Note on the semiclassical approximation in quantum gravity

We reexamine the semiclassical approximation to quantum gravity in the canonical formulation, focusing on the definition of a quasiclassical state for the gravitational field. It is shown that a state with classical correlations must be a superposition of states of the form ${\mathit{e}}^{\mathit{i}\mathit{S}}$. In terms of a reduced phase space formalism, this type of state can be expresesd as a coherent superposition of eigenstates of operators that commute with the constraints and so correspond to constants of the motion. Contact is made with the usual semiclassical approximation by showing that a superposition of this kind can be approximated by a WKB state with an appropriately localized prefactor. A qualitative analysis is given of the effects of geometry fluctuations, and the possibility of a breakdown of the semiclassical approximation due to interference between neighboring classical trajectories is discussed. It is shown that a breakdown in the semiclassical approximation can be a coordinate-dependent phenomenon, as has beeen argued to be the case close to a black hole horizon. © 1996 The American Physical Society.