We continue our work on large Poincare systems (LPS) characterized by a continuous spectrum and a continuous set of resonances. Poincare's theorem prevents the existence of solutions of the eigenvalue problem associated with the Hamiltonian (as well as the Liouville operator) which would be analytic in the coupling constant. In simple cases, such as the Friedrichs model studied in a recent paper, Poincare's "divergences" can be avoided using a natural time ordering of the dynamical states in the Hilbert space. However, in general, the time ordering has to be performed in the Liouville space of density matrices. To do so, we use the idea of the "dynamics of correlations" introduced in nonequilibrium statistical mechanics by one of the authors (I.P.). In short, we perform a time ordering of correlations according to the number of particles (or degrees of freedom) involved. Transitions to higher correlations are considered as future-oriented transitions and to lower correlations as past-oriented. This leads to new rules of analytic continuation first formulated by Cl. George. We show that we can obtain a spectral resolution of the Liouville-von Neumann operator in terms of generalized eigenfunctions involving complex distributions and complex eigenvalues. These complex distributions have a broken time symmetry. An essential consequence of our theory is that the spectral resolution of the Liouville operator is non-factorizable in quantum theory and not implementable in classical theory. The time evolution in terms of the density matrix becomes irreducible, and cannot be expressed in terms of wave functions (in quantum theory) or in terms of trajectories (in classical mechanics). This is contrary to what happens for integrable systems and for a simple class of LPS treated in our previous paper where the density matrix is reducible. We therefore obtain for LPS an alternative statistical formulation of classical and quantum mechanics on the level of Gibbs ensembles. In this way, we generalize the very meaning of integrability. Our formulation of dynamics includes the second law of thermodynamics, as we may display explicitly and H-function of the Boltzmann type, which decreases monotonously when the system evolves towards equilibrium. The equilibrium states are formed by the set of "eigenmodes" of the Liouville operator characterized by zero eigenvalue. This subspace of the Liouville-von Neumann space acts as an attractor. LPS are mixing in both classical and quantum mechanics. Our approach is also in line with our previous work on nonequilibrium statistical mechanics, which has led to the formulation of a non-Markovian master equation, but it goes much further, as it allows the construction of the spectral representation of the Liouville operator. It is a remarkable fact that the Liouville operator remains Hermitian, and the time evolution unitary, in spite of damping. This is possible because of the introduction of an extended Liouville space (it is a "rigged" Liouville space) spanned by complex time-symmetry broken distributions. Our approach puts the breaking of time-symmetry at the very basis of the microscopic dynamical description. The appearance of broken time-symmetry on the macroscopic level becomes the result of averaging of the microscopic equations of motion. In classical mechanics, LPS are chaotic systems. Our work therefore illustrates the effect of dynamical chaos on the formulation of the basic laws of physics.
