Quantum dynamics of Lorentzian spacetime foam
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 mode of Lorentziansignature 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 Schrödinger equation for wormhole wave functions, i.e., the wavefunction 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 quantummechanically unstable: A FeynmanKac decomposition of the WKB propagator yields no spectrum of bound states. Although an initially localized wormhole wave function may oscillate for many classical expansion and recontraction periods, it must eventually leak to large radius values. The possibility of such a mode unstable against growth, combined with the observed absence of macroscopic wormholes, suggests that stability considerations may place constraints on the nature or even the existence of Planckscale foamlike structure, at least of Lorentzian signature.
 Publication:

Physical Review D
 Pub Date:
 May 1994
 DOI:
 10.1103/PhysRevD.49.5199
 arXiv:
 arXiv:grqc/9309017
 Bibcode:
 1994PhRvD..49.5199R
 Keywords:

 04.60.Kz;
 Lower dimensional models;
 minisuperspace models;
 General Relativity and Quantum Cosmology
 EPrint:
 37 pages, 937