Non-Localizability and Asymptotic Commutativity

TL;DR

This paper introduces support cones for nonlocalizable distributions, refines local commutativity conditions in nonlocal quantum fields, and derives a Källen-Lehmann-like representation for string propagators, highlighting exponential spectral growth.

Contribution

It develops the concept of support cones for nonlocalizable distributions and refines nonlocal quantum field commutativity conditions, with applications to string theory.

Findings

Support cones exist for distributions on Gelfand-Shilov spaces with 0<β<1.

Refined nonlocality conditions for quantum fields are established.

Spectral densities in nonlocal theories exhibit exponential growth.

Abstract

The mathematical formalism commonly used in treating nonlocal highly singular interactions is revised. The notion of 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 K\"{a}llen-Lehmann is proposed. It is applicable to any initial and final string configurations and manifests exponential growth of spectral densities intrinsic in nonlocalizable theories.