Microcausality Violations via Dressing in Gauge & Gravity

• Dressing-induced microcausality violations refer to the introduction of nonlocal dressing operations that are essential for constructing gauge-invariant observables, leading to subtle breaches in spacelike commutativity.
• Methodologies in both QED and gravity employ nonlocal dressings such as Wilson lines and Coulomb-type integrations to counteract gauge transformations while inherently compromising strict locality.
• These violations, though numerically suppressed (e.g., ~10⁻³⁶), have significant implications for the algebraic foundations of quantum field theory, affecting subsystem independence and quantum gravity experiments.

Dressing-induced microcausality violations arise in gauge and gravitational theories from the necessity of defining physically meaningful, gauge-invariant observables via "dressing" operations. This inherently introduces a nonlocal structure to observables, leading to subtle, generally parametrically suppressed but conceptually important violations of microcausality—the property that spacelike separated operators commute. Such violations are a fundamental consequence of gauge symmetry, diffeomorphism invariance, and the nonlocal constraints that these symmetries impose on the kinematical algebra of observables. While negligible in practical laboratory contexts, these violations are central for the algebraic and operational foundations of quantum field theory in gauge and gravitational settings, and have direct repercussions for subsystem independence, experimental tests of quantum-gravity-induced entanglement, and the formulation of locality itself.

1. Nonlocal Dressing and its Role in Gauge Invariance

Gauge theories, including both electromagnetism (QED) and general relativity, require that physically meaningful observables be gauge-invariant. For local matter fields such as scalars or spinors, this demands the construction of "dressed" operators, in which the field is supplemented by a nonlocal functional of the gauge or metric field.

In linearized gravity, a scalar field $\phi(x)$ becomes gauge-invariant only upon addition of a nonlocal "dressing" $V^\mu[x; h]$, yielding $$\Phi(x) = \phi(x) + V^\mu(x) \partial_\mu \phi(x) + O(\kappa^2)$$, where $V^\mu$ is constructed to precisely cancel the diffeomorphism transformation of $\phi$ under gauge-induced coordinate shifts. Typical examples include Coulomb-type dressings, which explicitly involve metric integration along semi-infinite spatial curves, thus encoding the required long-range gravitational field profile and introducing unavoidable nonlocality (Boulle et al., 18 Dec 2025). Similarly, in QED, physically charged operators are constructed using Wilson lines or asymptotic boundary functionals, e.g., $\Psi_f(x) = \exp(-iq \int_{\gamma_f(x)} A) \psi(x)$, with $\gamma_f(x)$ a curve to the boundary; the existence and path-dependence of this line integral again signals intrinsic nonlocality (Hoehn et al., 18 Dec 2025).

2. Breakdown of Microcausality: Classical and Quantum Structure

The canonical formalism of quantum field theory associates strict microcausality with vanishing commutators (or, classically, Poisson brackets) between observables whose supports are spacelike separated. For dressed, gauge-invariant operators, however, the relevant notion of "support" is relational and inherently nonlocal: it typically comprises both the points at which fields are evaluated and the spacetime regions traversed by the dressing (Marolf, 2015, Hoehn et al., 18 Dec 2025).

At the classical level, relationally localized dressings can still ensure vanishing Poisson brackets when their supports are spacelike. For example, single-integral ("relational") observables in gravity can be constructed so that their classical brackets vanish whenever the regions sampled by their dressing functions do not overlap causally (Marolf, 2015). In QED with suitable nonlocal boundary conditions, relational locality can also guarantee microcausality for boundary-dressed operators (Hoehn et al., 18 Dec 2025).

Upon quantization, the situation becomes more delicate. While at each finite order in perturbation theory ($\hbar$ expansion), commutators of spacelike-supported, relationally localized observables vanish—as follows from the same functional support arguments—in general, nonperturbative contributions can generate nonzero, albeit exponentially suppressed, commutators (Marolf, 2015). Furthermore, leading-order (in $G$ or $q$) commutators between two dressed operators at spacelike separation are nonzero, reflecting the overlap of their nonlocal dressings (Boulle et al., 18 Dec 2025).

3. Explicit Construction and Magnitude of Dressings-Induced Commutators

Expanding on the linearized gravitational case, the commutator between two gravitationally dressed fields $\Phi(x)$ and $\Phi(y)$ at equal times and spacelike separation acquires a parametric form

$$[\Phi(x), \Phi(y)] = -\frac{i\kappa^2\hbar}{64\pi c^4}\left[\dot\phi(x)\partial_i\phi(y) + \partial_i\phi(x)\dot\phi(y)\right]\frac{x^i-y^i}{|x-y|} + \mathcal{O}(\kappa^3)$$

with $\kappa^2=32\pi G$ (Boulle et al., 18 Dec 2025). This nonvanishing result is rooted in the nonlocality of the gravitational dressing, specifically, the fact that the "dressing strings" associated with each operator stretch to spatial infinity and physically overlap even for spacelike $x$ and $y$.

Parametric estimates in practically relevant regimes (such as BMV-type quantum-gravity entanglement tests, involving masses $m\sim 10^{-14}$ kg, localizations $L\sim 10^{-6}$ m) yield

$$\frac{|[\Phi(x), \Phi(y)]|}{|\phi(x)\phi(y)|} \sim \frac{G m}{c^2 L} \sim 10^{-36}$$

with time-derivative commutators displaying rates of $\sim 10$ s$^{-1}$ in the rest-frame phase convention, or $\sim 10^{-44}$ s$^{-1}$ if kinetic frequencies dominate, highlighting the utter numerical suppression but qualitative universality of these violations (Boulle et al., 18 Dec 2025).

