Quantum stereographic projection and the homographic oscillator

Abstract

The quantum deformation created by the stenographic mapping from S2 to C is studied. It is shown that the resulting algebra is locally isomorphic to su(2) and is an unconventional deformation of which the undeformed limit is a contraction onto the harmonic oscillator algebra. The deformation parameter is given naturally by the central invariant of the embedding su(2). The deformed algebra is identified as a member of a larger class of quartic q oscillators. We next study the deformations in the corresponding Jordan-Schwinger representation of two independent deformed oscillators and solve for the deforming transformation. The invertibility of this transformation guarantees an implicit coproduct law which is also discussed. Finally we discuss the analogy between Poincaré's geometric interpretation of the quantum Stokes parameters of polarization and the stereographic projection as an important physical application of the latter.