Semi-local Quantum Electromagnetism

• Semi-local quantum electromagnetism is a framework where electromagnetic fields are extended beyond point-local interactions, incorporating topological and boundary effects.
• It accounts for non-conserved quantum sources and emergent field variables by integrating retarded propagators, gauge constraints, and effective multipolar couplings.
• The approach applies to Aharonov–Bohm phenomena, lattice gauge theories, and condensed matter systems, offering fresh perspectives beyond standard Maxwell theory.

Semi-local quantum electromagnetism is a family of formulations in which electromagnetic phenomena are neither treated as strictly point-local in the standard Maxwell–QED sense nor as unconstrained action-at-a-distance. Across the cited literature, the expression is used for several related situations: electromagnetic coupling to quantum sources with $\partial_\mu J^\mu\neq 0$; local, gauge-independent accounts of the Aharonov–Bohm effect whose observable content is topological or branch-relative; algebraic formulations in which boundary fluxes, edge modes, or unbounded spacelike regions carry physical information; and effective or emergent descriptions in which the relevant electromagnetic degrees of freedom are localized only up to multipole, lattice, or medium-response structure (Modanese, 2017, Saldanha, 2019, Sanders et al., 2012, Fewster et al., 28 Aug 2025). The recurring theme is not the abandonment of causality or structure, but a weakening of strict locality that remains controlled by gauge constraints, retarded propagators, topology, or effective coarse graining.

1. Terminological scope and recurring structures

The literature does not use “semi-local quantum electromagnetism” for a single canonical formalism. Rather, the term covers several technically distinct constructions that share a common feature: observables, sources, or field variables acquire spatially extended, topological, or boundary-sensitive content that cannot be reduced to strictly local bulk data.

Setting	Semi-local feature	Representative paper
Nonconserved quantum sources	Secondary nonlocal charge and current built from $I=\partial_t\rho+\nabla\!\cdot\mathbf J$	(Modanese, 2017)
Aharonov–Bohm phase generation	Local field interaction on each branch, global relative phase from enclosed flux or potential history	(Saldanha, 2019)
Generally covariant/algebraic electromagnetism	Gauss-law observables and Aharonov–Bohm-sensitive gauge classes produce controlled non-locality	(Sanders et al., 2012)
Boundary/corner AQFT	Surface sector, edge modes, and large gauge transformations enlarge the observable algebra	(Fewster et al., 28 Aug 2025)
Emergent and effective realizations	Lattice, medium, or multipolar variables replace strictly point-local fields	(Benton et al., 2012)

In the source-extension literature, semi-locality appears through retarded nonlocal corrections to Gauss’s law and Ampère’s law. In Aharonov–Bohm analyses, it appears because each branch of a wavefunction acquires phase locally while the measurable phase difference depends on a global flux or potential history. In algebraic formulations, it appears because charge observables, electric fluxes, and large gauge transformations are not exhausted by compactly supported interior fields. In condensed-matter and multipolar settings, it appears because the operationally relevant electromagnetic variables are emergent lattice gauge fields, non-local response kernels, or polarization and magnetization densities rather than strictly local microscopic currents (Modanese, 2017, Saldanha, 2019, Fewster et al., 28 Aug 2025, Kattan et al., 2022).

A common misconception is that all such uses refer to the same nonlocal modification of Maxwell theory. The cited work does not support that identification. Some papers alter the field equations, some retain ordinary Maxwell theory but enlarge the observable content, and some reinterpret locality through quantized mediators, boundary sectors, or effective field variables.

2. Nonconserved quantum sources and extended Maxwell structure

A central use of semi-local quantum electromagnetism arises when the quantum source is not locally conserved. For ordinary Schrödinger dynamics,

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$

and minimal coupling is consistent. The nonlocal and fractional extensions discussed in the literature instead satisfy

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$

or, for the fractional case,

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$

Because the standard Maxwell equation

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$

implies $\partial_\nu J^\nu=0$ by antisymmetry of $F^{\mu\nu}$, ordinary minimal coupling fails precisely when $I\neq0$ (Modanese, 2017).

The replacement used in the extended Aharonov–Bohm electrodynamics of that paper is

$$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$

with $I=\partial_t\rho+\nabla\!\cdot\mathbf J$0 the retarded inverse d’Alembertian. In three-vector form, the homogeneous equations remain unchanged, while the inhomogeneous equations become

$I=\partial_t\rho+\nabla\!\cdot\mathbf J$1

$I=\partial_t\rho+\nabla\!\cdot\mathbf J$2

Equivalently,

$I=\partial_t\rho+\nabla\!\cdot\mathbf J$3

The extra terms are described as a secondary charge density and a secondary current. They are nonlocal, spread via the retarded Green’s function, and lead to the “censorship” effect emphasized in the paper: a microscopic source can violate local conservation while the measured field may still resemble the field of an apparently conserved source (Modanese, 2017).

