Spatial Hypersurfaces in Causal Set Cosmology

TL;DR

This paper introduces a new geometric structure called thickened antichains in causal set cosmology, which captures richer topological information and allows for covariant observables, offering a novel perspective on quantum gravity sum-over-histories.

Contribution

It constructs a thickening of antichains in causal sets that encodes geometric and topological data, and relates covariant measures to completed histories rather than spacetime regions.

Findings

Covariant observables linked to thickened antichains.

Difference between covariant measure and sum-over-histories.

Potential applications to other quantum gravity approaches.

Abstract

Within the causal set approach to quantum gravity, a discrete analog of a spacelike region is a set of unrelated elements, or an antichain. In the continuum approximation of the theory, a moment-of-time hypersurface is well represented by an inextendible antichain. We construct a richer structure corresponding to a thickening of this antichain containing non-trivial geometric and topological information. We find that covariant observables can be associated with such thickened antichains and transitions between them, in classical stochastic growth models of causal sets. This construction highlights the difference between the covariant measure on causal set cosmology and the standard sum-over-histories approach: the measure is assigned to completed histories rather than to histories on a restricted spacetime region. The resulting re-phrasing of the sum-over-histories may be fruitful in other approaches to quantum gravity.