2001/09/03 by H.A. Kastrup, H. A. Kastrup, Kastrup, H. A.
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods #Quantum Mechanics and Non-Hermitian Physics #hep-th #math-ph #math.MP #physics.atom-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0109013
27 pages, Latex; Refs. added, additional remarks on the harmonic oscillator
arxiv created 2001/09/18 · arxiv updated 2009/11/30
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.