Wigner functions for curved spaces I: On hyperboloids

Details Wigner functions for curved spaces I: On hyperboloids Wigner functions for curved spaces I: On hyperboloids

2002-05-08

arxiv.org/abs/quant-ph/0205041v1

Physics

Quantum Physics

25 pages (included one figure), Latex file

Scientific paper

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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