par Tomaschitz, Roman 
Référence International journal of theoretical physics, 31, 2, page (187-210)
Publication Publié, 1992-02
Référence International journal of theoretical physics, 31, 2, page (187-210)
Publication Publié, 1992-02
Article révisé par les pairs
| Résumé : | 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 space H3 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 quotient C(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 products I×S, I a finite open interval, the fibers S compact Riemann surfaces. We give a short derivation of the Patterson formula connecting the ground-state energy with the Hausdorff dimension δ of λ, and give various examples for the calculation of δ from the tessellations of the boundary of H3, induced by the universal coverings of the manifolds. © 1992 Plenum Publishing Corporation. |
