Green functions in particle physics

Abstract

We review recent developments in the theory of Green functions (off-shell S-matrix elements) and their applications in elementary particle physics (particularly strong interactions). We concentrate on dynamical problems, taking the underlying particle symmetries for granted. Green functions were first considered as an offshoot of quantum field theory as applied in quantum electrodynamics, and our introduction sketches how this historical background has affected the development of both on-shell and off-shell methods for the strong interactions. Most of the detailed applications of Green function methods are in two-body elastic scattering at moderate relativistic energies (neglecting production processes). The main tool here is the Bethe-Salpeter equation in the ladder approximation. We derive the Bethe-Salpeter equation by summing Feynman graphs, and give a critical review of its application to various scattering and bound-state problems (nucleon-nucleon, pion-nucleon and pion-pion systems), including where available detailed comparisons with the results of dispersive and potential-theoretic calculations. We review very briefly the use of the Bethe-Salpeter equation as a `model' for the Regge-pole hypothesis and the quark hypothesis. We then go on to consider processes involving more than two particles, and the `internal' multiparticle (or many-body) structure of the corresponding Green functions. This structure can be simply and plausibly expressed by a set of generalized Bethe-Salpeter equations which together represent an off-shell version of unitarity. We rederive the graphical restrictions previously imposed on the kernles of two-body Bethe-Salpeter equations, and we derive the relativistic Faddeev-type three-body linear equations.