5 - The analytic S matrix

Summary

The lines of research outlined in Chapters 3 and 4, while never completely independent, merged into a common subject of study at the end of the 1950s. Since quantum field theory had come to a calculational impasse in strong-interaction physics and since people could not prove on the basis of general field theory axioms all of the analytic properties of scattering amplitudes needed to calculate relations among physically measurable quantities, it became the fashion to postulate or guess (with various degrees of justification) general properties and to make up the rules of the game as one went along. Goldberger was one of the most prolific practioners of this art. He recalls as a particularly appealing aspect of that project:

I think the fact that one could deal with physical matrix elements in a largely model independent way was the most attractive aspect to me of S-matrix theory

Let us indicate how this analytic S-matrix program came into existence.

By 1956 Gell-Mann was of the opinion that quantum field theory had no hope of explaining high-energy phenomena and he sketched an alternative program based on general principles and employing unitarity as a central tool in calculations. He acknowledged Heisenberg's original S-matrix program as the ancestor of this project. A crucial step for the dramatic forward progress of this alternative approach to conventional quantum field theory was taken by Mandelstam when he proposed a specific form for a relativistic (double) dispersion relation in the momentum transfer (essentially the scattering angle) variable as well as in the energy variable.