Jordan-Schwinger map, 3D harmonic oscillator constants of motion, and classical and quantum parameters characterizing electromagnetic wave polarization

In this work we introduce a generalization of the Jauch and Rohrlich quantum Stokes operators when the arrival direction from the source is unknown a priori. We define the generalized Stokes operators as the Jordan-Schwinger map of a triplet of harmonic oscillators with the GellMann and Ne’eman SU(3) symmetry group matrices. We show that the elements of the Jordan-Schwinger map are the constants of motion of the three-dimensional isotropic harmonic oscillator. Also, we show that generalized Stokes Operators together with the Gell-Mann and Ne’eman matrices may be used to expand the polarization density matrix. By taking the expectation value of the Stokes operators in a three-mode coherent state of the electromagnetic field, we obtain the corresponding generalized classical Stokes parameters. Finally, by means of the constants of motion of the classical three-dimensional isotropic harmonic oscillator we describe the geometric properties of the polarization ellipse PACS numbers: 42.50.-p, 42.25.-p, 42.25.Ja, 11.30.-j, 03.65.Fd This work was prepared during my posdoctoral year at Departamento de Matemáticas del Centro de Investigación y Estudios Avanzados del IPN, México D F, 07000, Mexico.