On the Localizability Problem of Relativistic Quantum Systems

The Localizability Problem addresses the absence of a theoretical counterpart to the position observable in relativistic quantum systems.

Historically, Newton and Wigner were the first to propose a solution by defining (spatially) localized one-particle states (Newton,Wigner-1949). However, this approach faced intrinsic problems concerning the causality structure of Minkowski spacetime. Since then, various proposals have emerged, primarily based on the definition of a covariant positive operator-valued measure. In this seminar, we introduce a new approach grounded in the logic structure of position measurements performed in Minkowski spacetime. This is achieved using techniques from Tomita-Takesaki's Modular Theory and the formalism of Modular Localization (Brunetti, Guido, Longo, 2002). Our method allows us to define a probability measure in the logic-theoretic sense, providing the probability of detecting relativistic quantum systems in causally complete regions in Minkowski spacetime. We also construct a logic-theoretic observable modeling position measurements.