EHRENFEST PRINCIPLE IN QUANTUM-GRAVITY

The Ehrenfest principle partial-t<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/d-tau = del-theta . del, where del is the gradient operator in superspace, and theta is the phase of the Wheeler-DeWitt wavefunction-psi; 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 (partial-t-theta) >> (partial-t ln(psi*psi)) and (partial-t-theta)2 >> (partial-t(2)-theta).