Nonlocalizability and asymptotical commutativity

The mathematical formalism commonly used in treating nonlocal highly singular interactions is revised. The notion of a support cone is introduced, which replaces that of support for nonlocalizable distributions. Such support cones are proven to exist for distributions defined on the Gelfand–Shilov spaces$S^\beta $, where $0<\beta <1$.This result leads to a refinement of previous generalizations of the local commutativity condition to nonlocal quantum fields. For string propagators, a new derivation of a representation similar to that of Källen–Lehmann is proposed. It is applicable to any initial and final string configurations and manifests exponential growth of spectral densities intrinsic in nonlocalizable theories.