Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 405
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Fig. 21.3. An augmented Hasse diagram of “poscau”, the partially ordered set of
finite causets. The elements of this set are the finite causets. To the “future” of
each causet are all the causets that can be generated from it by adding elements
to the future of or spacelike to its elements (the numbers on some of the links
represent the number of different ways the new element can be added, owing to
automorphisms of the “parent” causet). Only causets of up to size 4 are shown
here. An upwards path in poscau represents a sequence of transitions in a growth
process. Each such path is given a probability by a CSG model. Because of the
general covariance condition, the probabilities of paths ending at the same causet
are the same. (Note that the apparent “left–right symmetry” of poscau does not
survive above the 4-element causets.)
infinite-element causal sets. From the transition probabilities, a probability measure
on the space of all infinite-element past-finite causal sets can be constructed.
The order of birth can be viewed as a labelling of the elements of the growing
causet. A natural implementation of the principle of general covariance is that this
labelling should not be physically significant. Another physical principle is introduced
to ban superluminal influence, in a way appropriate to stochastic systems.
With these constraints, the free parameters of the model are reduced to a series of
real numbers.
The CSG models have made a useful testing ground for causal set dynamics,
allowing some questions to be answered that would have relevance for quantum
theories developed using the same method. For instance, the “typical” large causal
set (i.e. the type that is most likely to be found from a uniform probability distribution
over causal sets with some large number of elements) does not look like
a manifold, but instead has a “flat” shape described more fully in [60]. It might
be wondered what kind of a dynamics could overcome the great numbers of these