如何量化相位和模！
How to Quantize Phases and Moduli!
摘要：
强度为I_1，I_2且相对相位为φ = φ_2-φ_1的两个波的典型经典干涉图样可由3个观测量p = sqrt{I_1I_2}，pcos θ和-psin θ表征。例如，它们是Noh，Fougeres和Mandel（NFM）对量子光学中古老而臭名昭着的相位问题的半经典操作方法的起点。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有争议的复数模和相位的量子化问题：经典观测量pcos，-psin和p > 0的泊松括号构成SO（1，2）群的李代数。相应的自伴生成元K_1，K_2和K_3可由它的不可约酉表示得到。对于正离散级数，模算子K_3具有谱{k+n，n = 0，1，2，…; k > 0}。cos φ和sin φ的自伴算子可以定义为（（1/K_3）K_1 + K_1/K_3）/2和-（（1/K_3）K_2 + K_2/K_3）/2，当k ≥ 0.5时，它们具有理论上所要求的性质。这里提倡的方法解决了例如谐振子的模相位量子化问题，并且似乎为NFM形式主义提供了完整的量子理论基础。
英文摘要：
A typical classical interference pattern of two waves with intensities I_1, I_2 and relative phase phi = phi_2-phi_1 may be characterized by the 3 observables p = sqrt{I_1 I_2}, p cosϕand -p sinϕ. They are, e.g. the starting point for the semi-classical operational approach by Noh, Fougeres and Mandel (NFM) to the old and notorious phase problem in quantum optics. 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 applies those results to that controversial problem of quantizing moduli and phases of complex numbers: The Poisson brackets of the classical observables p cosϕ, -p sinϕand p > 0 form the Lie algebra of the group SO(1,2). The corresponding self-adjoint generators K_1, K_2 and K_3 of that group may be obtained from its irreducible unitary representations. For the positive discrete series the modulus operator K_3 has the spectrum {k+n, n = 0, 1,2,...; k > 0}. Self-adjoint operators for cos phi and sin phi can be defined as ((1/K_3)K_1 + K_1/K_3)/2 and -((1/K_3)K_2 + K_2/K_3)/2 which have the theoretically desired properties for k > or = 0.5. The approach advocated here solves, e.g. the modulus-phase quantization problem for the harmonic oscillator and appears to provide a full quantum theoretical basis for the NFM-formalism.
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