The complex spectral representation of the Liouville–von Neumann operator outside Hilbert space is applied to the decoherence problem in quantum Brownian motion. In contrast to the path-integral method, often used in the context of quantum decoherence for the case where the environment surrounding the Brownian particle (subsystem) is in thermal equilibrium, our spectral representation is applicable to systems far from equilibrium, including a pure state for the surrounding bath. Starting with this pure initial condition, the subsystem evolves in time obeying a diffusion-type kinetic equation. Hence, the collapse of wave functions is a dynamical phenomenon occurring outside Hilbert space, and is not simply a contamination of the subsystem, a popular view accepted in the so-called “environmental” approach, by the mixed nature of the thermal bath. The essential element in the understanding of quantum decoherence is the “extensivity” of quantities characterizing the thermodynamic limit. Quantum Zeno time is shown to be a lower bound of the decoherence time.

• Received 21 December 2000

DOI:https://doi.org/10.1103/PhysRevE.65.046102