Approaches to Quantum Gravity

156 Questions and answersNow a good cover has the combinatorial structure of a simplicial complex.Thus an “n-dimensional” site would be describable as a partially orderedset of simplicial n-complexes. This would begin to make contact with the spinfoam models for gravity.An interesting fact about causal sites is that their Hausdorff dimensioncould easily not equal their topological dimension. Here by Hausdorff dimensionI mean a measure of the growth in the number of regions of a givendiameter, whose actual definition is rather technical. This would depend onthe extent to which different maximally refined covers shared larger regions.A causal site with such properties would be a natural setting for dimensionalregularization. Given the importance of dimensional regularization inparticle physics, this seems to be a candidate for a “physical regularization.”I am in the process of studying a gedanken experiment for QuantumGravity which may shed light on this question. Consider a finite region inwhich a reproducible state of Quantum Gravity can be created, surroundedby observers that can be considered to live in Minkowski space. If we createthe state of Quantum Gravity, and probe it with rockets which set off bursts oflight at set times, we can consider the times and angles where the observerssee the bursts as measurements of the quantum state.Now it is quite possible that a single observer will see a given burst eitheras a set of images, as in the case of gravitational lensing, or not at all, in thecase of an event horizon. Let us ignore these possibilities for the moment.If the state in the region is essentially a single classical state, the simultaneousobservation of bursts will give consistent identifications of the apparentMinkowski pasts of the observers, giving a single set of regions which wouldamount to an approximate description of a manifold.Now suppose, for simplicity, that the state is a superposition of two classicalmetrics. The observation of the burst at a certain angle and time by oneobserver would appear correlated sometimes with one apparent region andsometimes with another to a second observer.Apparent subregions of the experimental region would appear in “sheets”that could not be directly compared.The set of all observable subregions would appear as a sheaf over the set ofall observers. Consistency relationships between observers would enrich thisto a site of observers. The observable subregions would also fiber over theset of metrics on the observed region. Subregions corresponding to differentmetrics would have no privileged identification, since different observerswould relate them differently.If we take the position that geometry in Quantum Gravity means observedgeometry (necessary if we want an operational view of quantum mechanics