How to Quantize Phases and Moduli !

A typical classical interference pattern of two waves with intensities I 1 , I 2 and relative phase ϕ = ϕ 2 − ϕ 1 may be characterized by the 3 observables p = √ I 1 I 2 , p cos ϕ, and −p sin ϕ. They are, e.g. the starting point for the semi-classical operational approach by Noh, Fougères and Mandel (NFM) to the old and notorious phase problem in quantum optics. Following a recent group theoretical quantization of the symplectic space S = {(ϕ ∈ R mod 2π, 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 cos ϕ, −p sin ϕ and p > 0 form the Lie algebra of the group SO ↑ (1, 2). The corresponding self-adjoint generators p cos ϕ = K 1 , − p sin ϕ = K 2 andˆp = 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 {n