DECOHERENCE, CHAOS, QUANTUM-CLASSICAL CORRESPONDENCE, AND THE ALGORITHMIC ARROW OF TIME

The environment - external or internal degrees of freedom coupled to t he system - can, in effect, monitor some of its observables. As a resu lt, the eigenstates of these observables decohere and behave like clas sical states: Continuous destruction of superpositions leads to enviro nment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macro scopic (but, ultimately, quantum) system is not just difficult to pred ict (requiring an accuracy exponentially increasing with time) but qui ckly ceases to be deterministic in principle as a result of the Heisen berg indeterminacy (which limits the resolution available in the initi al conditions). This happens after a time t((h) over bar) which is onl y logarithmic in the Planck constant. A definitely macroscopic (if som ewhat outrageous) example is afforded by various components of the sol ar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary sy stem as a whole). On the timescale t((h) over bar) the initial minimum uncertainty wavepackets corresponding to celestial bodies would be sm eared over distances of the order of radii of their orbits into ''Schr odinger cat - like'' states, and the concept of a trajectory would cea se to apply. In reality, such paradoxical states are eliminated by dec oherence which helps restore quantum-classical correspondence. The pri ce for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not ju st with the smallness of (h) over bar) the rate of increase of the von Neumann entropy of the decohering system is independent of the streng th of the coupling to the environment, and equal to the sum of the pos itive Lyapunov exponents. Algorithmic aspects of entropy production ar e briefly explored to illustrate the effect of decoherence from the po int of view of the observer. We show that ''decoherence strikes twice' ': introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for t he classicality of his records.