How to Quantize Phases and Moduli!

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A typical classical interference pattern of two waves with intensities I₁, I₂ and relative phase phi = phi₂-phi₁ may be characterized by the 3 observables p = sqrtI₁ I₂, 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 K₁, K₂ and K₃ of that group may be obtained from its irreducible unitary representations. For the positive discrete series the modulus operator K₃ 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/K₃)K₁ + K₁/K₃)/2 and -((1/K₃)K₂ + K₂/K₃)/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.