Extended Aharonov-Bohm Electrodynamics

• Extended Aharonov-Bohm electrodynamics is a framework that treats the four-divergence of the electromagnetic potential as a physical scalar mode, allowing the theory to incorporate non-locally conserved currents.
• The formulation introduces a non-local compensating current that reconciles anomalous charge distributions while preserving an effective local conservation law in the observable fields.
• The theory connects Maxwellian electromagnetism to topics such as fractional quantum mechanics, anomalous media, superconductivity, and even links to the Bohm quantum potential via bosonic Schrödinger models.

Extended Aharonov-Bohm electrodynamics is a reduced-gauge extension of Maxwell theory in which the four-divergence of the electromagnetic four-potential,

$S \equiv \partial_\mu A^\mu ,$

is treated as a genuine dynamical scalar mode rather than as a removable gauge quantity. In this framework the electromagnetic field can be coupled to currents for which local conservation fails,

$\partial_\mu j^\mu = I \neq 0,$

while the observable field can still be written as generated by a conserved effective source through the introduction of a non-local compensating current. The theory has been developed in covariant form, applied to fractional quantum mechanics and coherent tunnelling, extended to anomalous media and bosonic Schrödinger models, and used as a basis for phenomenological constructions involving superconductors, scalar-tensor gravity, and rotating observer splittings (Modanese, 2017, Modanese, 2017, Minotti et al., 2023, Pullano et al., 9 May 2026, Iadicicco et al., 24 Apr 2026, Minotti et al., 27 Mar 2026).

1. Formal structure of the theory

In standard Maxwell electrodynamics, the field equation

$\partial_\mu F^{\mu \nu}=j^\nu$

requires local current conservation,

$\partial_\mu j^\mu = 0,$

because the field tensor $F^{\mu\nu}$ is antisymmetric. Extended Aharonov-Bohm electrodynamics relaxes this restriction by modifying the potential-based theory so that non-locally conserved currents can be admitted consistently (Modanese, 2017, Minotti et al., 2023).

A central formulation uses the Aharonov-Bohm Lagrangian

$$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$

where $j^\mu$ is not required to be locally conserved and $\lambda$ is an arbitrary parameter. Variation yields

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$

and taking the divergence gives

$$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$

For $\partial_\mu j^\mu = I \neq 0,$0, described as the most natural choice for locality, the equations simplify to

$\partial_\mu j^\mu = I \neq 0,$1

but the Lorenz condition $\partial_\mu j^\mu = I \neq 0,$2 can no longer be imposed if $\partial_\mu j^\mu = I \neq 0,$3. The consequence is that $\partial_\mu j^\mu = I \neq 0,$4 becomes a physical degree of freedom, and gauge invariance survives only in reduced form, through transformations $\partial_\mu j^\mu = I \neq 0,$5 with $\partial_\mu j^\mu = I \neq 0,$6 (Minotti et al., 2023).

An equivalent covariant rewriting emphasizes the observable source. The field equations can be cast as

$\partial_\mu j^\mu = I \neq 0,$7

with $\partial_\mu j^\mu = I \neq 0,$8 the inverse d'Alembertian. The combination $\partial_\mu j^\mu = I \neq 0,$9 is always locally conserved: $\partial_\mu F^{\mu \nu}=j^\nu$0 If $\partial_\mu F^{\mu \nu}=j^\nu$1 is conserved, then $\partial_\mu F^{\mu \nu}=j^\nu$2 and Maxwell electrodynamics is recovered (Modanese, 2017).

2. Scalar sector, effective sources, and the “censorship” mechanism

The most characteristic structural feature of extended Aharonov-Bohm electrodynamics is the coexistence of a non-conserved primary source and a non-local compensating sector. In the source-based form,

$\partial_\mu F^{\mu \nu}=j^\nu$3

the secondary current is minus the gradient of a retarded potential sourced by the non-conservation of the primary current. Because the measurable field depends only on $\partial_\mu F^{\mu \nu}=j^\nu$4, local charge non-conservation is “censored” or compensated by the secondary current: electric and magnetic field measurements cannot distinguish the presence of a non-conserved current in isolation (Modanese, 2017).

