  This paper sets up a perturbative treatment of the evolving quantum state of
a gravitational system, in a Schr\"odinger-like picture, working about a
general background. This connects gauge symmetry, the constraints,
gravitational dressing, and evolution. Starting with a general time slicing, we
give a simple derivation of the relation between the constraints, the
hamiltonian, and its well-known boundary term. Among different approaches to
quantization with constraints, we focus on a "gauge-invariant canonical
quantization," which is developed perturbatively in the gravitational coupling.
The leading-order solution of the constraints (including the Wheeler-DeWitt
equation) for perturbations about the background is given in terms of an
explicit construction of gravitational dressings built using certain
generalized Green functions; different such dressings corresponding to adding
propagating gravitational waves to a particular solution of the constraints.
Dressed operators commute with the constraints, expressing their gauge
invariance, and have an algebraic structure differing significantly from the
undressed operators of the underlying field theory. These operators can act on
the vacuum to create dressed states, and evolution of general such states is
then generated by the boundary hamiltonian, and alternately may be
characterized using other relational observables. This provides a concrete
approach to studying perturbative time evolution, including the leading
gravitational backreaction, of quantum states of black holes with flat or anti
de Sitter asymptotics, for example on horizon-crossing slices. This description
of evolution in turn provides a starting point for investigating possibly
important corrections to quantum evolution, that go beyond quantized general
relativity.
