Three Roads To Quantum Gravity

WHAT CHOOSES THE LAWS OF NATURE?
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directly contradicts a prediction of Newton's theory, we can
deduce that, with probability 1, Newton's theory is false.
It is a little harder to pose the question in Einstein's theory
of spacetime, as that theory has an in®nite number of
solutions. In many of them space is approximately ¯at, but
in many of them it is not. Given that there are an in®nite
number of examples of each, it is not straightforward to ask
how probable it would be, were the solution chosen at
random, that the resulting universe would look almost like
three-dimensional Euclidean space.
It is easier to ask the question in a quantum theory of
gravity. To ask it we need a form of the theory that does not
assume the existence of any classical background geometry
for space. Loop quantum gravity is an example of such a
theory. As I explained in Chapters 9 and 10, it tells us that
there is an atomic structure to space, described in terms of the
spin networks invented by Roger Penrose. As we saw there,
each possible quantum state for the geometry of space can be
described as a graph such as that shown in Figures 24 to 27.
We can then pose the question this way: how probable is it
that such a graph represents a geometry for space that would
be perceived by observers like us, living on a scale hugely
bigger than the Planck scale, to be an almost Euclidean threedimensional
space? Well, each node of a spin network graph
corresponds to a volume of roughly the Planck length on each
side. There are then 10 99 nodes inside every cubic centimetre.
The universe is at least 10 27 centimetres in size, so it contains
at least 10 180 nodes. The question of how probable it is that
space looks like an almost ¯at Euclidean three-dimensional
space all the way up to cosmological scales can then be posed
as follows: how probable is it that a spin network with 10 180
nodes would represent such a ¯at Euclidean geometry?
The answer is, exceedingly improbable! To see why, an
analogy will help. To represent an apparently smooth, featureless
three-dimensional space, the spin network has to have
some kind of regular arrangement, something like a crystal.
There is nothing special about any position in Euclidean space
that distinguishes it from any other position. The same must
be true, at least to a good approximation, of the quantum