QUANTUM STEREOGRAPHIC PROJECTION AND THE HOMOGRAPHIC OSCILLATOR

The quantum deformation created by the stereographic mapping from S-2 to C is studied. It is shown that the resulting algebra is locally iso morphic to su(2) and is an unconventional deformation of which the und eformed limit is a contraction onto the harmonic oscillator algebra. T he deformation parameter is given naturally by the central invariant o f the embedding su(2). The deformed algebra is identified as a member of a larger class of quartic q oscillators. We next study the deformat ions in the corresponding Jordan-Schwinger representation of two indep endent deformed oscillators and solve for the deforming transformation . The invertibility of this transformation guarantees an implicit copr oduct law which is also discussed. Finally we discuss the analogy betw een Poincare's geometric interpretation of the quantum Stokes paramete rs of polarization and the stereographic projection as an important ph ysical application of the latter.