Persistence amplitudes from numerical quantum gravity

The Euclidean quantum amplitude to go between data specified on an initial and a final hypersurface may be approximated by the tree amplitude exp(-I_classical/ℏ), (1) where I_classical is the Euclidean action of the classical solution joining the initial and final data. In certain cases the tree amplitude is exact. We study I_classical, and hence the quantum amplitude, in the case of a spherically symmetric Riemannian gravitational field coupled to a spherically symmetric scalar field. The classical scalar field obeys an elliptic equation, which we solve using relaxation techniques, in conjunction with the field equations giving the gravitational field. An example of the transition from linearity to nonlinearity is presented and power-law behaviour of the action is demonstrated.