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

February 21, 2024

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 construted 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.

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