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.
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.
Abstract 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. Covariant 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.
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.
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.
Causal localizations describe the position of quantum systems moving not faster than light. They are constructed for the systems with finite spinor dimension. At the center of interest are the massive relativistic systems. For every positive mass, there is the sequence of Dirac tensor-localizations, which provides a complete set of inequivalent irreducible causal localizations. They obey the principle of special relativity and are fully Poincaré covariant. The boosters are determined by the causal position operator and the other Poincaré generators. The localization with minimal spinor dimension is the Dirac localization. Thus, the Dirac equation is derived here as a mere consequence of the principle of causality. Moreover, the higher tensor-localizations, not known so far, follow from Dirac’s localization by a simple construction. The probability of localization for positive energy states results to be described by causal positive operator valued (PO-) localizations, which are the traces of the causal localizations on the subspaces of positive energy. These causal Poincaré covariant PO-localizations for every irreducible massive relativistic system were, all the more, not known before. They are shown to be separated. Hence, the positive energy systems can be localized within every open region by a suitable preparation as accurately as desired. Finally, the attempt is made to provide an interpretation of the PO-localization operators within the frame of conventional quantum mechanics attributing an important role to the negative energy states.
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 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.