On the semiclassical limit of the Wheeler-DeWitt equation

Abstract

We continue our investigation of approximation schemes for obtaining semi-classical Einstein equations with a backreaction, starting from the Wheeler-DeWitt equation. The analysis is carried out using a toy model with two degrees of freedom, which represents a matter field interacting with gravity. We argue that the backreaction is to be found using the phase of the matter part of the wavefunction. Using a semiclassical Wigner function we find the general condition for the validity of a semiclassical theory: the dispersion in the metric derivative of the phase of the matter wavefunction should be negligible. We then consider a special case of the toy Lagrangian, that of a time-dependent harmonic oscillator, and show that the backreaction is equal to the expectation value of the matter Hamiltonian only if the background 'metric' varies slowly with time. The Wigner function, when applied to a semiclassical cosmological model, shows that the semiclassical approximation is valid only when the quantum contribution to the energy-momentum tensor is small compared to the classical contribution.