Gomes (2021) proved that in electromagnetism coupled to a charged scalar on a simply-connected universe divided into two regions, when charged matter vanishes on the shared interface but not in the bulks, the pair of regional physical states fails to determine a unique global physical state: a residual variety parametrized by U(1) survives, which he identifies with the direct empirical significance (DES) of regional gauge transformations — a theorem-level version of 't Hooft's beam-splitter. For non-Abelian structure groups only an infinitesimal, perturbative characterization is available (the variety's parametrization recovers the Lie algebra of the charge group about homogeneous backgrounds), and the paper states it is not clear how to obtain the finite version. This problem asks for the finite, non-perturbative non-Abelian analogue: a characterization of the set of physically distinct gluings of two regional states — whether it is parametrized by the charge group G, a subgroup or quotient, or a state-dependent structure, with inequivalence certified by gauge-invariant quantities — together with the finite-level completion of the DES classification, namely the status of boundary stabilizers that do not extend into the bulk and the cases where charged matter does not vanish at the interface.

Background

In Newtonian mechanics the empirical significance of a symmetry is familiar from Galileo's ship: a boost changes no event inside the cabin but is registered relationally, in the ship's velocity relative to the shore. Whether local gauge transformations — usually regarded as pure descriptive redundancy of a gauge theory — can carry an analogous relational empirical significance has been debated in philosophy of physics since the 2000s. Brading and Brown (2004) argued that regional gauge transformations must match across the boundary between a subsystem and its environment and hence are never observable; Greaves and Wallace (2014) defended the opposite: a symmetry has direct empirical significance (DES) when a transformation acts as a symmetry of a subsystem while the state of the total system changes in an empirically distinguishable way. The cleanest illustration is 't Hooft's beam-splitter: two arms of an interferometer carrying electron fields can be given independent constant phase shifts, transformations that leave each arm's internal physics unchanged but are visible in the interference pattern.

Gomes (2021) made this precise for Yang–Mills theory. Take a simply-connected universe Σ\Sigma decomposed into two bounded regions Σ+\Sigma_+ and Σ−\Sigma_- meeting on an interface S=∂Σ+=∂Σ−S=\partial\Sigma_+=\partial\Sigma_-. Write [φ][\varphi] for the physical content of a field configuration, i.e. its gauge-equivalence class restricted to the region in question, and [φ+]∪S(i)[φ−][\varphi_+]\cup_S^{(i)}[\varphi_-] for a physically legitimate composition (gluing) of the two regional physical contents. Global supervenience on subsystems (GSS) holds when the pair ([φ+],[φ−])([\varphi_+],[\varphi_-]) determines a unique global physical content; its failure (¬\negGSS) means there is a residual variety of physically distinct universes built from the same regional states, indexed by a set II of gluings. Gomes identifies ¬\negGSS with DES of regional gauge transformations: the variety is encoded in relations between subsystems, not in their intrinsic states — a form of holism.

His main result, Theorem 1, concerns electromagnetism (structure group U(1)U(1)) coupled to a charged Klein–Gordon scalar: if the charged matter vanishes on the interface SS but not in the bulk of the regions, the glued physical state is underdetermined and the residual variety is parametrized by U(1)U(1), realized as regional rigid (constant) phase shifts acting on the matter but not on the gauge potential — the beam-splitter situation derived as a theorem. For a non-Abelian structure group GG the situation differs qualitatively. The analogue runs through stabilizers: regional gauge transformations ξ\xi with Dξ=0D\xi=0 (DD the gauge-covariant derivative), which leave the gauge potential invariant but can still transform the matter. In the Abelian case every configuration has the same stabilizers, namely the constant transformations; for a non-Abelian group, generic configurations have trivial stabilizers, and non-trivial stabilizers exist only on special homogeneous strata of field space (e.g. the vacuum A=0A=0, whose stabilizers are the constant maps into GG). About such backgrounds Gomes recovers only the infinitesimal version: the variety is parametrized by elements of the Lie algebra g\mathfrak{g} (his Eq. A.12), and he states that 'for non-Abelian groups, it is not clear how to obtain the finite version', i.e. a parametrization by the charge group itself — the group of constant gauge transformations carrying the regional charges, isomorphic to the structure group GG (his Eq. A.13, I≃GI\simeq G). The integration tools that work in the Abelian case — Coulomb-type gauge fixings and dressings (gauge-covariant constructions that attach a charged field to a reference structure via a Wilson line — the path-ordered exponential of the gauge potential along a curve — so as to build gauge-invariant variables) — do not globalize: as the same author records in a companion preprint, the obstruction is tied to the Gribov problem, i.e. the absence of any global gauge-fixing section for non-Abelian theories (Gribov ambiguities). Related results in the edge-modes program — the study of the extra boundary degrees of freedom that appear when a gauge theory is restricted to a finite region (Gomes, Hopfmüller and Riello 2019; Gomes and Riello 2021) — split regional degrees of freedom into radiative and Coulombic parts and prove, via a field-space connection (a connection on the space of field configurations, used to compare fields across the region boundary), that regional radiative data determine the glued state uniquely whenever stabilizers are absent, and give a connection-dependent notion of non-Abelian gluing; but they do not deliver a canonical, group-parametrized residual variety, which is exactly what the identification of DES requires. The theorem-level non-Abelian analogue of Theorem 1, and with it the completion of the classification of when regional gauge transformations have DES, is what remains open.