This censorship property does not trivialize the theory. The secondary current is spatially extended rather than localized where the primary current is supported, so the effective source can have non-trivial spatial structure. In this sense, the theory preserves total local conservation of the effective source while allowing microscopic or mesoscopic source anomalies to have real electromagnetic consequences (Modanese, 2017, Modanese, 2017).

The scalar mode also changes the status of the potentials. In Maxwell theory, pure gauge degrees of freedom are not measurable. In the extended theory, the potentials $\partial_\mu F^{\mu \nu}=j^\nu$5 become directly measurable by probes for which the probe extra-current $\partial_\mu F^{\mu \nu}=j^\nu$6 is nonzero. The Lorentz force density on such a probe is

$\partial_\mu F^{\mu \nu}=j^\nu$7

or, in vector notation,

$\partial_\mu F^{\mu \nu}=j^\nu$8

This is the basis for the statement that the potentials become physical observables in the presence of anomalous matter sectors (Minotti et al., 2023).

A common misconception is that admitting $\partial_\mu F^{\mu \nu}=j^\nu$9 immediately implies arbitrary violation of global charge conservation. The literature does not make that claim. Rather, the framework is described as compatible with currents conserved globally but not locally, and it is explicitly stated not to apply where total global charge is not conserved (Modanese, 2017).

3. Bosonic Schrödinger models and the link to the Bohm quantum potential

A recent development analyzes a formal and physical connection between the Bohm quantum potential and the scalar mode $\partial_\mu j^\mu = 0,$0 in an effective non-relativistic bosonic model proposed by Minotti and Modanese (Pullano et al., 9 May 2026). In the Madelung representation,

$\partial_\mu j^\mu = 0,$1

the Bohm quantum potential is

$\partial_\mu j^\mu = 0,$2

so $\partial_\mu j^\mu = 0,$3 is determined by the relative curvature $\partial_\mu j^\mu = 0,$4 of the amplitude profile $\partial_\mu j^\mu = 0,$5 (Pullano et al., 9 May 2026).

In the same bosonic model, the scalar electromagnetic mode is sourced by the extra-current

$\partial_\mu j^\mu = 0,$6

Thus the source for $\partial_\mu j^\mu = 0,$7 is the spatial divergence of the density-weighted vector potential. The paper’s key clarification is that there is no direct source relation $\partial_\mu j^\mu = 0,$8. Instead, both quantities are functionals of the same amplitude variable $\partial_\mu j^\mu = 0,$9, but with different sensitivities: $F^{\mu\nu}$0 is sensitive to the relative curvature of $F^{\mu\nu}$1, whereas the source of $F^{\mu\nu}$2 is sensitive to the density profile $F^{\mu\nu}$3 and its gradient content through $F^{\mu\nu}$4 (Pullano et al., 9 May 2026).

Given appropriate boundary and normalization conditions, the differential equation for $F^{\mu\nu}$5 can be inverted to write $F^{\mu\nu}$6, leading to a mediated functional dependence

$F^{\mu\nu}$7

The relation is not one-to-one: constant rescalings of $F^{\mu\nu}$8 leave $F^{\mu\nu}$9 invariant but not $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$0. Using $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$1, the same structure becomes

$$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$2

$$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$3

which makes explicit that $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$4 depends on the absolute normalization and first-derivative structure of the amplitude, not only on its relative curvature (Pullano et al., 9 May 2026).

The same work gives a precise energetic interpretation of $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$5. Although $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$6 is state-dependent and should not be interpreted as an autonomous external potential, its density-weighted integral satisfies

$$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$7

This identifies $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$8 as a compact diagnostic of quantum pressure, rigidity, and inhomogeneity, equivalently a Fisher-information contribution. The proposed relation between $$L_{A.-B.} = \frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - \frac{\lambda}{2\mu_0} (\partial_\mu A^\mu)^2 + A_\mu j^\mu ,$$9 and $j^\mu$0 is therefore structural and mediated by the condensate amplitude profile rather than directly causal (Pullano et al., 9 May 2026).

