Three Exactly Soluble Quantum Field Theory Models in 2-,3- and 4-Dimensional Space Time and Some General Questions They Suggest

As a result of three decades of hard work on the local algebra formalism, we now have a general theory of quantized fields that provides a satisfactory framework for field theory. On the other hand, constructive quantum field theory has made rather limited progress toward the objective of characterizing and constructing all field theories that satisfy the axioms of the general theory. We do have the nontrivial examples P(ϕ)₂,Y₂,ϕ₃4,Y₃,Higgs₂,Higgs₃ and fragments of Y M₃ and Y M₄. However, these examples do not provide enough information to suggest reasonable guesses for the answers to general questions. For example, how do the perturbatively non-renormalizable theories fit into the general picture? The recent results of da Veiga and coworkers establishing the existence of tempered solutions of the Gross-Neveu model in three dimensional Euclidean space-time, show that a non-perturbative treatment of a perturbatively non-renormalizable theory is possible using rigorous renormalization group methods. What distinguishes such theories? Is the applicability of renormalization group methods to be regarded as always a reliable guide to the existence of non-trivial solutions of theories?