Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 399fluctuations” to be physically significant, and so we may need a condition of“manifoldlikeness” that is more forgiving than faithful embedding. A possiblemethod is given by coarse-graining [2]: removal of some points from the causalset C forming a new causal set C ′ , before testing C ′ for faithful embeddability atthe appropriate lower density of sprinkling ρ ′ . For example, this might reasonablybe done at random with the same probability p for removal of each element, andρ ′ = ρ(1 − p). This basically amounts to looking for a faithfully embeddable subsetof a causal set, following a certain set of rules. Below, the criterion of faithfulembeddability will be the one used, but it should be kept in mind that the causetsbeing talked about could be coarse-grainings of some larger causet.21.1.4 Reconstructing the continuumThe concept of faithful embedding gives the criterion for a manifold to approximateto a causet. But it is important to realise that the only use of sprinkling is toassign continuum approximations; the causal set itself is the fundamental structure.How then can this approximate discrete/continuum correspondence be used? Thatis, given a causal set that approximates a spacetime, how do we find an approximationto some particular property x of that spacetime? We need to find a propertyof the causal set itself, x(C), that approximates the value of x(M) with high probabilityfor a sprinkling of a spacetime M. Such estimators exist for dimension[38; 24; 39] timelike distance between points [40], and of course volumes. Asanother example, methods have been developed to retrieve topological informationabout spatial hypersurfaces in approximating Lorentzian manifolds, by referenceonly to the underlying causet [41].A simple example of how such estimators work is given by one of the estimatorsof timelike distance. Firstly the volume of the interval causally between two elementscan be easily estimated from the causal set (it is approximately proportionalto the number of elements in that causally defined region). In Minkowski space,this volume is related to the distance between the points, in a way that dependson dimension. Therefore, given the dimension, this timelike distance can also beestimated. See [40] for a different distance measure conjectured to hold for curvedspacetimes [24], and a way to identify approximations to timelike geodesics. Asthis other distance measure does not depend on the dimension, the two can becompared to give a dimension estimator.Given a causal set C without an embedding (this is after all our fundamentalstructure) it would be of great utility to be able to say if it was faithfully embeddableinto some spacetime or not – a criterion of “manifoldlikeness” – and if soto provide an embedding. The discrete-continuum correspondence given abovedoes not directly answer this question; it would be highly impractical to carry