6 - Quantum and Classical in Phase Space: Decoherence and the Second Law

Summary

The study of the quantum–classical correspondence has been focused on the quantum measurement problem. However, most of the discussion in the preceding chapters is motivated by a broader question: Why do we perceive our quantum Universe as classical? Therefore, emergence of the classical phase space and Newtonian dynamics from the quantum Hilbert space must be addressed. Chapter 6 starts by re-deriving decoherence rate for non-local superpositions using the Wigner representation of quantum states. We then discuss the circumstances that, in some situations, make classical points a useful idealization of the quantum states of many-body systems. This classical structure of phase space emerges along with the (at least approximately reversible) Newtonian equations of motion. Approximate reversibility is a non-trivial desideratum given that the quantum evolution of the corresponding open system is typically irreversible. We show when such approximately reversible evolution is possible. We also discuss quantum counterparts of classically chaotic systems and show that, as a consequence of decoherence, their evolution tends to be fundamentally irreversible: They produce entropy at the rate determined by the Lyapunov exponents that characterize classical chaos. Thus, quantum decoherence provides a rigorous rationale for the approximations that led to Boltzmann’s H-theorem.