Quantum Relativistic Spatial Localization and Representation of the Causal Logic of Minkowski Spacetime for a Massive Scalar Boson

Abstract This communication presents a comprehensive framework for constructing a notion of covariant achronal localizations for relativistic quantum systems from conserved currents, based on the works by D. Castrigiano, by the authors of this communication and a further work arising from their recent collaboration. The approach requires and provides an extension of the divergence theorem to open sets with almost Lipschitz boundary. Building upon prior results on localization on spacelike Cauchy surfaces, the construction is extended to arbitrary achronal sets. Applying this method to probability density currents with causal kernel for the massive scalar boson yields the first explicit covariant representation of the causal logic for an elementary system with definite mass. A parallel construction derives from the stress-energy tensor. The results establish a rigorous, causality-respecting notion of localization fulfilling the foundational spirit of quantum theory.