Problem Statement

For Yang–Mills theory with compact non-Abelian structure group GG, coupled to charged matter, on a simply-connected manifold Σ=Σ+∪Σ−\Sigma=\Sigma_+\cup\Sigma_- with regions sharing the interface S=∂Σ+=∂Σ−S=\partial\Sigma_+=\partial\Sigma_- (under the source's idealizations: Euclidean signature, closed Σ\Sigma, configurations admitting stabilizers such as homogeneous backgrounds): obtain the finite, non-perturbative analogue of the Abelian Theorem 1 — a characterization of the residual gluing variety

[φ(i)]=[φ+]∪S(i)[φ−],i∈I,[φ(i)]≠[φ(i′)] iff i≠i′.[\varphi(i)] = [\varphi_+] \cup_S^{(i)} [\varphi_-], \qquad i \in I, \qquad [\varphi(i)] \neq [\varphi(i')]\ \text{iff}\ i \neq i'.

Specifically: (1) determine the index set II for the non-Abelian theory — whether it is parametrized by the charge group GG itself (the finite counterpart of the U(1)U(1) theorem and of the infinitesimal Eq. A.12), by a subgroup or quotient of GG, or by some state-dependent structure — and prove the inequivalence (or collapse) of the resulting glued states using gauge-invariant quantities; (2) settle, within the same classification and at the finite, non-perturbative level — the source treats both (2) and (3) only perturbatively for non-Abelian groups — the status of boundary stabilizers, i.e. transformations fixing the fields on SS that do not extend to stabilizers of the bulk: do they label physically distinct gluings, and can they be associated with DES or must they be given another interpretation; and (3) settle the boundary-matter cases at the same finite level, classifying which conditions on the charged matter and field data at SS (vanishing, compatible non-vanishing, or partial) leave residual variety. The expected completion is a theorem-level non-Abelian counterpart of Theorem 1 together with the resulting classification of the sources of regional physical variety; extensions to Lorentzian signature, non-simply-connected topology, and asymptotic regimes are further directions of the source and are not part of this problem's completion boundary.

References

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  2. Gomes, H. (2021). Noether charges: the link between empirical significance of symmetries and non-separability. arXiv:2110.07208. https://arxiv.org/abs/2110.07208
  3. Gomes, H. (2021). The role of representational conventions in assessing the empirical significance of symmetries. arXiv:2110.14563. https://arxiv.org/abs/2110.14563
  4. Gomes, H., Hopfmüller, F., & Riello, A. (2019). A unified geometric framework for boundary charges and dressings: non-Abelian theory and matter. Nuclear Physics B, 941, 249–315. DOI: 10.1016/j.nuclphysb.2019.02.020. https://doi.org/10.1016/j.nuclphysb.2019.02.020
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  7. Borsboom, S. G. A. (2026). Boundaries in the instantaneous formulation of field theories. arXiv:2606.31265. https://arxiv.org/abs/2606.31265
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  9. Gomes, H. (2026). Ghosts that connect. arXiv:2607.27297. https://arxiv.org/abs/2607.27297
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