ESI The Erwin Schr odinger

The axiomatic formulation of quantum eld theory (QFT) of the 1950's in terms of elds deened as operator valued Schwartz distributions is reexamined in the light of subsequent developments. These include, on the physical side, the construction of a wealth of (2-dimensional) soluble QFT models with quadratic exchange relations, and, on the mathematical side, the introduction of the Colombeau algebras of generalized functions. Exploiting the fact that energy pos-itivity gives rise to a natural regularization of Wightman distributions as analytic functions in a tube domain, we argue that the exible notions of Colombeau theory which can exploit particular regularizations is better suited (than Schwartz distributions) for a mathematical formulation of QFT.