Ehrenfest's principle in quantum gravity

Abstract

The Ehrenfest principle δt〈q〉 = 〈i[H, q]〉 is proposed as 〈part of 〉 a definition of the time variable in canonical quantum gravity. This principle selects a time direction in superspace, and provides a conserved, positive definite probability measure. An exact solution of the Ehrenfest condition is obtained, which leads to constant-time surfaces in superspace generated by the operator d/dτ = Δϑ·Δ, where Δ is the gradient operator in superspace, and ϑ is the phase of the Wheeler-DeWitt wavefunction ψ; the constant-time surfaces are determined by this solution up to a choice of initial t = 0 surface. This result holds throughout superspace, including classically forbidden regions and in the neighborhood of caustics; it also leads to ordinary quantum field theory and classical gravity in regions of superspace where the phase satisfies |δ,ϑ| ≫ |δt In(ψ≠ψ)| and (δtϑ)2 ≫ |δ2tϑ|.