Three Roads To Quantum Gravity

188 THREE ROADS TO QUANTUM GRAVITY
one can measure only one of these components precisely at any
one time. When it is impossible to measure two quantities
simultaneously, they are said not to commute, and this idea
leads to a new kind of geometry which is labelled noncommutative.
In such a world one cannot even de®ne the
idea of a point where something may be exactly located.
Alain Connes's non-commutative geometry thus gives us
another way to describe a world in which the usual notion of
space has broken down. There are no points, so it does not even
make sense to ask if there are an in®nite number of points in a
given region. What is really wonderful, though, is that Connes
has found that large pieces of relativity theory, quantum theory
and particle physics can be carried over into such a world. The
result is a very elegant structure that seems also to penetrate to
several of the deepest problems in mathematics.
At ®rst, Connes's ideas were developed independently of
the other approaches. But in the last few years people have
been surprised to discover that both loop quantum gravity
and string theory describe worlds in which the geometry is
non-commutative. This gives us a new language in which to
compare the two theories.
One way to test the hypothesis that strings and loops are
different ways of describing the same physics is to attack a
single problem with both methods. There is an obvious target:
the problem of giving a description of a quantum black hole.
From the discussion in Chapters 5 to 8, we know that the main
objective is to explain in terms of some fundamental theory
where the entropy and temperature of a black hole come from,
and why the entropy is proportional to the area of the black
hole's horizon. Both string theory and loop quantum gravity
have been used to study quantum black holes, with spectacular
results coming on each side in the last few years.
The main idea on each side is the same. Einstein's theory of
general relativity is to be thought of as a macroscopic
description, obtained by averaging over the atomic structure
of spacetime, in exactly the same way that thermodynamics is
obtained by applying statistics to the motion of atoms. Just as
a gas is described roughly in terms of continuous quantities
such as density and temperature, with no mention of atoms,