Non-Commutative Quantum Field Theories in Terms of Tempered Ultrahyperfunctions

In the present paper, we wish to consider the quantum field theory on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, Wm, have the same form as the ordinary Wightman functions, Wm. We conjecture that the functions Wm satisfy a set of properties which actually will characterize a non-commutative quantum field theory in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the Spin-Statistics connection are the best known. We assume the validity of these theorems for noncommutative quantum field theories in the case of spatial non-commutativity only.