DISCRETE-SERIES OF SU(1, 1) AND GAUSSIAN DEFORMATION

Starting from the holomorphic discrete series of SU(1, 1), we construc t a hyperbolic analogue of a quantum mechanical harmonic oscillator wi th a Poincare disc as nonlinear phase space. A Bargmann-type transform is established which interwines the real wave and complex wave repres entation. The probability distribution of the position observable in v acuum is calculated explicitly, it admits the Gaussian law of the ordi nary harmonic oscillator as zero-curvature limit.