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Wigner functions for curved spaces I: On hyperboloids

Miguel Angel Alonso·George S. Pogosyan·Kurt Bernardo Wolf
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Abstract

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 Poschl-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.

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Miguel Angel Alonso, George S. Pogosyan, Kurt Bernardo Wolf. 2002-05-08. Wigner functions for curved spaces I: On hyperboloids. https://doi.org/10.1063/1.1518139

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