INTRINSIC IRREVERSIBILITY AND INTEGRABILITY OF DYNAMICS

Irreversibility as the emergence of a priviledged direction of time arises in an intrinsic way at the fundamental level for highly unstable dynamical systems, such as Kolmogorov systems or large Poincare systems. The presence of resonances in large Poincare systems causes a breakdown of the conventional perturbation methods analytic in the coupling parameter. These difficulties are manifestations of general limitations to computability for unstable dynamical systems. However, a natural ordering of the dynamical states leads to a well-defined prescription for the regularization of the propagators which lifts the divergence and gives rise to an extension of the eigenvalue problem to the complex plane. The extension acquires meaning in suitable rigged Hilbert spaces which are constructed explicitly for the Friedrichs model. We show that the unitary evolution group, when extended, splits into two semigroups, one decaying in the future and the other in the past. Irreversibility emerges as the selection of the semigroup compatible with our future observations. In this way the problems of integration and irreversibility both enjoy a common solution in the extended space.