  Feynman's sum-over-histories formulation of quantum mechanics is reviewed as
an independent statement of quantum theory in spacetime form. It is different
from the usual Schr\"odinger-Heisenberg formulation that utilizes states on
spacelike surfaces because it assigns probabilities to different sets of
alternatives. Sum-over-histories quantum mechanics can be generalized to deal
with spacetime alternatives that are not "at definite moments of time". An
example in field theory is the set of alternative ranges of values of a field
averaged over a spacetime region. An example in particle mechanics is the set
of the alternatives defined by whether a particle never crosses a fixed
spacetime region or crosses it at least once. The general notion of a set of
spacetime alternatives is a partition (coarse-graining) of the histories into
an exhaustive set of exclusive classes. With this generalization the
sum-over-histories formulation can be said to be in fully spacetime form with
dynamics represented by path integrals over spacetime histories and
alternatives defined as spacetime partitions of these histories. When
restricted to alternatives at definite moments of times this generalization is
equivalent to Schr\"odinger-Heisenberg quantum mechanics. However, the quantum
mechanics of more general spacetime alternatives does not have an equivalent
Schr\"odinger-Heisenberg formulation. We suggest that, in the quantum theory of
gravity, the general notion of "observable" is supplied by diffeomorphism
invariant partitions of spacetime metrics and matter field configurations. By
generalizing the usual alternatives so as to put quantum theory in fully
spacetime form we may be led to a covariant generalized quantum mechanics of
spacetime free from the problem of time.
