  Using the Gell-Mann and Hartle formalism of generalized quantum mechanics of
closed systems, we study the classical limit of coarse-grained spacetime
histories and their decoherence. The system under consideration is
one-dimensional and consists of a particle coupled to a von Neumann apparatus
that performs a measurement of the position of the particle during the finite
time interval during which the histories of this system take place. We consider
two cases: a free particle and a harmonic oscillator. The real line is divided
into intervals of the same length, and coarse-grained histories are defined by
the time average of the position of the particle on a given Feynman path to be
within one of these intervals. The position of the pointer in each Feynman path
correlates with this time average. The class operators for this system have
been evaluated, and the decoherence functional shows that these coarse-grained
histories do not decohere, not even when initially either the particle or the
pointer is in an eigenstate of position. Decoherence is obtained only when the
classical limit is taken. Qualitative arguments for decoherence in the
classical limit are presented for the case of a general particle potential.
