QUANTUM CHAOS, COMPLEX SPECTRAL REPRESENTATIONS AND TIME-SYMMETRY BREAKING

As has been shown in recent publications. classical chaos leads to com plex irreducible representation of the evolution operator (such as the Perron-Frobenius operator for chaotic maps). Complex means that time symmetry is broken (appearance of semi-groups) and irreducible that th e representation can only be implemented by distribution functions (an d not by trajectories). A somewhat similar situation occurs in Hamilto nian nonintegrable systems with continuous spectrum (''Large Poincare Systems'' LPS), both in classical and quantum mechanics.The eliminatio n of Poincares divergences requires an extended formulation of dynamic s on the level of distribution functions (or density matrices). This a pplies already to simple situations such as potential scattering in th e case of persistent interactions. There appear characteristic delta-f unction singularities in the density matrices in momentum representati on. Our theory predicts then dissipative processes corresponding to th e destruction of invariants of motion through Poincare resonances. Thi s prediction is quantitative agreement with extensive numerical simula tions presented here. We concentrate in this paper on potential scatte ring. We discuss also briefly a simple model of a many-body system (th e so called ''perfect Lorentz gas''). In both cases we obtain irreduci ble spectral representations which we consider as the signature of ''c haos''. We solve the eigenvalue problem for the Liouville-von Neumann operator for the class of singular density matrices corresponding to p ersistent scattering. This leads to a complex spectral representation in which cross sections appear as eigenvalues. Our previous results [s ee our previous paper in Chaos, Solitons & Fractals (1991)] obtained b y ''subdynamics theory'' are now derived through the solution of the e igenvalue problem for singular distributions. Our results can be teste d by numerical simulations. Again the agreement is excellent. Note tha t our results cannot be derived from conventional quantum theory for p robability amplitudes. We have therefore here a simple example of a qu antum theory which goes beyond the traditional Schrodinger formulation . As already mentioned, our theory is formulated on the level of densi ty matrices. Wave functions corresponding to persistent scattering (an d therefore to singular density matrices) ''collapse'' as the result o f Poincare divergences. We obtain therefore a unified formulation of q uantum theory without any appeal to extra dynamical concepts (such as many worlds, influence of environment,...). The appearance of chaos fo r LPS through the formulation of complex irreducible representations o n the level of density matrices solves therefore not only the ''time p aradox'' as it introduces time symmetry breaking on the microscopic le vel, but eliminates also the old standing epistemological problems of quantum theory associated to measurement and to decoherence.