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A Note on the Semi-Classical Approximation in Quantum Gravity

We re-examine 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 $e^{iS}$. In terms of a reduced phase space formalism, this type of state can be expressed 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 localised 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 neighbouring classical trajectories is discussed. It is shown that a breakdown in the semiclassical approximation can be a coordinate dependent phenomenon, as has been argued to be the case close to a black hole horizon.