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arXiv:2604.04173 · Letters in Mathematical Physics

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

Valter Moretti

math-phhep-thquant-phSubmitted Apr 5, 2026• Announced Apr 7, 2026• 0 citations
v1 · Apr 5, 2026v2 · Apr 7, 2026v3 · Jun 18, 2026v4 · Jul 7, 2026
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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.

Journal reference
Lett Math Phys 116, 89 (2026)
DOI
10.1007/s11005-026-02120-5
Comments
83 pages, no figures

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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.
We investigate whether commutativity is necessary to represent relativistic locality for localization observables of relativistic quantum systems in Minkowski spacetime. A well known no-go theorem by Halvorson and Clifton shows that commutativity of localization effects for causally separated regions is incompatible with other seemingly natural assumptions about spatial localization. Since commutativity is taken to represent locality in the Araki-Haag-Kastler framework of QFT, this prompts the question whether it follows from more elementary locality principles of quantum theory. Using Busch's operational analysis in terms of no-signaling and relativistic consistency, we argue that for particle-like systems commutativity is not implied by these principles. Assuming a natural local detectability principle, elementary localization observables are not localized in arbitrarily small spacetime neighborhoods of the relevant spatial regions, but rather in regions containing the entire rest space (a Cauchy surface) on which the measurement is performed. This reflects the particle picture itself, where localization occurs at a unique place on a rest space filled with ideal detectors, and therefore does not directly conflict with the Araki-Haag-Kastler notion of locality. We also show that commutativity and localization can coexist for less idealized localization procedures. To this end, we introduce conditional localization POVMs associated with bounded spatial regions interpreted as laboratories. By the gentle measurement lemma, these observables describe conditional localization probabilities and can, in principle, satisfy commutativity for causally separated laboratories. They may therefore be represented by local observables in the Araki-Haag-Kastler sense. Explicit examples are presented in a second work within local QFT.
Abstract We introduce and study a general notion of spatial localization on spacelike smooth Cauchy surfaces of quantum systems in Minkowski spacetime. The notion is constructed in terms of a coherent family of normalized POVMs, one for each said Cauchy surface. We prove that a family of POVMs of this type automatically satisfies a causality condition which generalizes Castrigiano’s one and implies it when restricting to flat spacelike Cauchy surfaces. As a consequence, no conflict with Hegerfeldt’s theorem arises. We furthermore prove that such families of POVMs do exist for massive Klein–Gordon particles, since some of them are extensions of already known spatial localization observables. These are constructed out of positive definite kernels or are defined in terms of the stress–energy tensor operator. Some further features of these structures are investigated, in particular the relation with the triple of Newton–Wigner selfadjoint operators and a modified form of Heisenberg inequality in the rest 3-spaces of Minkowski reference frames.
In this work, we study the Localizability Problem for relativistic quantum systems.We do it on two fronts.In the first part of this thesis, we extend Newton-Wigner's localization approach to homogeneous globally hyperbolic spacetimes, defining generalized (local) Newton-Wigner position operators.We also give criteria to classify which unitary representations of the spacetime isometry group give origin to localizable representations, showing that the stabilizer group of the spatial isometry group plays a fundamental role.In the second part, we present a novel approach to the Localizability Problem, utilizing techniques from the Modular Theory of Tomita-Takesaki.We argue that position measurements must follow logical principles, incorporated in a mathematical structure referred to as a logic.The core idea is to include the causality structure of Minkowski spacetime in this logic so that the causality problems inherent in Newton-Wigner's approach are solved.We do it through the Modular Localization map [1] for arbitrary massive representations of P + .Our main contribution is the construction of a (quasi-) probability measure on the logic structure of spacetime for each algebraic state (which we interpret as the probability of detection of the system in these spacetime regions), and of a position observable in the logic-theoretic sense.Additionally, we compare our new approach with Newton-Wigner's localization, showing they are approximate in certain regimes.
We review what it means for a relativistic particle to be localized and argue that the most useful approach is through local measurement. This perspective ties localization directly to relativistic causality and measurable outcomes in spacetime. Strict localization in quantum field theory is formalized by states that give the same expectation value as vacuum for all local observables outside some spacetime region. With this formalism, single photons cannot be strictly localized, but coherent states can. The structure and interpretation of local observables and local operator algebras are discussed in connection with the Reeh-Schlieder theorem. Since probabilities of local measurement outcomes cannot vanish in vacuum, particle detection events must be phrased carefully. Key characteristics of strictly localized states are surveyed alongside concrete strategies for constructing them. The relationship between localization, causality, and on-demand sources is analyzed, finding that strictly localized states can represent localized, propagating information in quantum field theory. Explicit examples of localized photonic states are presented with an emphasis on quantum optics, including states that approximate single photons and (entangled) Bell states.
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.
Constructing observables that describe the localization of relativistic particles is an important foundational problem in relativistic quantum field theory (QFT). The description of localization in terms of single-time observables leads to conflict with the requirement of causality. In this paper, we describe particle localization in terms of time-of-arrival observables, defined in terms of the interaction between a quantum field and a measuring apparatus. The resulting probabilities are linear functionals of QFT correlation functions. Specializing to the case of a scalar field, we identify several time-of-arrival observables differing on the way that the apparatus localizes particle-detection records. Maximum localization is obtained for a unique observable that is related to the Newton-Wigner position operator. Finally, we define a measure of localizability for relativistic particles in terms of a novel time-energy uncertainty relation for the variance of the time of arrival.