The arrow of time, irreversibility, equilibrium and measurement in quantum mechanics

Quantum mechanics is often criticized for its inability to account for the second law of thermodynamics and the measurement process, leading to ad hoc assumptions regarding wave function collapse and decoherence. The authors address this by examining the statistical mechanics of large quantum systems. They argue that the standard formulation of quantum mechanics is incomplete because it fails to account for the thermodynamic limit. By utilizing the algebraic formulation of quantum mechanics and the Liouville-von Neumann equation, they analyze the spectral properties of the Liouvillian operator to show how irreversibility and equilibrium emerge from the underlying dynamics.

The study reveals that for isolated quantum systems with an absolutely continuous spectrum—a condition naturally satisfied in the thermodynamic limit—the time-reversal symmetry of the equations of motion is broken. This symmetry breaking leads to a semi-group evolution where the system approaches a microcanonical equilibrium state over time. In this process, quantum coherence is lost, and pure states are transformed into mixtures, which corresponds to an increase in entropy. Furthermore, the authors show that when a macroscopic measurement apparatus is included, the outcome of a measurement aligns with the von Neumann projection postulate, with probabilities governed by the Born rule.

This work provides a rigorous, internal solution to the measurement problem and the arrow of time within quantum mechanics, without requiring external interpretations or subjective assumptions. By demonstrating that irreversibility is a consequence of the continuous spectrum of the Liouvillian in large systems, the authors bridge the gap between microscopic quantum dynamics and macroscopic thermodynamic behavior. This approach aligns quantum theory with classical ergodic theory, offering a consistent framework for understanding how classically observable states emerge from quantum systems.