4. Wave propagation, anomalous media, and fluctuations

When extended Aharonov-Bohm electrodynamics is applied to explicitly non-locally conserved quantum sources, the resulting radiative structure differs from Maxwellian expectations. For a special dipolar source motivated by fractional quantum mechanics, propagating fields may have non-transverse components, and the distinction between near-field zone and wave zone is blurred. The reason given is that the secondary current is spatially extended and contributes at far distances, so the radiated field is computed as if both the primary source and its retarded secondary current act as sources (Modanese, 2017).

The theory also admits “gauge waves,” defined as plane-wave solutions for the potentials,

$j^\mu$1

These satisfy $j^\mu$2 and produce no electric or magnetic field,

$j^\mu$3

In Maxwell theory such configurations would be regarded as non-physical, but in extended Aharonov-Bohm electrodynamics they can, in principle, be detected through the $j^\mu$4 coupling to anomalous probes (Minotti et al., 2023).

A phenomenological $j^\mu$5-model for anomalous media is introduced through

$j^\mu$6

Within this model, transverse electromagnetic waves propagate as usual with slightly modified phase velocities, while a longitudinal gauge-wave branch emerges, corresponding to pure potentials with zero fields and accessible only via anomalous probes. The modified energy density is

$j^\mu$7

and the power density transferred to matter is

$j^\mu$8

A proposed detector is a DC circuit with an anomalous segment, for which an incident gauge wave produces

$j^\mu$9

A linear dipole antenna made from material with non-zero $\lambda$0 is also proposed as a possible detector (Minotti et al., 2023).

Thermal fluctuation theory has been extended to the same framework. For non-conserved charges at thermal equilibrium, the spectral distribution of electromagnetic energy remains the same as in Maxwell theory, but the electric contribution doubles while the magnetic contribution remains the Maxwell one; the electric excess is compensated by a negative contribution from the scalar field $\lambda$1. The total equilibrium spectrum still satisfies Planck’s law,

$\lambda$2

For a conductor described by the $\lambda$3-model, the first-order current correlation spectrum contains a violet-noise contribution in addition to the classical Johnson-Nyquist white noise result, and the low-frequency voltage fluctuation spectrum becomes

$\lambda$4

These results preserve the standard total equilibrium spectrum while predicting a modified partition of field energy and a specific high-frequency correction to thermal voltage noise (Minotti et al., 17 Apr 2026).

5. Related Aharonov-Bohm phase phenomena and quantum-field interpretations

Work on the broader Aharonov-Bohm effect provides an adjacent conceptual setting for the extended electrodynamics, especially because both bodies of literature emphasize potentials, topology, and nonlocal structure. A proposed electrodynamic Aharonov-Bohm scheme shows that a nonzero phase difference can appear even if interferometer paths do not enclose magnetic flux and experience negligible scalar potential differences during propagation. In that setup a time-varying solenoid current acts while the particle is inside Faraday cages, so the particle is subjected to negligible electromagnetic fields, yet the total phase contains a nonzero term coming from the change in vector potential during the particle’s stay in the cages (Saldanha, 2023).

A more general spacetime-surface formula writes the phase as

$\lambda$5

In the topological schemes developed for the electrodynamic Aharonov-Bohm effect, the phase depends on the topology of the electric and magnetic fields in spacetime rather than on local field values along the particle path. Concrete schemes include a no-wire case with

$\lambda$6

wire-connected cages yielding $\lambda$7 or $\lambda$8 depending on which side the wire is attached, and an external-solenoid arrangement with a wire wound $\lambda$9 times giving

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$0

These are presented as topological consequences of the spacetime connectivity of field-free regions (Saldanha et al., 2024).

Quantum-electrodynamic treatments in the Lorenz gauge provide a local, gauge-independent account of phase generation in which virtual longitudinal and scalar photons mediate the interaction. In this framework the relevant energy shift is path-dependent,

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$1

with

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$2

The resulting phase difference is gauge-invariant for closed paths, but may depend on the gauge for nonclosed paths, reinforcing the statement that it can be measured only in closed paths (Saldanha, 2019, Saldanha, 2024).

