We use a one-dimensional model system to compare the predictions of two different yardsticks to compute the position of a particle from its quantum field theoretical state. Based on the first yardstick (defined by the Newton-Wigner position operator), the spatial density can be arbitrarily narrow, and its time evolution is superluminal for short time intervals. Furthermore, two spatially distant particles might be able to interact with each other outside the light cone, which is manifested by an asymmetric spreading of the spatial density. The second yardstick (defined by the quantum field operator) does not permit localized states, and the time evolution is subluminal.
Abstract I rigorously analyze a proposal, introduced by D.R. Terno, about a spatial localization observable for a Klein–Gordon massive real particle in terms of a Poincaré-covariant family of POVMs. I prove that these POVMs are actually a kinematic deformation of the Newton–Wigner PVMs. The first moment of one of these POVMs, however, exactly coincides with a restriction (on a core) of the Newton–Wigner self-adjoint position operator, though the second moment does not. This fact permits to preserve all nice properties of the Newton–Wigner position observable, dropping the unphysical features arising from the Hegerfeldt theorem. The considered POVM does not permit spatially sharply localized states, but it admits families of almost localized states with arbitrary precision. Next, I establish that the Terno localization observable satisfies part of a requirement introduced by D.P.L. Castrigiano about causal temporal evolution concerning the Lebesgue measurable spatial regions of any Minkowskian reference frame. The validity of the complete Castrigiano’s causality requirement is also proved for a notion of spatial localization which generalizes Terno’s one in a natural way.
This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every
n-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities.
The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions.
Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects.
Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.