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arXiv:2604.03729 · Letters in Mathematical Physics

Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part I: A General Analysis

Valter Moretti

math-phhep-thquant-phSubmitted Apr 4, 2026• Announced Apr 7, 2026• 0 citations• Revised Apr 7, 2026 (v2)
v1 · Apr 4, 2026v2 · Apr 7, 2026v3 · Sep 10, 2026
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Abstract

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 will be presented in forthcoming work within local QFT.

Journal reference
Lett Math Phys 116, 122 (2026)
DOI
10.1007/s11005-026-02159-4
Comments
38 Pages, no figures

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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 will be presented in forthcoming work within local QFT.
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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 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.
An operational development of quantum theory is presented, from general correlations through single-particle theories to elementary quantum field theory, with a focus throughout on causal relations. The objective is to establish the extent to which quantum theories are consistent with the principle of relativistic causality (that no two events separated by a distance greater than their separation in time multiplied by the speed of light may have a causal influence on each other) and to examine the assumptions that this analysis requires. To do this, it is necessary to pay particular attention to the notions of spatial location and measurement. It is found that in a relativistically causal theory, any measurement made in a finite spatial region must have the capacity for particle creation. A number of derivations are presented, including some relativistic single-particle propagators and a Hamiltonian based on the square root of the Klein Gordon equation.