This source structure makes the analysis technically harder because the potentials involve a double retarded integral. That feature is central in later numerical studies. In the high-frequency dipole calculation for a non-local source with

$I=\partial_t\rho+\nabla\!\cdot\mathbf J$4

the reported parameters are $I=\partial_t\rho+\nabla\!\cdot\mathbf J$5, $I=\partial_t\rho+\nabla\!\cdot\mathbf J$6, and $I=\partial_t\rho+\nabla\!\cdot\mathbf J$7. For observation at $I=\partial_t\rho+\nabla\!\cdot\mathbf J$8 to the dipole axis, the longitudinal and transverse components are defined by

$I=\partial_t\rho+\nabla\!\cdot\mathbf J$9

and the longitudinal field is reported to be typically $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$0–$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$1 times larger than the standard transverse benchmark (Modanese, 2019). The same work interprets the effect as radiation from a distributed “cloud” of secondary source terms, so that $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$2 can hold in vacuum.

A quasi-static counterpart was proposed for Josephson tunnelling in YBCO. There the anomalous source is modeled by a current that disappears on one side of a barrier and reappears on the other, producing an extra-source $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$3 with sink and source separated by $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$4. The numerical result for the idealized thin-wire geometry is not a large additional magnetic field but a nearly exact cancellation, $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$5, interpreted as a missing-field phenomenon generated by the compensating secondary current cloud. The paper proposes differential detection with three parallel wires and short current pulses $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$6, and estimates a macroscopic field difference of order $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$7 relative to a normal material if roughly $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$8 of the current in roughly $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=0,\qquad \rho=|\Psi|^2,\qquad \mathbf J=\frac{-i\hbar}{2m}\left(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*\right),$$9 of the sample volume is affected (Modanese, 2018).

These extended-source theories stand in explicit tension with formulations that retain a strictly local spin-1 exchange picture. Under the assumptions of a massless spin-1 field with only two physical polarizations and a tree-level exchange action free of inverse spatial Laplacians, locality forces a conserved visible-sector current,

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$0

and an associated internal $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$1 symmetry (Hertzberg et al., 2020). The contrast is structural: one line of work preserves a mild notion of locality and thereby recovers ordinary electromagnetic coupling, while the other admits a non-local extension of the source sector.

3. Aharonov–Bohm locality, Gauss law, and noncompact localization

The Aharonov–Bohm effect is a second major setting in which semi-local quantum electromagnetism is articulated. In the Lorentz-gauge QED treatment, the effect is described as a fully local process mediated by exchange of virtual photons between the charged particle and the solenoid or electrodes. For the magnetic case, the couplings are

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$2

and the second-order vacuum-energy shift is

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$3

After the polarization sum one obtains

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$4

and therefore

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$5

The paper’s point is that the phase is generated locally, continuously, and path by path, even though the relative phase is topological. The use of Lorentz gauge with virtual longitudinal photons makes the treatment manifestly gauge-independent and valid for arbitrary interferometer geometries, not only circular ones. The electric Aharonov–Bohm effect is treated analogously through virtual scalar photons and

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$6

That work explicitly describes the effect as a semi-local quantum electromagnetic phenomenon: each branch couples locally to the field, while the interference compares the branches globally (Saldanha, 2019).

In algebraic and generally covariant formulations, the same phenomenon is expressed differently. The generally covariant quantization of the vector potential on globally hyperbolic spacetimes treats $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$7 as a $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$8-connection and argues that the physically correct gauge equivalence is by exact forms rather than all closed forms, because potentials differing by closed but non-exact forms can be distinguished by Aharonov–Bohm phase shifts. The Peierls bracket on local affine observables is degenerate, and the resulting non-local behavior is traced not to pathology but to Gauss’ law. The theory is therefore described as generally covariant but not strictly locally covariant in the Brunetti–Fredenhagen–Verch sense; the failure of locality is presented as physically meaningful because it encodes electric charge and flux observables (Sanders et al., 2012).

An asymptotic QED model sharpens the localization issue further. There the net of algebras is localized not in bounded double cones but in regions with unbounded spacelike extension, such as fattened time-slices and symmetrical spacelike cones. Electromagnetic fields can be localized in arbitrarily thin fattened symmetrical spacelike cones, but dressed charged fields cannot be localized in any fixed symmetrical spacelike cone because the Coulomb tail extends in all spatial directions. Only charge-neutral composites, such as products of appropriately dressed fermion fields with compensating charges, become local or bi-localized observables (Herdegen et al., 2011).

