On Recovering Continuum Topology from a Causal Set

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.

The statement of a lemma has been simplified and the proof rewritten. Errors have been corrected. One figure has been removed. All results remain unchanged. Accepted for publication in JMP. 31 pages, 4 figs