General properties of non-commutative field theories

In this paper we study general properties of non-commutative field theories obtained from the Seiberg-Witten limit of string theories in the presence of an external B-field. We analyze the extension of the Wightman axioms to this context and explore their consequences, in particular, we present a proof of the CPT theorem for theories with space-space non-commutativity. We analyze as well questions associated to the spin-statistics connections, and show that non-commutative N=4, U(1) gauge theory can be softly broken to N=0 satisfying the axioms and providing an example where the Wilsonian low energy effective action can be constructed without UV/IR problems, after a judicious choice of soft breaking parameters is made. We also assess the phenomenological prospects of such a theory, which are in fact rather negative.