On the Meaning of Localization in Non-Local Quantum Field Theory

We formulate and derive an uncertainty principle for ultraviolet-complete non-local quantum field theory regulated by an entire function of the d’Alembertian. Under explicit hypotheses on the induced equal-time detector response kernel we prove that the observed localization width obeys an exact variance-addition law. Combining this with the ordinary Heisenberg inequality gives us a non-local uncertainty relation. The bound reduces to the usual local relation in the infrared or local limit when E_M → ∞, while in the ultraviolet it implies a minimal localization length of order L_M. We then explain what this means for locality, microcausality, the interpretation of spacetime points, and the ultraviolet structure of quantum field theory. In this framework spacetime remains a Lorentz-covariant continuum at the level of the manifold description but pointlike localization ceases to be a physically realizable observable notion below the intrinsic non-locality scale.