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

TL;DR

This paper demonstrates that in gauge theories like QED, certain non-local boundary conditions can render some charged observables effectively local within the bulk, preserving microcausality and influencing algebraic quantum field theory structures.

Contribution

It introduces a method to impose boundary conditions that allow charged observables to be viewed as local, maintaining microcausality in gauge theories like QED.

Findings

Non-local boundary conditions enable local interpretation of charged observables.

The approach preserves microcausality within perturbative QED.

Implications for algebraic quantum field theory and potential extension to gravity.

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.