Complex Systems

Quantum Chaos on Hyperbolic Manifolds: A New Approach to Cosmology Download PDF

Roman Tomaschitz
Dipartimento di Matematica Pura ed Applicata
dell' Università degli Studi di Padova,
Via Belzoni 7, I-35131 Padova, Italy
Instituts Internationaux de Physique et de Chimie Solvay
Université Libre de Bruxelles,
C.P. 231--Campus Plaine, Bd. du Triomphe,
B-1050 Brussels, Belgium


We consider classical and quantum motion on multiply connected hyperbolic spaces, which appear as space-like slices in Robertson-Walker cosmologies. The topological structure of these manifolds creates on the one hand bounded chaotic trajectories, and on the other hand quantal bound states whose wave functions can be reconstructed from the chaotic geodesics. We obtain an exact relation between a probabilistic quantum mechanical wave field and the corresponding classical system, which is likewise probabilistic because of the instabilities of the trajectories with respect to the initial conditions. The central part in this reconstruction is played by the fractal limit set of the covering group of the manifold. This limit set determines the bounded chaotic trajectories on the manifold. Its Hausdorff measure and dimension determine the wave function of the quantum mechanical bound state for geodesic motion.

We investigate relativistic scalar wave fields in de Sitter cosmologies, coupled to the curvature scalar of the manifold. We study the influence of the topological structure of space-time on their time evolution. Likewise we calculate the time asymptotics of their energies in the early and late stages of the cosmic expansion.While in the late stages both bounded and unbounded states approach the same rest energy, they show significantly different behavior at the beginning of the expansion. While the stable bound states have simple power law behavior, extended states show oscillations in their energy, with a frequency and an amplitude both diverging to infinity, indicating the instability of the quantum field at the beginning of the cosmic expansion.