  Path integrals appear to offer natural and intuitively appealing methods for
defining quantum-mechanical amplitudes for questions involving spacetime
regions. For example, the amplitude for entering a spatial region during a
given time interval is typically defined by summing over all paths between
given initial and final points but restricting them to pass through the region
at any time. We argue that there is, however, under very general conditions, a
significant complication in such constructions. This is the fact that the
concrete implementation of the restrictions on paths over an interval of time
corresponds, in an operator language, to sharp monitoring at every moment of
time in the given time interval. Such processes suffer from the quantum Zeno
effect - the continual monitoring of a quantum system in a Hilbert subspace
prevents its state from leaving that subspace. As a consequence, path integral
amplitudes defined in this seemingly obvious way have physically and
intuitively unreasonable properties and in particular, no sensible classical
limit. In this paper we describe this frequently-occurring but
little-appreciated phenomenon in some detail, showing clearly the connection
with the quantum Zeno effect. We then show that it may be avoided by
implementing the restriction on paths in the path integral in a "softer" way.
The resulting amplitudes then involve a new coarse graining parameter, which
may be taken to be a timescale \epsilon, describing the softening of the
restrictions on the paths. We argue that the complications arising from the
Zeno effect are then negligible as long as \epsilon >> 1/ E, where E is the
energy scale of the incoming state.
