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arXiv:1303.6649 · Journal of Physics A Mathematical and General

Unsharp localization and causality in relativistic quantum theory

Paul Busch

Published Sep 6, 1999• 47 citations
arXivHTMLPDF

Abstract

The conflict between relativistic causality and localizability is analysed in the light of the existence of unsharp localization observables. A theorem due to 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 Lüders 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 of whether localization observables can be measured by means of local operations.

DOI
10.1088/0305-4470/32/37/305

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•Jan 1, 1998•75 citations
The conflict between relativistic causality and localizability is analysed in the light of the existence of unsharp localization observables. A theorem due to 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 Lüders 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 of whether localization observables can be measured by means of local operations.
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 are presented in a second work within local QFT.
We review what it means for a relativistic particle to be localized and argue that the most useful approach is through local measurement. This perspective ties localization directly to relativistic causality and measurable outcomes in spacetime. Strict localization in quantum field theory is formalized by states that give the same expectation value as vacuum for all local observables outside some spacetime region. With this formalism, single photons cannot be strictly localized, but coherent states can. The structure and interpretation of local observables and local operator algebras are discussed in connection with the Reeh-Schlieder theorem. Since probabilities of local measurement outcomes cannot vanish in vacuum, particle detection events must be phrased carefully. Key characteristics of strictly localized states are surveyed alongside concrete strategies for constructing them. The relationship between localization, causality, and on-demand sources is analyzed, finding that strictly localized states can represent localized, propagating information in quantum field theory. Explicit examples of localized photonic states are presented with an emphasis on quantum optics, including states that approximate single photons and (entangled) Bell states.
In this paper, I provide a formal set of assumptions and give a natural criterion for a quantum field theory to admit particles. I construct a naïve approach to localization for a free bosonic quantum field theory and show how this localization scheme, as a consequence of the Reeh-Schlieder theorem, fails to satisfy this criterion. I then examine the Newton-Wigner concept of localization and show that it fails to obey strong microcausality and thus is subject to a more general version of the Reeh-Schlieder theorem. I review approaches to quantum field theoretic explanations of particle detection events and explain how particles can be regarded as emergent phenomena of a relativistic field theory. In particular, I show that effective localization of Hilbert space vectors is equivalent to an approximate locality of observable algebras.
The localization problem in relativistic quantum theory has persisted for more than seven decades, yet it is largely unknown and continues to perplex even those well-versed in the subject. At the heart of this problem lies a fundamental conflict between localizability and relativistic causality, which can also be construed as part of the broader dichotomy between measurement and unitary dynamics. This article provides a historical review of the localization problem in one-particle relativistic quantum mechanics, clarifying some persistent misconceptions in the literature, and underscoring the antinomy between causal dynamics and localized observables.
This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every nnn-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities. The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions. Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects. Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.