Abstract
Geodesic motion in infinite spaces of constant negative curvature provides for the first time an example where a basically quantum mechanical quantity, a ground-state energy, is derived from Newtonian mechanics in a rigorous, nonsemiclassical way. The ground state energy emerges as the Hausdorff dimension of a quasi-self-similar curve at infinity of three-dimensional hyperbolic spaceH 3 in which our manifolds are embedded and where their universal covers are realized. This curve is just the locus of the limit set λ(γ) of the Kleinian group γ of covering transformations, which determines the bounded trajectories in the manifold; all of them lie in the quotientC(λ)γ, C(λ) being the hyperbolic convex hull of λ(γ). The three-dimensional hyperbolic manifolds we construct can be visualized as thickened surfaces, topological productsI×S, I a finite open interval, the fibersS compact Riemann surfaces. We give a short derivation of the Patterson formula connecting the ground-state energy with the Hausdorff dimensionδ of λ, and give various examples for the calculation of δ from the tessellations of the boundary ofH 3, induced by the universal coverings of the manifolds.
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Tomaschitz, R. An alternative to wave mechanics on curved spaces. Int J Theor Phys 31, 187–210 (1992). https://doi.org/10.1007/BF00673252
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DOI: https://doi.org/10.1007/BF00673252