Three Roads To Quantum Gravity

HOW TO COUNT SPACE
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theorists. The Russian theorist Alexander Polyakov was
visiting and was to give a talk that afternoon. In those days
there were great schools of theoretical physics in the Soviet
Union, but their members were seldom allowed to travel to
the West. Polyakov was the most creative and most charismatic
of them, and we all went along to his seminar. I recall
someone with disarming warmth and informality, under
which there was hidden (but not too well) someone with
unlimited con®dence.
He began by telling us that he had dedicated his life to
pursuing a foolish and quixotic vision, which was to reexpress
QCD in a form in which the theory could be solved
exactly. His idea for doing this was to recast QCD completely
as a theory of the dynamics of lines and loops of colourelectric
¯ux. These were the same as Wilson's loops, and
indeed Polyakov had independently invented the picture of
QCD on a discrete lattice. But in this seminar at least he
worked without the lattice, to try to pull out from the theory a
description in which the quantized loops of electric ¯ux
would be the fundamental entities. A physicist working
without a lattice is something like a trapeze artist working
without a net. There is an ever present danger that a false
move will lead to a fatal result. In physics the fatalities arise
from confrontations with in®nite and absurd mathematical
expressions. As we mentioned earlier, such expressions arise
in all quantum theories based on continuous space and time.
In his seminar Polyakov showed that despite these in®nities,
one could give physical meaning to loops of electric ¯ux. If he
did not succeed completely in solving the resulting equations,
his seminar was altogether an assertion of faith in the
hypothesis of duality ± that the strings are as fundamental as
the electric ®eld lines.
The idea of duality is still a major driving force behind
research in elementary particle physics and string theory.
Duality is the very simple view that there are two ways of
looking at the same thing ± either in terms of strings or in
terms of ®elds. But so far no one has been able to show that
duality is applicable in ordinary QCD. It has been shown to
be valid in very specialized theories which depend on very