Quantum fluctuations, conformal deformations, and Gromov's topology --- Wheeler, DeWitt, and Wilson meeting Gromov

Abstract

The moduli space of isometry classes of Riemannian structures on a smooth manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum gravity. A natural question concerning it is: What is a natural topology on such moduli space that reflects best quantum fluctuations of the geometries within the Planck's scale? This very question has been addressed by B.DeWitt and others. In this article we introduce Gromov's $\varepsilon$-approximation topology on the above moduli space for a closed smooth manifold. After giving readers some feel of this topology, we prove that each conformal class in the moduli space is dense with respect to this topology. Implication of this phenomenon to quantum gravity is yet to be explored. When going further to general metric spaces, Gromov's geometries-at-large-scale based on his topologies remind one of K.Wilson's theory of renormalization group. We discuss some features of both and pose a question on whether both can be merged into a single unified theory.