  We describe a simple gauge-fixing that leads to a construction of a quantum
Hilbert space for quantum gravity in an asymptotically Anti de Sitter
spacetime, valid to all orders of perturbation theory. The construction is
motivated by a relationship of the phase space of gravity in asymptotically
Anti de Sitter spacetime to a cotangent bundle. We describe what is known about
this relationship and some extensions that might plausibly be true. A key fact
is that, under certain conditions, the Einstein Hamiltonian constraint equation
can be viewed as a way to gauge fix the group of conformal rescalings of the
metric of a Cauchy hypersurface. An analog of the procedure that we follow for
Anti de Sitter gravity leads to standard results for a Klein-Gordon particle.
