Three Roads To Quantum Gravity

KNOTS, LINKS AND KINKS
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didn't you get mine?' He was given the best room in the hotel,
and Air India put him in ®rst class on his ¯ight home to
Rome.
Thus was born what is now called loop quantum gravity. It
took several years of work, ®rst with Carlo and then as part of
a growing community of friends and colleagues, to unravel
the meaning of the solutions to the quantum gravity equations
we had found. One straightforward consequence is that
quantum geometry is indeed discrete. Everything we had
done had been based on the idea of a discrete line of force, as
in a magnetic ®eld in a superconductor. Translated into the
loop picture of the gravitational ®eld, this turns out to imply
that the area of any surface comes in discrete multiples of
simple units. The smallest of these units is about the Planck
area, which is the square of the Planck length. This means that
all surfaces are discrete, made of parts each of which carries a
®nite amount of area. The same is true of volume.
To arrive at these results we had to ®nd a way to eliminate
the in®nities that plague all expressions in quantum theories
of ®elds. I had an intuition, stemming from my past conversations
with Julian Barbour, and the work I had done with
Louis Crane, that the theory should have no in®nities. Many
physicists have speculated that the in®nities come from some
mistaken assumption about the structure of space and time on
the Planck scale. From the older work it was clear to me that
the wrong assumption was the idea that the geometry of
spacetime was ®xed and non-dynamical. When calculating
the measures of geometry, such as area and volume, one had
to do it in just the right way to eliminate any possible contamination
from non-dynamical, ®xed structures. Exactly
how to do this was a technical exercise that cannot be
explained here. But in the end it did turn out that as long as
one asks a physically meaningful question, there will be no
in®nities.
In my experience it really is true that as a scientist one has
only a few good ideas. They are few and far between, and
come only after many years of preparation. What is worse,
having had a good idea one is condemned to years of hard
work developing it. The idea that area and volume would be