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International Journal of Theoretical Physics

Quantum Relativistic Spatial Localization and Representation of the Causal Logic of Minkowski Spacetime for a Massive Scalar Boson

Carmine De Rosa, Valter Moretti

Published Feb 25, 2026• 0 citations
Publisher

Abstract

Abstract This communication presents a comprehensive framework for constructing a notion of covariant achronal localizations for relativistic quantum systems from conserved currents, based on the works by D. Castrigiano, by the authors of this communication and a further work arising from their recent collaboration. The approach requires and provides an extension of the divergence theorem to open sets with almost Lipschitz boundary. Building upon prior results on localization on spacelike Cauchy surfaces, the construction is extended to arbitrary achronal sets. Applying this method to probability density currents with causal kernel for the massive scalar boson yields the first explicit covariant representation of the causal logic for an elementary system with definite mass. A parallel construction derives from the stress-energy tensor. The results establish a rigorous, causality-respecting notion of localization fulfilling the foundational spirit of quantum theory.

DOI
10.1007/s10773-026-06291-8

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Abstract This communication presents a comprehensive framework for constructing a notion of covariant achronal localizations for relativistic quantum systems from conserved currents, based on the works by D. Castrigiano, by the authors of this communication and a further work arising from their recent collaboration. The approach requires and provides an extension of the divergence theorem to open sets with almost Lipschitz boundary. Building upon prior results on localization on spacelike Cauchy surfaces, the construction is extended to arbitrary achronal sets. Applying this method to probability density currents with causal kernel for the massive scalar boson yields the first explicit covariant representation of the causal logic for an elementary system with definite mass. A parallel construction derives from the stress-energy tensor. The results establish a rigorous, causality-respecting notion of localization fulfilling the foundational spirit of quantum theory.
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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.
We construct a covariant representation of the causal logic for the Dirac system and the electron, based on the conserved Dirac probability current and a general current-to-localization procedure for achronal regions. A polynomial decay estimate, derived by a non-stationary phase argument, verifies the conditions required to define covariant achronal localization for the full Dirac system. Extending this localization to complete spacetime regions yields the corresponding causal-logic representation; restriction to the positive-energy invariant subspace gives the analogous construction for the electron. We establish several structural properties. On Euclidean space, Dirac localization agrees with canonical projection-valued localization, while its position operator differs from the Newton--Wigner operator by an explicitly determined bounded self-adjoint correction. Full Dirac achronal localization is projection valued and represents achronal separateness by orthogonality, thus satisfying microscopic causality. After compression to the electron subspace, it becomes positive-operator valued while preserving causality. We further extend known results from spatial to general achronal regions, prove separation and norm-one criteria, and analyze the high-boost limit, obtaining Lorentz contraction in a precise probabilistic sense. Finally, we discuss the electron--positron decomposition and the state transformation induced by position measurements, showing that measurement-induced positron production is universal and independent of the specific measuring device.
On plain physical grounds localization of relativistic quantum particles is extended to the achronal regions of Minkowski spacetime. Achronal localization fulfills automatically the requirements of causality. It constitutes the frame which complies most completely with the principle of causality for quantum mechanical systems. Achronal localization is equivalent to the localization in the regions of the causal logic. Representations of the causal logic are constructed for the systems with mass spectrum of positive Lebesgue measure and every definite spin. Apparently no representation of the causal logic for a real mass system has been achieved in the past.
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 nnn-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.
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.