Quantum Dynamics of Lorentzian Spacetime Foam

TL;DR

This paper models Lorentzian spacetime foam using a simple wormhole that evolves classically and quantum mechanically, revealing quantum instability and wave function leakage, with implications for understanding microscopic spacetime fluctuations.

Contribution

It introduces a minisuperspace model for Lorentzian spacetime foam based on a simple wormhole, analyzing its classical and quantum dynamics with novel insights into its quantum instability.

Findings

WKB approximation used to evaluate the wormhole wave function propagator

Classically stable wormhole is quantum-mechanically unstable

Wave function leaks to large radius values over time

Abstract

A simple spacetime wormhole, which evolves classically from zero throat radius to a maximum value and recontracts, can be regarded as one possible mode of fluctuation in the microscopic ``spacetime foam'' first suggested by Wheeler. The dynamics of a particularly simple version of such a wormhole can be reduced to that of a single quantity, its throat radius; this wormhole thus provides a ``minisuperspace model'' for a structure in Lorentzian-signature foam. The classical equation of motion for the wormhole throat is obtained from the Einstein field equations and a suitable equation of state for the matter at the throat. Analysis of the quantum behavior of the hole then proceeds from an action corresponding to that equation of motion. The action obtained simply by calculating the scalar curvature of the hole spacetime yields a model with features like those of the relativistic free particle. In particular the Hamiltonian is nonlocal, and for the wormhole cannot even be given as a differential operator in closed form. Nonetheless the general solution of the Schr\"odinger equation for wormhole wave functions, i.e., the wave-function propagator, can be expressed as a path integral. Too complicated to perform exactly, this can yet be evaluated via a WKB approximation. The result indicates that the wormhole, classically stable, is quantum-mechanically unstable: A Feynman-Kac decomposition of the WKB propagator yields no spectrum of bound states. Though an initially localized wormhole wave function may oscillate for many classical expansion/recontraction periods, it must eventually leak to large radius values. The possibility of such a mode unstable against growth, combined with