  The origin of classical predictability is investigated for the one
dimensional harmonic chain considered as a closed quantum mechanical system. By
comparing the properties of a family of coarse-grained descriptions of the
chain, we conclude that local coarse-grainings in this family are more useful
for prediction than nonlocal ones. A quantum mechanical system exhibits
classical behavior when the probability is high for histories having the
correlations in time implied by classical deterministic laws. But approximate
classical determinism holds only for certain coarse-grainings and then only if
the initial state of the system is suitably restricted. Coarse-grainings by the
values of the hydrodynamic variables (integrals over suitable volumes of
densities of approximately conserved quantities) define the histories usually
used in classical physics. But what distinguishes this coarse-graining from
others? This paper approaches this question by analyzing a family of
coarse-grainings for the linear harmonic chain. At one extreme in the family
the chain is divided into local groups of $N$ atoms. At the other extreme the
$N$ atoms are distributed nonlocally over the whole chain. Each coarse-graining
follows the average (center of mass) positions of the groups and ignores the
``internal'' coordinates within each group, these constituting a different
environment for each coarse-graining. We conclude that noise, decoherence, and
computational complexity favor locality over nonlocality for deterministic
predictability.
