与干涉相关的相位和模的群论量子化
Group Theoretical Quantization of Phase and Modulus Related to Interferences
摘要：
在最近的辛空间S={（phi in R mod 2 pi，p>0）}，本文将这些结果应用于干涉现象中的量化模和相位的老问题：该群的自伴李代数生成元K_1，K_2和K_3对应于经典观测量pcos（phi），-psin（phi）和p > 0，它们的泊松括号也服从该李代数。对于正项级数的不可约酉表示，模算子K_3具有正的离散谱{n+k，n= 0，1，2，.; k > 0}。cos（phi）和sin（phi）的自伴算子可以定义为（K_3^{-1} K_1 + K_1 K_3^{-1}）/2和-（K_3^{-1} K_2 + K_2 K_3^{-1}）/2，它们在k >0.32时具有理论上所需的性质。计算了关于本征态数和相干态的矩阵元。一个结论是群论量子化可以通过量子光学实验来检验。
英文摘要：
Following a recent group theoretical quantization of the symplectic space S={(phi in R mod 2pi, p>0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper proposes an application of those results to the old problem of quantizing modulus and phase in interference phenomena: The self-adjoint Lie algebra generators K_1, K_2 and K_3 of that group correspond to the classical observables p cos(phi), -p sin(phi) and p > 0 the Poisson brackets of which obey that Lie algebra, too. For the irreducible unitary representations of the positive series the modulus operator K_3 has the positive discrete spectrum {n+k, n=0,1,2,...; k > 0}. Self-adjoint operators for cos(phi) and sin(phi) can then be defined as (K_3^{-1}K_1 + K_1 K_3^{-1})/2 and - (K_3^{-1} K_2 + K_2 K_3^{-1})/2 which have the theoretically desired properties for k >0.32. Some matrix elements with respect to number eigenstates and with respect to coherent states are calculated. One conclusion is that group theoretical quantization may be tested by quantum optical experiments.
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