Abstract
Starting from the holomorphic discrete series of SU(1, 1), we construct a hyperbolic analogue of a quantum mechanical harmonic oscillator with a Poincare disc as nonlinear phase space. A Bargmann-type transform is established which interwines the real wave and complex wave representation. The probability distribution of the position observable in vacuum is calculated explicitly, it admits the Gaussian law of the ordinary harmonic oscillator as zero-curvature limit.
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