Persistence Amplitudes from Numerical Quantum Gravity

TL;DR

This paper investigates the Euclidean quantum amplitude in spherically symmetric gravitational and scalar fields, demonstrating how classical solutions influence quantum transition amplitudes and exploring the transition from linear to nonlinear regimes.

Contribution

It provides a detailed analysis of classical solutions for spherically symmetric fields and their impact on quantum amplitudes, including numerical solutions and power law behavior of the action.

Findings

Classical solutions can be obtained using relaxation techniques.

Transition from linear to nonlinear behavior is demonstrated.

Power law behavior of the Euclidean action is shown.

Abstract

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}/\hbar), 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} 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 non-linearity is presented and power law behavior of the action is demonstrated.