A Note on the Semi-Classical Approximation in Quantum Gravity

TL;DR

This paper analyzes the semiclassical approximation in quantum gravity, exploring the nature of quasiclassical states, their superpositions, and conditions under which the approximation may break down, especially near black hole horizons.

Contribution

It clarifies the structure of quasiclassical states in quantum gravity and links the semiclassical approximation to superpositions of eigenstates, highlighting potential breakdown scenarios.

Findings

Superpositions of states of the form e^{iS} define classical correlations.

Semiclassical states can be approximated by localized WKB states.

Breakdown of the approximation can occur near black hole horizons.

Abstract

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.