Non-perturbative conformal quantum gravity

Abstract

A quantised metric is viewed here as a probability measure on the space of (possibly degenerate) metrics on a manifold. For simplicity, attention is confined to the conformal class of a fixed 'classical' Riemannian metric g on a compact manifold. Using the theory of probability measures on Hilbert spaces, the authors construct (on Sobolev spaces of conformally coupled scalar fields) a very natural family of genuine probability measures that depends on c, G, h(cross), Boltzmann's constant and a temperature parameter. The conformal class of g is parametrised by these scalar fields in a way that permits the evaluation of moments of functionals associated with the scalar curvature and the volume element. The classical metric g can essentially be recovered by taking the low-temperature limit of the pointwise mean of the quantum metric. However, such limits are possible only when the scalar curvature of g is non-negative in an average sense, thus suggesting a natural origin for the positive-energy condition. High-temperature asymptotics for scalar curvature and volume functionals are studied in detail. The formalism permits a quantitative treatment of the possibly foamy nature of space. In several places the mathematics singles out the physically relevant dimensions, three and four, as being the most natural.