Quantum electrodynamics in curved spacetime presents a related, yet physically distinct, form of apparent microcausality violation manifesting as superluminal low-frequency phase velocities and local amplification of the photon field. Both effects derive from the tidal distortion of the electron-positron cloud dressing the photon and the resultant modification of the refractive index. The apparent superluminality at low frequency, $v_{\text{phase}}(0)>1$, and local field amplification, $|\mathcal{A}(u)|$ increasing in some domains, both superficially violate causality/unitarity but are resolved via full QFT analysis: the high-frequency wavefront speed remains luminal and the integrated probability flux complies with global constraints (Hollowood et al., 2012).

4. Relational Locality, Boundary Conditions, and Recovery of Microcausality

In abelian gauge theories, nontrivial global charges and their associated dressings extend to infinity or the boundary, making strictly local characterization of observables impossible under standard boundary conditions. This challenge can be addressed by imposing suitably nonlocal boundary conditions on the large gauge sector and adopting a relational notion of locality (Hoehn et al., 18 Dec 2025).

For instance, by imposing "Dirichlet up to large gauge" conditions on the asymptotic boundary, and requiring that the boundary phase $\alpha(y)$ (constructed from dressed fields at the bulk-boundary interface) satisfy boundary field equations, one can reconstruct a subsystem structure where charged, boundary-dressed observables commute at spacelike separation on the relevant boundary surface. The construction ensures that the net of local algebras, $\mathcal{A}_U$, defined over spacetime regions $U$, preserves locality in terms of relational support, provided all such relational supports remain spacelike. Nonetheless, such recovery of microcausality is intrinsically frame-dependent: each choice of boundary-gauge frame induces a different causal structure for the algebra of observables, the identification of vacuum, and the localization of operators (Hoehn et al., 18 Dec 2025).

5. Operational and Algebraic Consequences for Subsystem Independence

The existence of dressing-induced noncommutativity has immediate implications for quantum information applications, especially protocols designed to probe the quantum nature of gravity via entanglement between spatially separated subsystems (e.g., the BMV and broader GIE protocols). Standard analyses assume exact subsystem independence, enforced by strict commutation of observables associated to each laboratory or mass, which is no longer axiomatically viable once gauge-invariant dressings are incorporated (Boulle et al., 18 Dec 2025).

The nonvanishing $[ \Phi(x), \Phi(y) ]$ at spacelike $x, y$ undermines the strict split property, Hilbert space factorization, and prepares states or measurements genuinely independent of other regions. While the scale of the violation is negligible for practical purposes, conceptually it mandates a reassessment of such protocols: claimed inferences regarding the quantum status of gravity rest on an approximate, perturbative, and subsystem-effective independence valid only when dressing effects are suppressed. Conversely, the magnitude of these commutators themselves could, in principle, become experimental probes: robust upper bounds on the noncommutativity parameter $\epsilon$ would directly constrain the nonlocal algebraic structure required by diffeomorphism or gauge invariance (Boulle et al., 18 Dec 2025).

6. Nonperturbative Effects, Limitations, and Theoretical Outlook

At the perturbative level (finite orders in $\hbar$ or $G$), microcausality can be maintained to arbitrarily high accuracy. However, nonperturbative quantum-gravitational effects—for example, Poincaré recurrences, finite Bekenstein-Hawking entropy, or global gauge anomalies—can induce corrections to commutators suppressed by $e^{-S/\hbar}$ or similar exponents of the entropy/action (Marolf, 2015). In spacetimes with infinite state density (asymptotically flat), such recurrences may not arise, although they are expected for black holes in AdS or de Sitter backgrounds.

Ultraviolet singularity structure is controlled by standard locally covariant renormalization procedures, which do not spoil causal factorization. The key open technical assumption remains the full justification that renormalization and dressing constructions can always be implemented in manifestly covariant gauges such that the functional-support arguments for spacelike commutativity persist—this issue is settled in Yang-Mills (by BRST), but requires further work in the nonlinear gravity case (Marolf, 2015).

Infrared features, including boundary-relational modifications and quasi-periodicity in AdS/dS, necessitate small causal modifications to dressings, inducing exponentially small (but nonzero) commutators. This entangles the algebraic structure of local observables with global properties (such as the system's entropy or recurrence time), implying that the classical ideal of strict, absolute locality can only be retrieved as an approximation within a specific dynamical and boundary-gauge context (Marolf, 2015, Hoehn et al., 18 Dec 2025).

7. Comparison Across Gauge and Gravitational Theories

Dressing-induced microcausality violations manifest generically wherever Gauss-law-type constraints require observables to encode their nonlocal field content. In QED, the adoption of nontrivial boundary conditions and relational localization enables the partial restoration of microcausal behavior within a given frame, at the cost of making locality itself frame-dependent (Hoehn et al., 18 Dec 2025). In gravity, relational observables provide a systematic construction with exact microcausality at the classical level, approximate microcausality within the perturbative regime, and only minuscule but principled violations induced by quantum gravity effects (Marolf, 2015).

In both settings, the violation of naive, pointwise locality arises not from acausal signal propagation but as a consequence of nonlocal constraint structure—dressing operators imprint information about the global configuration of gauge and gravitational fields, thereby tying local measurements to extended geometric and topological data. The analytic structure of these commutators, as well as their magnitude and operational significance, depend crucially on the chosen gauge frame, boundary and asymptotic conditions, and the physical scale at which violations are probed.

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References:

• "Comments on Microcausality, Chaos, and Gravitational Observables" (Marolf, 2015)
• "Fighting non-locality with non-locality: microcausality and boundary conditions in QED" (Hoehn et al., 18 Dec 2025)
• "The Unbearable Beingness of Light, Dressing and Undressing Photons in Black Hole Spacetimes" (Hollowood et al., 2012)
• "Subsystems (in)dependence in GIE proposals" (Boulle et al., 18 Dec 2025)