Three Roads To Quantum Gravity

156 THREE ROADS TO QUANTUM GRAVITY
they can have any energy, as the conservation of energy is
suspended during their brief lives. This creates big problems.
One has to add up all the diagrams to get the overall
probability for the process to occur, but if some particles can
have any energy between zero and in®nity, then the list of
possible processes one has to add up will be in®nite. This
leads to mathematical expressions that are no more than
complicated ways of writing the number in®nity. As a result,
Feynman's method seems at ®rst to give nonsensical answers
to questions about the interactions of electrons and photons.
Quite ingeniously, Feynman and others discovered that the
theory was giving silly answers to only a few questions, such
as `What is the mass of the electron?' and `What is its charge?'
The theory predicts that these are in®nite! Feynman ®gured
out that if one simply crosses out these in®nite answers
wherever they appear, and substitutes the right, ®nite answer,
the answers to all other questions become sensible. All the
in®nite expressions can be removed if one forces the theory to
give the right answer for the mass and the charge of the
electron. This procedure is called renormalization. When it
works for a theory, that theory is called renormalizable. The
procedure works very well for quantum electrodynamics. It
also works for quantum chromodynamics, and for the Weinberg±Salam
theory, which is our theory of radioactive decay.
When this procedure does not work, we say that a theory is
not renormalizable ± the method fails to give a sensible
theory. This is actually the case for most theories; only certain
special ones can be made sense of by these methods.
The most important theory that cannot be made sense of in
this way is Einstein's theory of gravity. The reason has to do
with the fact that arbitrarily large energies can appear in the
particles moving inside the diagrams. But the strength of the
gravitational force is proportional to the energy, because
energy is mass, from Einstein, and gravity pulls on mass,
from Newton. So the diagrams with larger energies give
correspondingly larger effects. But according to the theory,
the energies inside the diagrams can be arbitrarily large. The
result is a kind of runaway feedback process in which we lose
all control over what is happening inside the diagrams. No