  The axiomatic formulation of quantum field theory (QFT) of the 1950's in
terms of fields defined as operator valued Schwartz distributions is
re-examined 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 positivity gives rise to a natural regularization of Wightman
distributions as analytic functions in a tube domain, we argue that the
flexible notions of Colombeau theory which can exploit particular
regularizations is better suited (than Schwartz distributions) for a
mathematical formulation of QFT.