At the algebraic quantum field theory level, the Aharonov-Bohm effect has also been described in terms of superselection sectors on globally hyperbolic spacetimes. In that description, the topological quantum number is the holonomy representation of the fundamental group,

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$3

and, in the Abelian case,

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$4

with $$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$5 a flat background potential. This identifies the Aharonov-Bohm phase with the label of a topological superselection sector (Dappiaggi et al., 2019).

6. Rotating-frame closures, gravity couplings, and superconducting applications

Several recent works extend the formalism beyond flat-space source theory, but they do so with different status. In rotating frames and stationary gravitomagnetic backgrounds, a no-go result holds at the microscopic level for standard generally covariant, locally $$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$6-invariant matter: the physical four-current remains covariantly conserved, so neither rotation nor stationary gravitomagnetism by themselves generate a genuine source for the scalar sector. After a $$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$7 decomposition with respect to a rotating observer congruence, however, the observer-measured transport variables satisfy

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$8

with exact split source

$$\partial^2 A^\mu + (\lambda-1)\partial^\mu \partial_\nu A^\nu = \mu_0 j^\mu ,$$9

In the weak-field rigid-rotation limit,

$$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$0

and for localized non-axisymmetric transients,

$$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$1

The resulting AB-type closure,

$$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$2

is explicitly described as effective rather than fundamental, observer-tied rather than local-inertial, and experimentally meaningful only at mesoscopic or macroscopic scales (Iadicicco et al., 24 Apr 2026).

Coupling the theory to scalar-tensor gravity produces a different extension. Because the extended electromagnetic sector has a nonvanishing trace, unlike Maxwell theory, scalar gravitational fields can couple directly to the electromagnetic sector. In one tensor-scalar construction the total action includes the Brans-Dicke scalar $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$3, an additional scalar $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$4, and the electromagnetic term

$$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$5

while the scalar source $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$6 contains contributions from $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$7 and $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$8. The same framework emphasizes that the principal phenomenological parameters are the vacuum expectation value $$\partial^2(\partial_\nu A^\nu)=\frac{\mu_0}{\lambda}I,\qquad I\equiv \partial_\mu j^\mu .$$9 of the second gravitational scalar and the local non-conservation level $\partial_\mu j^\mu = I \neq 0,$00, and states that there is presently much space left for uncertainty (Minotti et al., 2024).

For bosonic matter described by a macroscopic wavefunction, especially superconductors, the coupling to the electromagnetic potential generates $\partial_\mu j^\mu = I \neq 0,$01 already at the semiclassical level. In this setting the source equation reduces to

$\partial_\mu j^\mu = I \neq 0,$02

and at a normal-superconducting junction the scale is given as

$\partial_\mu j^\mu = I \neq 0,$03

When the scalar-tensor sector is included, a nonlinear bulk system for $\partial_\mu j^\mu = I \neq 0,$04 and a gravitational scalar combination $\partial_\mu j^\mu = I \neq 0,$05 admits the saturation solution

$\partial_\mu j^\mu = I \neq 0,$06

This produces a threshold condition for macroscopic effects and is used to derive scaling relations for pulsed discharges across normal-superconducting junctions, reported as consistent with threshold behavior in two independent experimental configurations for a single microscopic parameter. The same work also presents time-dependent propagating solutions in the weak-field regime and a class of one-dimensional traveling exact solutions of the nonlinear vacuum Einstein equations (Minotti et al., 27 Mar 2026).

These developments sharpen an important distinction within the literature. In the source-based electrodynamics of non-conserved currents, the scalar sector is part of the fundamental field equations. In the rotating-frame closure, by contrast, the source is a bookkeeping term arising from observer-adapted transport variables. In the gravity-coupled superconducting models, the phenomenology depends on additional scalar fields and on parameters whose values are not fixed by the theory.