Wigner functions for curved spaces I: On hyperboloids

We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature —in this paper on hyperboloids—, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D, 1) transformations. To the free systems obeying the Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the 1-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl-Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature. PACS: 03.65.Fd, 03.65.Pm, 11.30.Cp Permanent address: Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia and International Center for Advanced Studies, Yerevan State University, Yerevan, Armenia