Spatial Hypersurfaces in Causal Set Cosmology

Within the causal set approach to quantum gravity, a discrete analog of a spacelike region is an ``antichain'', which is a set of unrelated elements. In the continuum spacetime approximation of the theory, a moment-of-time hypersurface is well represented by an ``inextendible'' antichain. We define a richer structure corresponding to a ``thickening'' of this antichain which contains non-trivial geometric and topological information. We find that covariant observables can be associated with such thickened antichains and transitions between them, within 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 restricted spacetime regions. The resulting re-phrasing of the sum-over-histories formulation may be fruitful in other approaches to quantum gravity