On the Localizability Problem of relativistic quantum systems - Oliveira, Ivan Romualdo de (2024)
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 \\textit{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 \\cite{brunetti_modular_2002} for arbitrary massive representations of $\\mathcal{P}_+$. 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.