On the Meaning of Localization in Non-Local Quantum Field Theory and On the Limits of a Space-Time Description and the Physical Meaning of Phase Space in a Nonlocal Continuum

Abstract

First: In this paper we explore and derive an uncertainty principle for an ultraviolet complete nonlocal quantum field theory where under our hypothesises of an induced equal time detector response kernel, we then prove that the observed localization width obeys an exact variance addition law. Then when we combine this with the ordinary Heisenberg inequality and we obtain a nonlocal uncertainty relation. The bound reduces to the usual local relation in the infrared or local limit when $E_M \to \infty$, while in the ultraviolet it implies a minimal localization length of order $L_M$. We go on to explain what this means for locality, microcausality, the interpretation of spacetime points, and the ultraviolet structure of quantum field theory. In this formulation we note and prove that spacetime will remain a Lorentz covariant continuum at the level of the manifold description but pointlike localization ceases to be a physically realizable observable notion below the nonlocality scale. Second: In a previous paper we derived an uncertainty relation for nonlocal fields by showing that the physical localization width in nonlocal quantum field theory is broadened by the response kernel generated from the entire-function regulator. In this follow up we will reinterpret that result as just the position-sector limit of a more symmetric statement as we did not take into account the inherent nonlocality of momentum. We find under normalized, centered response kernels for both position and momentum, we prove the variance laws and derive the corresponding nonlocal phase-space uncertainty relation. The result still preserves the conclusions of the original paper all while strengthening the interpretation as nonlocal quantum field theory implies not merely a minimal measurable length, but a finite phase-space cell. We also explore experimental routes to test the theory.