Decoherence, chaos, quantum-classical correspondence and the arrow of time

The environment - external or internal degrees of freedom coupled to the object of interest - can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states. Continuous destruction of superpositions leads to the effective environment-induced superselection (einselection), which is beginning to be recognized as a key step in the transition from quantum to classical. We investigate it here in the context of quantum chaos. I show that the evolution of a chaotic macroscopic system is not just difficult to predict (requiring accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg; uncertainty (which limits the available resolution). This happens after a time tt, which is only logarithmic in the Planck constant. For example, various components of the solar system are chaotic, with the Lyapunov timescales ranging from a bit more then a month (Hyperion) to millions of years (planetary system as a whole). On the timescale th the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into "Schrodinger cat-like" states, and the very concept of a "trajectory" would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability. In the classical limit (associated not with the smallness of (h) over bar, but with decoherence) the rate of increase of entropy is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. I end by noting that the cost of information transfer between systems - of the action measured in units of (g) over bar's per bit - decreases with the increasing size. This suggests why information may seem to be so irrelevant for classical dynamics, and yet is obviously so crucial at the quantum level.