  One of the reasons we expect a standard quantum mechanics, which predicts
probabilities for alternatives defined on spacelike slices, to be inadequate
for quantum gravity is that the notion of ``spacelike'' is ill-defined in a
theory where the metric itself is behaving quantum-mechanically. Spacetime
coarse grainings--sets of alternatives defined with respect to a region
extended in time as well as space--have previously been considered in the
quantum mechanics of a single nonrelativistic particle. This work examines
spacetime coarse grainings in the quantum mechanics of a free *relativistic*
particle, not as a theory for real particles (which are described by quantum
field theory) but as a simple reparametrization-invariant theory which may
model some features of quantum gravity.
  We use Hartle's sum-over-histories generalized quantum mechanics formalism,
in which probabilities are assigned to a set of alternatives if they decohere,
i.e., if the quantum-mechanical interference between their branches of the
initial state is nearly zero. For one extremely simple set of alternatives, we
calculate these branches. Despite the nonlocality of the positive-definite
version of the Klein-Gordon inner product, some initial conditions are found to
give decoherence and allow the consistent assignment of probabilities.