Taken together, these works reject the common claim that the Aharonov–Bohm effect demonstrates a simple nonlocal influence of the potential. The stronger statement supported by the cited papers is more differentiated: the phase can be generated by local interaction vertices, but the gauge equivalence classes, flux observables, and charged sectors retain irreducibly global or semi-local content.

4. Boundary sectors, edge modes, and algebraic semi-local observables

A modern algebraic formulation makes semi-locality explicit on spacetimes with boundary and corner. On a finite Cauchy lens $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I,$$9, a compact Lorentzian spacetime region whose spacelike boundary splits into $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$0 meeting at a codimension-2 corner $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$1, the physically relevant observables are not exhausted by compactly supported interior fields. Electric fluxes, Wilson-line-like observables ending on the boundary, and edge modes survive reduction by gauge transformations and require a larger algebra (Fewster et al., 28 Aug 2025).

For the classical field $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$2, with action

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$3

the presymplectic form on a regular Cauchy surface $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$4 is

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$5

Its radical is

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$6

so only gauge transformations vanishing at the corner are degenerate directions. The reduced phase space

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$7

still contains nontrivial large gauge directions generated by exact forms whose boundary values do not vanish.

The initial-data analysis yields the Cauchy–Hodge–Helmholtz decomposition

$$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$8

where $$\frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf J=I_\alpha, \qquad I_\alpha = -i \hbar^{\alpha-1} D_\alpha \left[ \nabla \Psi^*\,(-\nabla^2)^{\alpha/2-1}\nabla\Psi -\text{c.c.} \right].$$9 is the closed loop sector and $\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$0 the surface sector. The full Poisson algebra of semi-local observables is generated by symplectically smeared fields

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$1

and decomposes into commuting bulk and surface subalgebras. Large gauge transformations act trivially on the local bulk algebra but nontrivially on the surface algebra, so the latter encodes edge-mode observables (Fewster et al., 28 Aug 2025).

Quantization proceeds through the Weyl algebra

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$2

with decomposition

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$3

To recover gauge invariance for observables that transform under large gauge transformations, the paper introduces auxiliary boundary degrees of freedom as quantum reference frames. The resulting relativisation map is defined directly on the $\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$4-algebra: $\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$5 This construction is state-independent and turns gauge-covariant bulk observables into gauge-invariant joint observables. The same framework identifies superselection sectors by an external flux function $\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$6 and gives a gluing prescription for two Cauchy lenses with common boundary, with the shared surface Weyl algebra carrying the gluing data (Fewster et al., 28 Aug 2025).

A plausible implication is that semi-locality here is not a defect of the observable algebra but the algebraic form of the statement that boundary flux and large gauge data are physical. That interpretation is consistent with the earlier Gauss-law analyses, although the present boundary formulation is technically more explicit about edge sectors and gauge dressing.

5. Emergent, lattice, and material realizations

Semi-local quantum electromagnetism also appears as an effective description in condensed matter and macroscopic media. In quantum spin ice, the local ice-rule constraint is written as

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$7

and quantum tunnelling between ice configurations, generated in the minimal model by the hexagonal plaquette flip term

$\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$8

produces a deconfined $\partial_\mu F^{\mu\nu}=\frac{4\pi}{c}J^\nu$9 quantum spin liquid. The coarse-grained theory is a lattice Maxwell theory,

$\partial_\nu J^\nu=0$0

with continuum action

$\partial_\nu J^\nu=0$1

The experimentally important predictions are that classical pinch points are progressively “bleached out” and disappear at $\partial_\nu J^\nu=0$2, while linearly dispersing emergent photons appear in inelastic neutron scattering but with vanishing low-energy intensity. The field theory is reported to agree quantitatively with zero-temperature QMC on 2000-site clusters, and at the physical point $\partial_\nu J^\nu=0$3 the estimated emergent light speed is $\partial_\nu J^\nu=0$4; using exchange parameters for Yb$\partial_\nu J^\nu=0$5Ti$\partial_\nu J^\nu=0$6O$\partial_\nu J^\nu=0$7, the paper quotes $\partial_\nu J^\nu=0$8 meV and $\partial_\nu J^\nu=0$9 (Benton et al., 2012).

An analog-gravity variant appears in string-net condensates. There, local tuning of couplings in space and time yields an emergent $F^{\mu\nu}$0 gauge field propagating on an effective curved geometry. In $F^{\mu\nu}$1 dimensions the gauge theory can be written with metric

$F^{\mu\nu}$2

subject to the metricity condition

$F^{\mu\nu}$3

By choosing

$F^{\mu\nu}$4

the emergent photon experiences a Schwarzschild background outside the horizon. The paper explicitly characterizes the construction as only approximately local at finite cutoff: continuum locality is valid in the long-wavelength limit, while the microscopic theory remains lattice-based (Prémont-Schwarz, 2011).

