How to Quantize Phases and Moduli!

Abstract

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.