Spatial localization of relativistic quantum systems: the commutativity requirement and the locality principle: part I—a general analysis

Source abstract

Abstract We study the role of commutativity in the representation of relativistic locality for localization observables of relativistic quantum systems in Minkowski spacetime. A celebrated no-go theorem by Halvorson and Clifton shows that the commutativity of localization effects associated with causally separated regions is incompatible with other apparently natural assumptions about spatial localization. In particular, commutativity is assumed to provide the mathematical representation of locality in the Araki–Haag–Kastler formulation of quantum field theory. This raises the question of whether commutativity follows from more elementary locality principles of quantum theory. Adopting Busch’s operational analysis in terms of no-signaling and relativistic consistency, we argue that, for a particle-like system, the commutativity requirement does not follow from these principles. Under a natural local detectability principle, elementary localization observables are not localized in arbitrarily small spacetime neighborhoods of the corresponding spatial regions, but rather in regions containing the entire rest space (a Cauchy surface) on which the measurement is performed. This is a consequence of the very nature of a particle, which is assumed to be localized at a unique position on a rest space completely filled with ideal detectors. In this respect, there is no direct conflict with the Araki–Haag–Kastler formulation of local quantum physics. On the other hand, we also show that commutativity and localization may coexist when one refers to less idealized localization procedures. Indeed, we introduce conditional localization POVMs associated with bounded spatial regions interpreted as laboratories. Owing to a result in quantum information theory known as the gentle measurement lemma, these observables describe conditional localization probabilities. In principle, these observables satisfy commutativity when associated with causally separated laboratories. As a consequence, they could be represented by local observables in the sense of Araki–Haag–Kastler. Explicit examples of such local observables satisfying the commutativity requirement are presented in a second paper within the framework of local quantum field theory.