Statistical mechanics and the duality of quantum mechanical time   evolution
  Through the H theorem, Bolzmann attempted to validate the foundations of statistical mechanics. However, it is incompatible with the fundamental laws of mechanics because its deduction requires the introduction of probability. In this paper we attempt a justification of statistical mechanics without deviating from the existing framework of quantum mechanics. We point out that the principle of equal a priori probabilities is easily proven in the dual space. The dual of the space of the quantum states is the space of the observations. We then prove that time evolution of the operators of observations obeys Boltzmann equation. This result implies that the difference of the states from equal probability becomes unobservable as time elapses. 
Tsuguo Mogami