On Recovering Continuum Topology from a Causal Set

TL;DR

This paper introduces a new topology on causal sets using thickened antichains, enabling the recovery of continuum spacetime homology from discrete causal sets, supporting the causal set approach to quantum gravity.

Contribution

It presents a novel topology on causal sets that allows for recovering the homology of the underlying spacetime from discrete structures.

Findings

Successfully recovers spacetime homology from causal sets

Supports the causal set approach to quantum gravity

Establishes a discrete-continuum correspondence

Abstract

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using ``thickened antichains'' is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or ``Hauptvermutung'' of causal set theory.