Quantum chaos, complex spectral representations and time symmetry breaking

As has been shown in recent publications, classical chaos leads to complex 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 the representation can only be implemented by distribution functions (and not by trajectories). A somewhat similar situation occurs in Hamiltonian nonintegrable systems with continuous spectrum (“Large Poincaré Systems” LPS), both in classical and quantum mechanics.The elimination of Poincarés divergences requires an extended formulationof dynamics on the level of distribution functions (or density matrices). This applies already to simple situations such as potential scattering in the case of persistent interactions. There appear characteristic delta-function singularities in the density matrices in momentum representation. Our theory predicts then dissipative processes corresponding to the destruction of invariants of motion throuh Poincaré resonances. This prediction is quantitative agreement with extensive numerical simulations presented here.We concentrate in this paper on potential scattering. We discuss also briefly a simple model of a many-body system (the so called “perfect Lorentz gas”). In both cases we obtain irreducible spectral representations which we consider as the signature of “chaos”.We solve the eigenvalue problem for the Liouville-von Neumann operator for the class of singular density matrices corresponding to persistent scattering. This leads to a complex spectral representation in which cross sections appear as eigenvalues. Our previous results [see our previous paper in Chaos, Solitons & Fractals (1991)] obtained by “subdynamics theory” are now derived through the solution of the eigenvalue problem for singular distributions. Our results can be tested by numerical simulations. Again the agreement is excellent. Note that our results cannotbe derived from conventional quantum theory for probability amplitudes. We have therefore here a simple example of a quantum theory which goes beyond the traditional Schrödinger formulation. As already mentioned, our theory is formulated on the level of density matrices. Wave functions corresponding to persistent scattering (and therefore to singular density matrices) “collapse” as the result of Poincaré divergences.We obtain therefore a unified formulation of quantum theory without any appeal to extra dynamical concepts (such as many worlds, influence of environment,…). The appearance of chaos for LPS through the formulation of complex irreducible representations on the level of density matrices solves therefore not only the “time paradox” as it introduces time symmetry breaking on the microscopic level, but eliminates also the old standing epistemological problems of quantum theory associated to measurement and to decoherence.