Phase space analysis and particle structure in field theory

Abstract

Progress in the last few years in constructive field theory has included (1) the construction of three-dimensional, non-Abelian gauge theories in a finite volume, (2) the construction of models that are asymptotically free or have a nontrivial fixed point, (3) a new presentation of perturbation theory yielding old and new large-order estimates, and (4) asymptotic completeness for renormalizable models in the two-particle region; the existence of multiparticle structure equations in the Euclidean region, provided that the coupling constant goes to zero as the energy increases; these equations yield formally the asymptotic completeness. In obtaining most of these results the phase space expansion pioneered by Glimm and Jaffe plays a crucial role. The first part of this review is a description of the phase space expansion in the “Ecole Polytechnique” version; the second part is devoted to the construction of models, the third part treats perturbation theory; and the last part deals with asymptotic completeness.