  In a theory of quantum gravity, states can be represented as wavefunctionals
that assign an amplitude to a given configuration of matter fields and the
metric on a spatial slice. These wavefunctionals must obey a set of constraints
as a consequence of the diffeomorphism invariance of the theory, the most
important of which is known as the Wheeler-DeWitt equation. We study these
constraints perturbatively by expanding them to leading nontrivial order in
Newton's constant about a background AdS spacetime. We show that, even within
perturbation theory, any wavefunctional that solves these constraints must have
specific correlations between a component of the metric at infinity and
energetic excitations of matter fields or transverse-traceless gravitons. These
correlations disallow strictly localized excitations. We prove perturbatively
that two states or two density matrices that coincide at the boundary for an
infinitesimal interval of time must coincide everywhere in the bulk. This
analysis establishes a perturbative version of holography for theories of
gravity coupled to matter in AdS.
