Statistical mechanics and the duality of quantum mechanical time evolution

TL;DR

This paper demonstrates that the Boltzmann equation governs the time evolution of quantum observation operators, providing a quantum-mechanical justification for statistical mechanics without introducing additional probabilistic assumptions.

Contribution

It establishes a duality framework where the principle of equal a priori probabilities is proven within quantum mechanics, linking quantum state evolution to classical statistical behavior.

Findings

Time evolution of observation operators obeys the Boltzmann equation

Differences from equal probability states become unobservable over time

Provides a quantum foundation for statistical mechanics

Abstract

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.