光相空间S^2 = {phi mod 2 pi，I > 0}在群SO（1，2）中的量子化
Quantization of the Optical Phase Space S^2 = {phi mod 2pi, I > 0} in Terms of the Group SO(1,2)
摘要：
量子化正则对角和作用量变量φ和I的问题几乎和量子力学本身一样古老，几十年来一直是量子光学中一个激烈争论但仍未解决的问题。本文提出了一种新的方法来解决这一问题，即SO（1，2）群量子化：关键是相空间S^2 = {phi mod 2 pi，I>0}具有全局结构S^1 x R^+（一个简单的锥），不能用传统的方法进行量子化。由于SO（1，2）群适当地作用于该空间，其正离散级数的幺正表示提供了正确的量子理论框架。空间S^2具有轨道折叠R^2/Z_2的圆锥结构。该结构与辛群Sp（2，R）的中心Z_2密切相关。S^2上的基本变量是函数h_0 = I，h_1 = I cosphi和h_2 = -I sinphi，它们的泊松括号服从李代数so（1，2）。在量子理论中，它们用么正表示的自伴生成元K_0、K_1和K_2来表示。一个重要的预言是经典的毕达哥拉斯关系式h_1^2+h_2^2 = h_0^2在量子理论中可能被违反。对于每一种表示，我们可以定义3种不同类型的相干态，它们的复相位可以只用K_1和K_2来“测量”，而不需要引入任何新的相位算符！详细分析了SO（1，2）结构的光压缩和干涉特性以及谐振子的光压缩和干涉特性。新的相干态可用于引入密度算符的（Husimi型）Q和（Sudarshan-Glauber型）P表示。K_0、K_1和K_2是基本的，因为我们可以用它们构造（复合）位置和动量算符！
英文摘要：
The problem of quantizing the canonical pair angle and action variables phi and I is almost as old as quantum mechanics itself and since decades a strongly debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space S^2 = {phi mod 2pi, I>0} has the global structure S^1 x R^+ (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts appropriately on that space its unitary representations of the positive discrete series provide the correct quantum theoretical framework. The space S^2 has the conic structure of an orbifold R^2/Z_2. That structure is closely related to the center Z_2 of the symplectic group Sp(2,R). The basic variables on S^2 are the functions h_0 = I, h_1 = I cosphi and h_2 = -I sinphi, the Poisson brackets of which obey the Lie algebra so(1,2). In the quantum theory they are represented by self-adjoint generators K_0, K_1 and K_2 of a unitary representation. A crucial prediction is that the classical Pythagorean relation h_1^2+h_2^2 = h_0^2 may be violated in the quantum theory. For each representation one can define 3 different types of coherent states the complex phases of which can be "measured" by means of K_1 and K_2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The new coherent states can be used for the introduction of (Husimi type) Q and (Sudarshan-Glauber type) P representations of the density operator. The 3 operators K_0, K_1 and K_2 are fundamental in the sense that one can construct (composite) position and momentum operators out of them!
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