Group Theoretical Quantization of Phase and Modulus Related to Interferences

Abstract

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 proposes an application of those results to the old problem of quantizing modulus and phase in interference phenomena: The self-adjoint Lie algebra generators K_1, K_2 and K_3 of that group correspond to the classical observables p cos(phi), -p sin(phi) and p > 0 the Poisson brackets of which obey that Lie algebra, too. For the irreducible unitary representations of the positive series the modulus operator K_3 has the positive discrete spectrum {n+k, n=0,1,2,...; k > 0}. Self-adjoint operators for cos(phi) and sin(phi) can then be defined as (K_3^{-1}K_1 + K_1 K_3^{-1})/2 and - (K_3^{-1} K_2 + K_2 K_3^{-1})/2 which have the theoretically desired properties for k >0.32. Some matrix elements with respect to number eigenstates and with respect to coherent states are calculated. One conclusion is that group theoretical quantization may be tested by quantum optical experiments.