A third material setting is macroscopic QED in non-local and Onsager-violating media. For the most general linear, absorbing medium, the constitutive law is written through the conductivity tensor

$F^{\mu\nu}$5

and the electromagnetic Green tensor solves

$F^{\mu\nu}$6

Quantization is fixed by the noise-current commutator

$F^{\mu\nu}$7

For local media the same framework can be rewritten in terms of $F^{\mu\nu}$8, $F^{\mu\nu}$9, $I\neq0$0, and $I\neq0$1, and a main conclusion is that continuous duality invariance survives only when non-reciprocal responses are allowed (Buhmann et al., 2011).

These realizations show that semi-locality can arise from several different mechanisms: emergent gauge constraints on a lattice, non-local constitutive response in a medium, or continuum approximations to a tunable many-body system. The shared point is that electromagnetic behavior is organized by collective or response kernels rather than by a strictly local microscopic Maxwell field alone.

6. Multipolar, measurement-theoretic, and configuration-space reconstructions

A different use of semi-locality appears in multipolar QED for localized charge-current distributions. Instead of minimal coupling to $I\neq0$2, the theory is rewritten in terms of microscopic polarization and magnetization fields $I\neq0$3 coupled to the physical fields $I\neq0$4. The defining relations are

$I\neq0$5

The passage from minimal coupling is achieved by the Power–Zienau–Woolley transformation

$I\neq0$6

which turns the transverse electric field into the displacement field,

$I\neq0$7

and yields a multipolar Hamiltonian with couplings $I\neq0$8 and $I\neq0$9. The formalism is designed for localized assemblies of atoms and molecules, possibly with net charge but absent free current, and the resulting renormalized energy shift generalizes the Lamb shift to arbitrary multipole order (Kattan et al., 2022).

Another reconstruction derives electromagnetism from operational quantum measurement theory. A generalized von Neumann interaction

$$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$0

produces a warped-convolution deformation of observables, and with a skew matrix encoding a magnetic field the momentum transforms as

$$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$1

so minimal coupling appears as a measurement-induced deformation. In the classical limit, expressed through Rieffel deformation, the first-order term becomes the Poisson bracket

$$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$2

from which the paper recovers Maxwell’s equations (Andersson, 2015). The locality claim here is operational rather than geometric: the field acts as an apparatus-like subsystem whose rapid interaction deforms the measured system.

A more radical reconstruction is the semi-relativistic $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$3-body theory of electrons and photons with fixed nuclei. There, the observed electromagnetic fields are expectation values of $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$4-fields depending on spacetime and full particle configuration,

$$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$5

while the empirical charge and current densities are configuration-space quantities averaged against $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$6. The underlying $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$7-fields satisfy Maxwell-type wave equations on time $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$8 configuration space, and the paper argues that photon creation and annihilation formalisms emerge from recursive source terms without second-quantizing the classical Maxwell field. The microscopic model is not Lorentz covariant, but the paper suggests that Lorentz covariance of macroscopic physics emerges through a law of large numbers (Kiessling, 2020).

A semi-classical reinterpretation of the Aharonov–Bohm effect pushes the same theme in another direction. It assigns wave amplitudes to electrostatic, magnetostatic, and vector-potential configurations, even in regions where the classical fields vanish: $$\partial_\mu F^{\mu \nu} = \frac{4\pi}{c}J^\nu - \frac{4\pi}{c}\partial^\nu\partial^{-2} \bigl(\partial_\gamma J^\gamma\bigr), \qquad I\equiv\partial_\gamma J^\gamma,$$9 These amplitudes act locally on the electron as unitary phases, and the paper uses this viewpoint to discuss the scales $I=\partial_t\rho+\nabla\!\cdot\mathbf J$00 V-s and $I=\partial_t\rho+\nabla\!\cdot\mathbf J$01 Wb in electrostatic phase generation and flux quantization, as well as rational charge quantization $I=\partial_t\rho+\nabla\!\cdot\mathbf J$02 (Bhattacharya, 2021).

The diversity of these reconstructions shows that “semi-local” can refer to several different technical compromises between strict locality and physically adequate electromagnetic description: localization around a center through multipole fields, deformation by an interaction apparatus, dependence on full configuration space, or persistent phase amplitudes in field-free regions. The cited literature does not collapse these into a single ontology, but it consistently treats them as structured alternatives to a purely point-local classical field picture.