Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part II: A Model from Local QFT

TL;DR

This paper constructs relativistic localization observables in quantum field theory that respect causality and locality, providing a rigorous framework for understanding position measurements in relativistic quantum systems.

Contribution

It introduces a class of positive-energy localization POVMs derived from local field quantities, ensuring relativistic causality and compatibility with the algebraic QFT framework.

Findings

Localization observables satisfy causality conditions preventing superluminal detection

First moments of the constructed POVMs reproduce the Newton-Wigner position operator

Conditional localization POVMs commute for causally separated regions, aligning with local algebraic QFT principles

Abstract

This paper completes a previous work by constructing a class of positive-energy relativistic spatial localization observables in Minkowski spacetime within quantum field theory, using the stress-energy-momentum tensor smeared with suitable test functions. For each timelike direction, the construction yields a family of positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every n-particle sector and satisfying a natural relativistic causality condition excluding superluminal propagation of detection probabilities. These observables arise from local or quasi-local field-theoretic quantities and provide a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable introduced previously, and its first moment reproduces the Newton-Wigner position operator under suitable normalization conditions. Because the normally ordered stress-energy-momentum tensor is not positive on the full Fock space, as implied by the Reeh-Schlieder theorem, we study quantum energy inequalities and derive lower bounds controlling deviations from positivity. This leads to regularized families of positive operators approximating the localization effects. We also construct conditional localization observables for finite laboratories using modified local energy operators and their Friedrichs self-adjoint extensions. Using Haag duality and Kadison's result on affiliation, we show that the resulting conditional POVMs belong to local von Neumann algebras and therefore commute for causally separated regions, in agreement with the Araki-Haag-Kastler framework. These results support the view that commutativity of localization observables is recovered at the level of conditional measurements in finite spacetime regions.