Quantum particle localization observables on Cauchy surfaces of Minkowski spacetime and their causal properties

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

Final version, 59 Pages, no figures, further references and comments added, several typos corrected, accepted for publication in Lett. Math. Phys

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Cited by in corpus (4)

• Localization in Quantum Field Theory
• Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
• Achronal Localization, Representations of the Causal Logic for massive systems
• Localization of the massive scalar boson on achronal hyperplanes, derivation of Lorentz contraction