Three Roads To Quantum Gravity

128 THREE ROADS TO QUANTUM GRAVITY
It took many years for us and others to work out the
implications of what we had found in those few days. But
even at the start we knew that we had in our grasp a quantum
theory of gravity that could do what no theory before it had
done ± it gave us an exact description of the physics of the
Planck scale in which space is constructed from nothing but
the relationships among a set of discrete elementary objects.
These objects were still Wilson's and Polyakov's loops, but
they no longer lived on a lattice, or even in space. Instead,
their interrelations de®ned space.
There was one step to go to complete the picture. We had to
prove that our solutions really were independent of the
background space. This required us to show that they solved
an additional set of equations, known as diffeomorphism
constraints, which expressed the independence of the theory
from the background. These were supposed to be the easy
equations of the theory. Paradoxically, the equations we had
solved so easily, the so-called Wheeler±DeWitt equations,
were supposed to be the hard ones. At ®rst I was very
optimistic, but it turned out to be impossible to invent
quantum states that solved both sets of equations. It was
easy to solve one or the other, but not both.
Back at Yale the next year, we spent many fruitless hours
with Louis Crane trying to do this. We pretty much convinced
ourselves it was impossible. This was very frustrating because
it was easy to see what the result would be: if we could only
get rid of the background, we would have a theory of nothing
but loops and their topological relationships. It would not
matter where in space the loops were, because the points in
space would have no intrinsic meaning. What would matter
would be how the loops intersected one another. It would also
matter how they knotted and linked.
I realized this one day while I was sitting in my garden in
Santa Barbara. Quantum gravity would be reduced to a theory
of the intersecting, knotting and linking of loops. These
would give us a description of quantum geometry on the
Planck scale. From the work I had done with Paul and Ted, I
also knew that the quantum versions of the Einstein equations
we had invented could change the way the loops linked and