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

TL;DR

This paper introduces Gromov's epsilon-approximation topology on the moduli space of Riemannian structures, explores its properties, and discusses potential implications for quantum gravity and connections to Wilson's renormalization theory.

Contribution

It defines and analyzes a new topology on the moduli space of Riemannian structures, showing conformal classes are dense, and explores links to quantum gravity and renormalization concepts.

Findings

Conformal classes are dense in the moduli space under Gromov's topology.

Gromov's geometries relate to Wilson's renormalization group theory.

Implications for quantum gravity remain to be explored.

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.