Approaches to Quantum Gravity

410 J. Henson
the type normally used on lattices) over the whole Lorentz group. Significantly, a
new “non-locality scale” must be introduced, above the Planck scale but macroscopically
small, to allow for the non-locality of the causal set. The analysis of this
discrete d’Alembertian has so far been carried out only in flat space, although it has
been tentatively conjectured that the scheme will also be successful for sprinklings
of curved spacetimes. Its discovery provides a way to define a classical dynamics
of scalar fields on a fixed causal set background, giving a causal but non-local field
theory, which may lead to hints on non-standard phenomenology. It would also
be an interesting exercise to find a way to quantise the field, and look for similar
results there.
One of the most intriguing uses is for causal set dynamics, as mentioned in
section 21.2.2. How can this discretised d’Alembertian help us to find an action for
causal 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 some
arbitrary origin of co-ordinates. It can be seen from some of the results in [75]that
the 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 of
estimating the d’Alembertian of fields in curved space times, we also have a way
of estimating the scalar curvature. If this method turns out to be correct, and the
values found are stable and practically calculable, it will be of great significance
for causal set dynamics.
These results are, hopefully, only the first handle on the problem of locality in
causal sets, and consideration of what has been learnt may lead to the development
of more techniques, as the reason for this success is more fully grasped. One
goal would be would be to find an expression for the action which is combinatorially
simple and compelling, and which gives sensible values for non-manifoldlike
causal sets. Work on these topics has only just begun.
21.4 Conclusions
Discreteness provides a solution for many of the problems we confront in our
attempts to construct a theory of Quantum Gravity. From the assumptions of discreteness
and standard Lorentz invariance, we find that our choices of fundamental
histories are extremely limited. Although this should not discourage other attempts