Wigner Functions for the Pair Angle and Orbital Angular Momentum
H. A. Kastrup
备注：
15 pages, 3 figures. Considerably expanded update of the previous Letter version, including discussions of the marginal distributions and of several examples. Accepted for publication in Phys. Rev. A
构造物理和数学定义良好的维格纳函数的正则对角和角动量的问题得到解决。在R^2中构造平面相空间（q，p）的维格纳函数的一个关键元素是海森堡-外尔群，而在S^1 x R中构造圆柱相空间（θ，p）的相应群是平面的欧几里得群E（2）及其酉表示。在这里，角θ被替换为唯一对应于单位圆上的点的对（cos θ，sin θ）。平面相空间和圆柱相空间的维格纳函数的主要结构性质是惊人的相似。一个至关重要的作用是sinc函数，它提供了插值的不连续的量子化角动量的连续经典的，根据著名的惠特克基数函数，众所周知的插值和采样理论。角动量（连续）和角动量（不连续）的量子力学边缘分布通常是通过（θ，p）维格纳函数的适当积分唯一获得的。在所讨论的例子中，有一个简单的“猫”态的基本系统。
英文摘要：
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner functions for the planar phase space (q,p) in R^2 is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space (theta,p) in S^1 x R is the Euclidean group E(2) of the plane and its unitary representations. Here the angle theta is replaced by the pair (cos theta, sin theta) which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role plays the sinc function which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function, well-known from interpolation and sampling theories. The quantum mechanical marginal distributions for angle (continuous) and angular momentum (discontinuous) are as usual uniquely obtained by appropriate integrations of the (theta,p) Wigner function. Among the examples discussed is an elementary system of simple "cat" states.
The framework of Wigner functions for the canonical pair angle and orbital angular momentum, derived and analyzed in 2 recent papers [H. A. Kastrup, Phys.Rev. A 94, 062113(2016) and Phys.Rev. A 95, 052111(2017)] is applied to elementary concepts of quantum information like qubits and 2-qubits, e.g., entangled EPR/Bell states etc. Properties of the associated Wigner functions of such superposed states (pure and mixed) are discussed and illustrated. The Wigner functions of ERP/Bell states are distinguished by their topologically "twisted" domain on the configuration subspace S^1xS^1, a torus, of…
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Institute for Theoretical Physics and Astrophysics, University of Würzburg（维尔茨堡大学理论物理与天体物理研究所）；Würzburg-Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群ctd.qmat）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Department of Physics, University of California, Berkeley（加州大学伯克利分校物理系）；Materials Sciences Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室材料科学部）；Department of Physics and Astronomy, Rice University（莱斯大学物理与天文系）
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