In the previous chapter, we were forced to face the fact that string theory, if it describes nature, is not weakly coupled. On the other hand, the very formulation we have put forward of the theory is perturbative. We have described the quantum mechanics of single strings, and given a prescription for calculating their interactions order by order in perturbation theory in a parameter gs. There is a parallel here to Feynman's early work on relativistic quantum theory: Feynman guessed a set of rules for computing perturbative amplitudes of electrons. In this case, however, one already had a candidate for an underlying description: quantum electrodynamics. It was Dyson who clarified the connection. For Abelian theories, the non-perturbative theory probably does not really exist, but in the case of non-Abelian gauge theories it does. The field theoretic formulation provides an understanding of the underlying symmetry principles, and access to a trove of theoretical information.

A string field theory would be a complicated object. The string fields themselves would be functionals of the classical two-dimensional fields which describe the string. The quantization of such fields is sometimes called “third quantization.” Much effort has been devoted to writing down such a field theory. For open strings, one can write relatively manageable expressions which reproduce string perturbation theory. For closed strings, infinite sets of contact interactions are required. But apart from their cumbersome structure, there are reasons to suspect that this is not a useful formulation. There would seem to be, for example, vastly too many degrees of freedom.