  We argue that under very general conditions, there is a significant
complication in amplitudes for spacetimes regions constructed from path
integrals. 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 $\eps$,
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 $\eps
>> 1/ E$, where $E$ is the energy scale of the incoming state. Our criticisms
of path integral constructions largely apply to approaches to quantum theory
such as the decoherent histories approach or quantum measure theory, which do
not specifically involve measurements. We address some criticisms of our
approach by Sokolovksi, concerning the relevance of our results to
measurement-based models.
