Emergent Space-Time via a Geometric Renormalization Method

TL;DR

This paper introduces a geometric renormalization scheme for metric spaces, including graphs, to model emergent space-time, analyzing properties like scale invariance, dimension stability, and potential fractal or wormhole structures.

Contribution

It proposes a novel geometric renormalization method for metric spaces, providing criteria for continuum limits and analyzing properties like scale invariance and dimension stability.

Findings

Coarse grained spaces can have different distance functions while being homeomorphic.

The dimension remains stable under local quasi-isometric coarse graining.

Limit spaces may be fractal if the dimension is non-integer.

Abstract

We present a purely geometric renormalization scheme for metric spaces (including uncolored graphs), which consists of a coarse graining and a rescaling operation on such spaces. The coarse graining is based on the concept of quasi-isometry, which yields a sequence of discrete coarse grained spaces each having a continuum limit under the rescaling operation. We provide criteria under which such sequences do converge within a superspace of metric spaces, or may constitute the basin of attraction of a common continuum limit, which hopefully, may represent our space-time continuum. We discuss some of the properties of these coarse grained spaces as well as their continuum limits, such as scale invariance and metric similarity, and show that different layers of spacetime can carry different distance functions while being homeomorphic. Important tools in this analysis are the Gromov-Hausdorff distance functional for general metric spaces and the growth degree of graphs or networks. The whole construction is in the spirit of the Wilsonian renormalization group. Furthermore we introduce a physically relevant notion of dimension on the spaces of interest in our analysis, which e.g. for regular lattices reduces to the ordinary lattice dimension. We show that this dimension is stable under the proposed coarse graining procedure as long as the latter is sufficiently local, i.e. quasi-isometric, and discuss the conditions under which this dimension is an integer. We comment on the possibility that the limit space may turn out to be fractal in case the dimension is non-integer. At the end of the paper we briefly mention the possibility that our network carries a translocal far-order which leads to the concept of wormhole spaces and a scale dependent dimension if the coarse graining procedure is no longer local.