Approaches to Quantum Gravity

398 J. Hensonthis way, no discreteness scale is set, and there might not be enough embedded elementsto “see” enough causal information. A further criterion is needed to ensurethe necessary density of embedded elements.So, to retrieve enough causal information, and to add the volume information,the concept of sprinkling is needed. A sprinkling is a random selection of pointsfrom a spacetime according to a Poisson process. The probability for sprinkling nelements into a region of volume V isP(n) = (ρV )n e −ρV. (21.1)n!Here, ρ is a fundamental density assumed to be of Planckian order. Note that theprobability depends on nothing but the volume of the region. The sprinkling definesan embedded causal set. The Lorentzian manifold (M, g) is said to approximatea causet C if C could have come from sprinkling (M, g) with relatively high probability.5 In this case C is said to be faithfully embeddable in M. Onaverage,ρVelements are sprinkled into a region of volume V , and fluctuations in the numberare typically of order √ ρV (a standard result from the Poisson statistics), becominginsignificant for large V . This gives the promised link between volume andnumber of elements.Can such a structure really contain enough information to provide a good manifoldapproximation? We do not want one causal set to be well-approximated bytwo spacetimes that are not similar on large scales. The conjecture that this cannothappen (sometimes called the “causal set haupvermutung”, meaning “fundamentalconjecture”) is central to the program. It is proven in the limiting case whereρ →∞[36], and there are arguments and examples to support it, but some stepsremain to be taken for a general proof. One of the chief difficulties has been thelack of a notion of similarity between Lorentzian manifolds, or more properly, adistance measure on the space of such manifolds. Progress on this has now beenmade [37], raising hopes of a proof of the long-standing conjecture.A further generalisation of this scheme may be necessary. Above, it was notedthat certain small causal sets cannot be embedded into Minkowski space of anyparticular dimension. This means that, for C a large causal set that is faithfullyembeddable into a region of n-dimensional Minkowski, by changing a small numberof causal relations in C we can form a causet that no longer embeds. Fromour experience with quantum theories, we most likely will not want such “small5 The practical meaning of “relatively high probability” has so far been decided on a case-by-case basis. It isusually assumed that the random variable (function of the sprinkling) in question will not be wildly far fromits mean in a faithfully embeddable causet. Beyond this, standard techniques involving χ 2 tests exist to test thedistribution of sprinkled points for Poisson statistics.