An alternative to wave mechanics on curved spaces
- Universite Libre de Bruxelles (Belgium)
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, non-semiclassical way. The ground state energy emerges as the Hausdorff dimension of a quasi-self-similar curve at infinity of three-dimensional hyperbolic space H{sup 3} in which our manifolds are embedded and where their universal covers are realized. This curve is just the locus of the limit set {Lambda}({Gamma}) of the Kleinian group {Gamma} of covering transformations, which determines the bounded trajectories in the manifold; all of them lie in the quotient C({Lambda})/{Gamma}, C({Gamma}) being the hyperbolic convex hull of {Lambda}({Gamma}). The three-dimensional hyperbolic manifolds we construct can be visualized as thickened surfaces, topological products IxS, 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 {delta} of {Lambda}, and give various examples for the calculation of {delta} from the tessellations of the boundary of H{sup 3}, induced by the universal coverings of the manifolds. 33 refs., 13 figs., 2 tabs.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 125437
- Journal Information:
- International Journal of Theoretical Physics, Vol. 31, Issue 2; Other Information: PBD: Feb 1992
- Country of Publication:
- United States
- Language:
- English
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