Approaches to Quantum Gravity

410 J. Hensonthe type normally used on lattices) over the whole Lorentz group. Significantly, anew “non-locality scale” must be introduced, above the Planck scale but macroscopicallysmall, to allow for the non-locality of the causal set. The analysis of thisdiscrete d’Alembertian has so far been carried out only in flat space, although it hasbeen tentatively conjectured that the scheme will also be successful for sprinklingsof curved spacetimes. Its discovery provides a way to define a classical dynamicsof scalar fields on a fixed causal set background, giving a causal but non-local fieldtheory, which may lead to hints on non-standard phenomenology. It would alsobe an interesting exercise to find a way to quantise the field, and look for similarresults there.One of the most intriguing uses is for causal set dynamics, as mentioned insection 21.2.2. How can this discretised d’Alembertian help us to find an action forcausal sets? Consider the field ⊔⊓σ(0, x), where σ(x, y) is Synge’s world function(i.e. half of the square of the geodesic distance between x and y) and 0 is somearbitrary origin of co-ordinates. It can be seen from some of the results in [75]thatthe d’Alembertian of this field at the origin gives the scalar curvature there:R(0) =⊔⊓⊔⊓ σ(0, x) ⏐⏐x=0. (21.3)The geodesic length between two timelike points in a causal set can be estimated(independently, it is conjectured, of curvature). Therefore, if we have a way ofestimating the d’Alembertian of fields in curved space times, we also have a wayof estimating the scalar curvature. If this method turns out to be correct, and thevalues found are stable and practically calculable, it will be of great significancefor causal set dynamics.These results are, hopefully, only the first handle on the problem of locality incausal sets, and consideration of what has been learnt may lead to the developmentof more techniques, as the reason for this success is more fully grasped. Onegoal would be would be to find an expression for the action which is combinatoriallysimple and compelling, and which gives sensible values for non-manifoldlikecausal sets. Work on these topics has only just begun.21.4 ConclusionsDiscreteness provides a solution for many of the problems we confront in ourattempts to construct a theory of Quantum Gravity. From the assumptions of discretenessand standard Lorentz invariance, we find that our choices of fundamentalhistories are extremely limited. Although this should not discourage other attempts