11 - Functional methods in the light-cone gauge

Summary

Perturbative calculations based on the operator formalism described in chapters 7 – 10 can be used when at least one of the strings at each interaction vertex is a physical on-shell state. That approach is effective for calculations of tree and one-loop amplitudes but is not applicable for multiloop amplitudes, since these necessarily involve at least one vertex coupling three internal (and therefore off-shell) strings.

In principle, the systematic derivation of rules for calculating arbitrary string-theory diagrams should follow from a second-quantized string field theory. This has been developed in the light-cone gauge for both bosonic strings and superstrings. A major effort is currently underway to develop covariant gauge-invariant action principles, as well. Such a formulation might provide a deeper understanding of the geometric significance of string theories.

We will not develop string field theory in this book, but in this chapter we do develop a “first-quantized” approach to the Feynman rules of light-cone string field theory. Apart from exhibiting at least some of the ingredients of string field theory, this approach has several other virtues. The equivalence of the first-quantized Feynman rules to world-sheet path integrals – the only approach to multiloop amplitudes that we have mentioned hitherto – is important in understanding the unitarity of the latter. Also, while we have extensively discussed conformal invariance and conformal mappings in previous chapters, the explicit and detailed applications of these concepts in this chapter should shed a new light on them.