Unsharp Localization and Causality in Relativistic Quantum Theory

TL;DR

This paper investigates the relationship between relativistic causality and localizability in quantum theory, demonstrating that unsharp localization observables are inherently non-sharp and that local commutativity is essential for Einstein causality.

Contribution

It generalizes Schlieder's theorem to show unsharp localization observables must be strongly unsharp under local commutativity and links local measurements with causality constraints.

Findings

Unsharp localization observables are strongly unsharp under local commutativity.

Local commutativity is necessary for Einstein causality with unsharp observables.

Raises questions about measuring localization observables via local operations.

Abstract

The conflict between relativistic causality and localizability is analyzed in the light of the existence of unsharp localization observables. A theorem due to S. Schlieder is generalized, showing that the assumption of local commutativity implies the localization observable in question to be unsharp in a strong sense. Furthermore, a recent generalization of a theorem of L\"uders is applied to demonstrate that local commutativity is a necessary consequence of Einstein causality even in the case of unsharp observables if they admit local measurements. These findings raise the question whether localization observables can be measured by means of local operations.