As was briefly mentioned
earlier, a useful way of studying theories that cannot be solved exactly
is by computing power series expansions in a small parameter. For example,
quantum electrodynamics has a small parameter, called the fine structure
constant, which is given by
Thus, if T(a)
denotes some physical quantity of interest, one computes
and the first few terms can give a very good approximation. This approach,
which is called perturbation theory, is the way superstring theories were
studied until recently. The problem is that in superstring theory there
is no reason that the expansion parameter a
should be small.
More significantly, there
are important qualitative phenomena that are missed in perturbation theory.
The reason is that there are non-perturbative contributions to many physically
interesting quantities that have the structure
Such a contribution is completely invisible in perturbation theory.
Perturbative quantum string
theory can be formulated by the Feynman sum-over-histories method. This
amounts to associating a genus h Riemann surface, which can be
visualized as a sphere with h handles attached to it, to the hth
term in the string theory perturbation expansion. The genus h surface
is identified as the corresponding string theory Feynman diagram.
The attractive features
of this approach are that there is just one diagram at each order of the
perturbation expansion and that each diagram represents an elegant (though
complicated) mathematical expression that is ultraviolet finite (no short-distance
The main drawback of this
approach is that it gives no insight into how to go beyond perturbation