11 research outputs found
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 {\it a
priori}. We define the generalized Stokes operators as the Jordan-Schwinger map
of a triplet of harmonic oscillators with the Gell-Mann 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 ellips
A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product
International audienceLet be a topological space. In this survey we consider several types of configuration spaces, namely the classical (usual) configuration spaces and , the orbit configuration spaces and with respect to a free action of a group on , and the graph configuration spaces and , where is a graph and is a suitable subgroup of the symmetric group . The ordered configuration spaces , , are all subsets of the -fold Cartesian product of with itself, and satisfy . If denotes one of these configuration spaces, we analyse the difference between and from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. We are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion , the homotopy type of the homotopy fibre of the map via certain constructions on various spaces that depend on , and the long exact sequence in homotopy of the fibration involving and arising from the inclusion . In this respect, if is either a surface without boundary, in particular if is the -sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space of the -dimensional sphere by a free action of a Lie Group , we present some recent results obtained in [23,24] for the first case, and in [18] for the second and third cases. We briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, we describe and prove a few of their basic properties. We finish the paper with a list of open questions and problems