Spatial Localization of Relativistic Quantum Systems: The Commutativity Requirement and the Locality Principle. Part I: A General Analysis

TL;DR

This paper examines the necessity of commutativity for representing relativistic locality in quantum systems, arguing that elementary localization does not imply commutativity and introducing conditional localization observables.

Contribution

It challenges the assumption that commutativity is required for locality, proposing a framework where localization and commutativity can coexist under certain conditions.

Findings

Elementary localization observables are not localized in arbitrarily small neighborhoods.

Localization occurs on a rest space, consistent with the particle picture.

Conditional localization POVMs can satisfy commutativity for causally separated labs.

Abstract

We investigate whether commutativity is necessary to represent relativistic locality for localization observables of relativistic quantum systems in Minkowski spacetime. A well known no-go theorem by Halvorson and Clifton shows that commutativity of localization effects for causally separated regions is incompatible with other seemingly natural assumptions about spatial localization. Since commutativity is taken to represent locality in the Araki-Haag-Kastler framework of QFT, this prompts the question whether it follows from more elementary locality principles of quantum theory. Using Busch's operational analysis in terms of no-signaling and relativistic consistency, we argue that for particle-like systems commutativity is not implied by these principles. Assuming a natural local detectability principle, elementary localization observables are not localized in arbitrarily small spacetime neighborhoods of the relevant spatial regions, but rather in regions containing the entire rest space (a Cauchy surface) on which the measurement is performed. This reflects the particle picture itself, where localization occurs at a unique place on a rest space filled with ideal detectors, and therefore does not directly conflict with the Araki-Haag-Kastler notion of locality. We also show that commutativity and localization can coexist for less idealized localization procedures. To this end, we introduce conditional localization POVMs associated with bounded spatial regions interpreted as laboratories. By the gentle measurement lemma, these observables describe conditional localization probabilities and can, in principle, satisfy commutativity for causally separated laboratories. They may therefore be represented by local observables in the Araki-Haag-Kastler sense. Explicit examples will be presented in forthcoming work within local QFT.