The discovery of string theory is a remarkable story that began as an attempt to understand hadronic dynamics. It was believed that quantum field theory could not account for the dynamics of hadrons and that one might be able to simply write down the S matrix which was to satisfy a set of properties one of which was called duality. An S matrix for initially four, and then any number, of spin-0 particles of the same mass which did satisfy all the desired properties was eventually found. Although it might seem that these S matrix elements could not be of much value, using physical and mathematical consistency, the early pioneers deduced from them the S matrix for the scattering of an infinite number of particles that were identified as those exchanged in the scattering of the spin-0 particles. Eventually these particles were identified with the states of a string. In telling this story in section 18.1 we will also get a good feeling for what string scattering amplitudes look like, what are their properties and, indeed, for string theory itself.

There is no known complete theory of strings that can be used to compute, even as a matter of principle, all their properties. However, there are a number of approaches that can be used to compute the perturbative scattering of strings. One of the earliest and most used approaches involves a sum over the world surfaces swept out by the string and this is the subject of section 18.2.