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

TL;DR

The paper investigates how gauge theories' inherent non-locality, particularly for charged observables, interacts with locality and microcausality when non-local boundary conditions are imposed. By focusing on scalar QED, it demonstrates that a substantial class of charged, boundary-dressed observables can be regarded as local to a bulk codimension-1 surface if one employs a non-local Goldstone boundary condition on the large gauge sector and a relational notion of observable support. This relational locality is analyzed at both classical and perturbative quantum levels, revealing a frame-dependent but internally consistent net of algebras and a frame-dependent vacuum; changes of the microcausal frame correspond to canonical deformations of the boundary data and generate observable effects. The work highlights a trade-off: boundary non-locality can enhance bulk microcausality for certain observables, while altering the global structure of the theory, with potential qualitative extensions to gravity and holography. Overall, it clarifies how boundary conditions and relational locality shape microcausality, nets of algebras, and the vacuum in gauge theories.

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.

Figures (8)

• Figure 1.1: Bulk-dressed observables like $\psi(x)\psi'(x)$ can have arbitrarily compact support. Boundary-dressed observables like $\Psi(y)$ must extend to the boundary, and are sensitive to large gauge transformations (i.e. those which act non-trivially at the boundary).
• Figure 1.2: On the left, standard Dirichlet boundary conditions lead to a scalar field obeying microcausality; signals are reflected off the timelike boundaries $\Gamma_{1,2}$. On the right, periodic boundary conditions lead to a scalar field that violates microcausality (with respect to the underlying causal structure); signals instantaneously propagate between the timelike boundaries.
• Figure 2.1: We consider spacetimes whose boundaries decompose into a timelike component $\Gamma$ and two spacelike components $\Sigma^\pm$.
• Figure 3.1: The non-local boundary conditions studied in this paper involve the choice of a timelike codimension 1 surface $\mathcal{N}\subset\mathcal{M}$, a bijection $f:\mathcal{N}\to\Gamma$ that preserves causal ordering, and for each $y\in\mathcal{N}$ a curve $\gamma(y)$ connecting $y$ to $f(y)$.
• Figure 4.1: For $O_1$, $O_2$ with spacelike separated supports, we can always choose a Cauchy surface $\Sigma$ with an open neighbourhood that only ever intersects one of the domains of influence of $\operatorname{supp}(O_1)$, $\operatorname{supp}(O_2)$.

Theorems & Definitions (6)

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