We extend a procedure for construction of the photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are re-established by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our position operator, like that proposed by Pryce, is a matrix that does not commute with the spin operator. Unlike the Pryce operator, however, our operator has commuting components, but the commutators of these components with the total angular momentum require an extra term to rotate the matrices for each vector component around the momentum direction. Several proofs of the nonexistence of a photon position operator with commuting components are based on overly restrictive premises that do not apply here.
导出了光子计数阵列探测器的正算子值测度（POVM），发现它等于在像素面积和测量时间上积分的光子通量密度。由于光子通量密度等于数密度乘以光速，这在理论上证明了光子计数阵列提供光子位置的粗粒度测量的观察。本文得到的POVM可以写成一组投影到局域态基上的投影，与最近量子成像方案中光子位置的描述一致[M. Tsang，Phys. Rev. Lett. \textbf{102}，253601（2009）]。描述光子计数实验的波函数是光子态矢量在局域基上的投影。坍缩是指向电磁真空而不是局域态，因此违反了量子力学的教科书规则，但与广义可观测量理论和入射光子的非局域性是相容的。
Control of particle propagation beyond the uncertainty limit by interference between position and momentum
如Phys. Rev. A 96，020101（R）（2017）所示，通过增加在"非此即彼"之外的位置和动量的狭窄间隔内找到粒子的概率，就有可能证明量子粒子在自由空间中不沿直线运动。极限为0.5，使用定位在位置上的分量和定位在动量上的分量之间的相长量子干涉。然后，在稍后的时间在相应的空间间隔中找到粒子的概率违反了基于牛顿第一定律的有效性的粒子传播不等式的下限。在本文中，本地化的两个状态分量在各自的目标区间的问题是解决通过引入一组的三个系数，定量地描述任意波函数的本地化。这种表征被应用到高斯的叠加，如果高斯波函数的宽度与位置和动量间隔的大小一起被优化，则粒子传播不等式的违反超过5%。结果表明，粒子传播不等式的违反源于量子干涉将粒子的初始位置和动量与未来位置联系起来的基本方式，表明违反是量子极限下因果关系的一个基本特征。
The explicit form of the unitary representation of the Poincaré group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators
