Model Green's Functions and the Axioms of Quantum Field Theory.

Model field theories, each of which satisfies most of the requirements, for a complete relativistic quantum field theory, are constructed. The hierarchy of Green's functions associated with each model are explicitly displayed as products of two-point functions so organized that all the linear Wightman axioms are easily checked. It then becomes possible to explore the difficulties that arise when one attempts to implement various nonlinear axioms, e.g. the cluster property, physical positivity, and Nelson's regularity conditions are implemented and limited forms of positivity, called essential positivities, are developed and explored. The models constructed also permit one to analytically continue from Minkowski to Euclidean space and to make connections with Euclidean field theory.