Renormalization of general relativity on a background of spacetime foam

Abstract

We show that a distribution of virtual black holes in the vacuum at shorter than Planck scales can have a profound effect on the perturbative high-energy behavior of a quantum field theory. This follows if we make the assumption that only perturbations which vanish inside the apparent horizons of virtual black holes can make a coherent contribution to scattering amplitudes. As a result there is a cutoff-dependent contribution to the density of states which comes from the scale dependence of the density of virtual black holes in the vacuum. This results in a modification in the divergence-structure of perturbation theory on a background of spacetime foam. The divergence structure then depends on how fast the volume of a typical spacelike surface fills up with virtual black holes as the cutoff scale is decreased to zero. If the distribution of virtual black holes is scale invariant then the result is to decrease the spectral dimension of spacetime to a nonintegral value less than four. In this case general relativity becomes renormalizable in a 1 N expansion without additional dimension-four counterterms. This case can also be understood in terms of recent work on density of states functions on fractals, because the set of points which are not contained within a scale-invariant distribution of black holes will be fractal. If the distribution of virtual black holes fills up faster, asymptotically, than the scale-invariant distribution, all sums over virtual states are cut off by exponential factors and are finite. In both cases, as the dynamics is generated by the hamiltonian of general relativity, the energy is bounded from below and the theory will be stable.