Perturbation Theory

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 infinities).
The main drawback of this approach is that it gives no insight into how to go beyond perturbation theory.

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