，归一化的单侧MS振幅在无关相位范围内与Kijowski到达时间分布中对应的动量 sector 振幅一致，因此Kijowski分布中的两个方向贡献可通过边界两侧定义的两个首到达问题重现，这建立了所得到达时间统计量之间的等价性，同时强调了不同的物理解释：MS描述具有相干动量分量的首到达过程，而Kijowski的POVM将两个动量 sector 视为独立的方向贡献。综上，归一化单向电流提供了在某点首到达的POVM描述，结合了基于探测器的动力学与正算子测量结构。
英文摘要：
Detector-based first-arrival models and operator-based arrival-time observables provide two conceptually distinct approaches to the quantum time-of-arrival problem. Here we connect these approaches by using the positive unidirectional first-arrival current of Marchewka and Schuss (MS) as the basis for constructing a family of positive arrival-time operators and, after normalization, an arrival-time POVM. The first-arrival character is inherited from the dynamics with a Dirichlet boundary, while the detector response is introduced at the amplitude level through a non-negative spectral function $λ(k)$. For a one-dimensional free particle, resolution of the identity uniquely selects $λ(k)=\fracπ{4k}$, yielding an arrival-time POVM. The normalized unidirectional current can therefore be interpreted as a generalized Born-rule probability density rather than as a hazard rate, as in the original MS formulation. We further show that the normalized one-sided MS amplitude coincides, up to an irrelevant phase, with the corresponding momentum-sector amplitude in Kijowski's arrival-time distribution. Consequently, the two directional contributions in Kijowski's distribution can be reproduced by two first-arrival problems defined on opposite sides of the boundary. This establishes an equivalence between the resulting arrival-time statistics while emphasizing the different physical interpretations: MS describes a first-arrival process with coherent momentum components, whereas Kijowski's POVM treats the two momentum sectors as separate directional contributions. Thus, the normalized unidirectional current provides a POVM description of first arrival at a point, combining detector-based dynamics with a positive-operator measurement structure.
