Spatial localization of relativistic quantum systems: the commutativity requirement and the locality principle—part II: a model from local QFT

Abstract This paper is the second and final part of a two-part study. We construct a class of positive-energy relativistic spatial localization observables in Minkowski spacetime within the standard framework of quantum field theory, based on the stress–energy–momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction yields a family of positive operator-valued measures (POVMs) defined on spacelike hypersurfaces, which are well defined on each n -particle sector and satisfy a natural relativistic causality condition ruling out superluminal propagation of detection probabilities. The proposed localization observables arise from local or quasi-local quantum-field-theoretic quantities, thereby providing a rigorous realization of previously heuristic constructions. In the one-particle sector, the scheme reduces to the observable introduced by the author in previous literature, and its first moment reproduces the Newton–Wigner position operator under suitable normalization and centering conditions. Since the Reeh–Schlieder theorem implies that the normally ordered stress–energy–momentum tensor need not be positive on the full Fock space, we analyze the role of quantum energy inequalities and establish lower bounds that allow us to control deviations from positivity. This leads to the introduction of regularized families of operators, bounded from below, that approximate the localization effects. We further construct conditional localization observables associated with finite laboratories by means of suitably modified local energy operators. In particular, by Haag duality, the resulting conditional POVMs are shown to belong to local von Neumann algebras and hence to commute when associated with causally separated regions, in agreement with the Araki–Haag–Kastler framework. Our results provide a quantum-field-theoretic implementation of the idea that commutativity of localization observables should be recovered at the level of conditional measurements in spacetime regions of finite extent.