Three Roads To Quantum Gravity

152 THREE ROADS TO QUANTUM GRAVITY
Physics, like the other sciences, is the art of the possible. So
I must add a rider here, which is that we do not really
understand QED. We know the principles of the theory and
we can deduce from them the basic equations that de®ne the
theory. But we cannot actually solve these equations, or even
prove that they are mathematically consistent. Instead, to
make sense of them we have to resort to a kind of subterfuge.
We make some assumptions about the nature of the solutions
± which, after more than ®fty years, are still unproved ± and
these lead us to a procedure for calculating approximately
what happens when photons and electrons interact. This
procedure is called perturbation theory. It is very useful in
that it does lead to answers that agree very precisely with
experiment. But we do not actually know whether the
procedure is consistent or not, or whether it accurately
re¯ects what a real solution to the theory would predict.
String theory is presently understood mainly in the language
of this approximation procedure. It was invented by modifying
the approximation procedure, rather than the theory. This
is how people were able to invent a theory which is understood
only as a list of solutions.
Perturbation theory is actually quite easy to describe.
Thanks to Feynman, there is a simple diagrammatic means
for understanding it. Picture a world of processes in which
three things can happen. An electron may move from point A
at one time to point B at another. We can draw this as a line, as
in Figure 33. A photon may also travel, which is indicated by
a dotted line in the ®gure. The only other thing that may
happen is that an electron and a photon interact, which is
indicated by the point where a photon line meets an electron
line. To compute what happens when two electrons meet, one
simply draws all the things that can happen, beginning with
two electrons entering the scene, and ending with two
electrons leaving. There are an in®nite number of such
processes, and we see a few of them in Figure 34. Feynman
taught us to associate with each diagram the probability
(actually the quantum amplitude, whose square is the
probability) of that process. One can then work out all the
predictions of the theory.