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arXiv:2512.16898

Fighting non-locality with non-locality: microcausality and boundary conditions in QED

Philipp A. Hoehn, Josh Kirklin

hep-thgr-qcquant-phSubmitted Dec 18, 2025• Announced Dec 19, 2025
arXivHTMLPDF

Abstract

In gauge theories, globally charged observables necessarily depend non-locally on the kinematical fields, with this dependence extending to the asymptotic boundary of spacetime. Despite this, we show that a subset of such observables can be consistently regarded as local to the bulk, in a manner that respects microcausality and leaves locality properties of uncharged observables untouched. A sufficient condition for this is to impose kinematically non-local boundary conditions on the large gauge sector of the theory, and to invoke a relational notion of localisation for observables. This reveals a relatively underappreciated link between boundary conditions, and different notions of microcausality and locality. We develop this point through a detailed case study in scalar QED, describing non-local boundary conditions that allow a large family of observables on a codimension-1 bulk surface to be viewed as local to that surface, despite being dressed by asymptotic Wilson lines. We show that this property continues to hold within a perturbative quantisation of the theory, and we argue that this leads to a consistent local net of algebras that includes these charged observables in bulk algebras. We explain how this setup may be understood in terms of a preferred dynamical reference frame for small gauge transformations appearing in the boundary conditions. Many features of the theory (such as microcausality, the vacuum state, and the net of algebras of observables) depend on the choice of this frame, and we briefly discuss some repercussions of this for algebraic formulations of QFT. While our analysis is performed in QED, we expect our results to carry over qualitatively to more complicated theories including gravity.

Comments
52 pages, 7 figures, comments welcome

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In gauge theories, globally charged observables necessarily depend non-locally on the kinematical fields, with this dependence extending to the asymptotic boundary of spacetime. Despite this, we show that a subset of such observables can be consistently regarded as local to the bulk, in a manner that respects microcausality and leaves locality properties of uncharged observables untouched. A sufficient condition for this is to impose kinematically non-local boundary conditions on the large gauge sector of the theory, and to invoke a relational notion of localisation for observables. This reveals a relatively underappreciated link between boundary conditions, and different notions of microcausality and locality. We develop this point through a detailed case study in scalar QED, describing non-local boundary conditions that allow a large family of observables on a codimension-1 bulk surface to be viewed as local to that surface, despite being dressed by asymptotic Wilson lines. We show that this property continues to hold within a perturbative quantisation of the theory, and we argue that this leads to a consistent local net of algebras that includes these charged observables in bulk algebras. We explain how this setup may be understood in terms of a preferred dynamical reference frame for small gauge transformations appearing in the boundary conditions. Many features of the theory (such as microcausality, the vacuum state, and the net of algebras of observables) depend on the choice of this frame, and we briefly discuss some repercussions of this for algebraic formulations of QFT. While our analysis is performed in QED, we expect our results to carry over qualitatively to more complicated theories including gravity.
Bell appealed to the theory of relativity in formulating his principle of local causality. But he maintained that quantum field theories do not conform to that principle, even when their field equations are relativistically covariant and their observable algebras satisfy a relativistically motivated microcausality condition. A pragmatist view of quantum theory and an interventionist approach to causation prompt the reevaluation of local causality and microcausality. Local causality cannot be understood as a reasonable requirement on relativistic quantum field theories: it is unmotivated even if applicable to them. But microcausality emerges as a sufficient condition for the consistent application of a relativistic quantum field theory.
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Non-locality and quantum appear together entitling scientific projects and papers quite usually.These are two hardly detachable concepts and modern quantum theories are built up taking into account this aspect explicitly.However, that characteristic turns challenging from a fundamental point of view and can be unhelpful for practical purposes.Here, we construct quantum field theory with deliberately local aspects for a 1+1 dimensional Dirac field in flat spacetime generalizing a recently developed formalism for scalar fields.That purpose is achieved exploiting the non-uniqueness of quantization process.This local construction leads to a natural notion of local quanta and provides a local algebra of operators.As an application, it is shown that the standard vacuum state is a boiling soup of local particles and this fact is connected with a still Gedankenexperiment consisting in slamming down and removing a mirror in an empty cavity. I. MOTIVATION AND BACKGROUNDNon-locality is a feature as intuitively challenging as genuine in the quantum world.The so-successful ordinary representation of quantum field theory (QFT) is based on the information carried by global operators (creation-annihilation operators) related to field excitations with well defined momentum, and therefore, completely delocalized in space.This picture accounts for an innumerable amount of phenomena with an astonishing accuracy, as particle physics or solid state physics shows.However, this conventional formalism can hardly deal with tasks related to local properties of quantum field states.Simple questions as "where is that particle?"or "how are these particles distributed in space?" are unsuccessfully answered in this background.What is more, it is difficult to reconcile the common understanding of a particle as a minute entity in space with the particle notion arising from the ordinary Fock construction, which rather describes excitations spreading in whole space.Furthermore, no localized states can be constructed in such QFT where occupation number representation of a quantum field can be understood as a "momentum" representation in the sense that the number of particles informs about the momentum carried by the field.The problem at this point is that it does not exist a reciprocal "position" representation, we have no local (in space) creation-annihilation operators.This result can be understood as a consequence of Hegerfeldt's theorem.
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