  Familiar textbook quantum mechanics assumes a fixed background spacetime to
define states on spacelike surfaces and their unitary evolution between them.
Quantum theory has changed as our conceptions of space and time have evolved.
But quantum mechanics needs to be generalized further for quantum gravity where
spacetime geometry is fluctuating and without definite value. This paper
reviews a fully four-dimensional, sum-over-histories, generalized quantum
mechanics of cosmological spacetime geometry. This generalization is
constructed within the framework of generalized quantum theory. This is a
minimal set of principles for quantum theory abstracted from the modern quantum
mechanics of closed systems, most generally the universe. In this
generalization, states of fields on spacelike surfaces and their unitary
evolution are emergent properties appropriate when spacetime geometry behaves
approximately classically. The principles of generalized quantum theory allow
for the further generalization that would be necessary were spacetime not
fundamental. Emergent spacetime phenomena are discussed in general and
illustrated with the example of the classical spacetime geometries with large
spacelike surfaces that emerge from the `no-boundary' wave function of the
universe. These must be Lorentzian with one, and only one, time direction. The
essay concludes by raising the question of whether quantum mechanics itself is
emergent.
