Thermodynamic limit, Hilbert space and breaking of time symmetry

The role of the thermodynamic limit for “large” non-integrable systems (so-called Large Poincaré systems) is investigated. As had been shown by Dirac in field theory, the interaction between the degrees of freedom destroy the Hilbert space structure. Here the situation is somewhat similar. The spectral analysis of the Liouville operator has therefore to be performed in a generalized function space (“rigid Hilbert” space) allowing the appearance of complex eigenvalues. As the result, the symmetry between past and future is broken. The laws of physics then express, in general, “possibilities” and no more “certitudes”. The fundamental description is in terms of probabilities and no more in terms of trajectories or wave functions which are associated with stochastic realizations.