Relativistic quantum field theory with a fundamental length

Since there are indications (from string theory and concrete models) that one must consider relativistic quantum field theories with a fundamental length the question of a suitable framework for such theories arises. It is immediately evident that quantum field theory in terms of tempered distributions and even in terms of Fourier hyperfunctions cannot meet the (physical) requirements. We argue that quantum field theory in terms of ultra-hyperfunctions is a suitable framework. For this we propose a set of axioms for the fields and for the sequence of vacuum expectation values of the fields, prove their equivalence, and we give a class of models (analytic, but not entire functions of free fields). (C) 2004 American Institute of Physics.