Three Roads To Quantum Gravity

KNOTS, LINKS AND KINKS
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What we had done was to apply the usual methods for
constructing a quantum theory to the simple form of the
equations for general relativity that Sen and Ashtekar had
discovered. These led to the equations for the quantum theory
of gravity. These equations had ®rst been written down in the
1960s, by Bryce DeWitt and John Wheeler, but we found new
forms for them which were dramatically simpler. We had to
plug into these equations formulas that described possible
quantum states of the geometry of space and time. On an
impulse I tried something that Louis Crane and I had played
with, which was to build these states directly from the
expressions Polyakov used to describe the quantized loops
of electric ®elds. What we found was that, as long as the loops
did not intersect, they satis®ed the equations. They looked
like the loops in Figure 23.
FIGURE 23
Quantum states of the geometry of space are expressed in loop quantum
gravity in terms of loops. These states are exact solutions to the equations of
quantum gravity, as long as there are no intersections or kinks in the loops.
It took us a few days of hard work to ®nd still more
solutions. We found that even if the loops intersected, we
could still combine them to make solutions provided certain
simple rules were obeyed. In fact, we could write down an
in®nite number of these states ± all we had to do was draw
loops and apply some simple rules whenever they intersected.