Nonlocal Extension of the Borchers Classes of Quantum Fields

We formulate an equivalence relation between nonlocal quantum fields, generalizing the relative locality which was studied by Borchers in the framework of local QFT. The Borchers classes are shown to allow a natural extension involving nonlocal fields with arbitrarily singular ultraviolet behavior. Our consideration is based on the systematic employment of the asymptotic commutativity condition which, as established previously, ensures the normal spin and statistics connection as well as the existence of PCT symmetry in nonlocal field theory. We prove the transitivity of the weak relative asymptotic commutativity property generalizing Jost-Dyson's weak relative locality and show that all fields in the same extended Borchers class have the same S-matrix.