First-Order Electromagnetic Operators

• First-Order Electromagnetic Operator Approach is a reformulation of Maxwell’s equations using first-order differential operators to directly encode electromagnetic degrees of freedom.
• It unifies various formulations—like DKP spin-1, Dirac-like, optical, and macroscopic QED—by handling constitutive, gauge, and boundary conditions within an operator framework.
• The approach offers structural simplicity and computational benefits in modeling perturbative, scattering, and boundary effects for diverse electromagnetic systems.

Searching arXiv for recent and relevant uses of “first-order electromagnetic operator” and closely related formulations. “First-Order Electromagnetic Operator Approach” denotes a family of formulations in which electromagnetic dynamics, constitutive response, or source coupling is organized around operators that are first order in derivatives or first order in the relevant perturbation, rather than being written solely as second-order equations for a potential or field component. In the literature, this includes massless Duffin–Kemmer–Petiau spin-1 systems on curved manifolds, Dirac-like matrix representations of Maxwell theory, Hilbert-space and density-operator formulations of classical electromagnetism, first-order Maxwell Green operators for macroscopic quantum electrodynamics, operator-valued constitutive laws that eliminate $\mathbf{D}$ and $\mathbf{H}$, and first-order boundary-operator linearizations for uncertainty quantification (Casana et al., 2016, Li et al., 2023, Oue, 2019, Gratus et al., 2019, Agarwal et al., 29 Mar 2026, Escapil-Inchauspé et al., 2023). Taken together, these works suggest a methodological class rather than a single canonical formalism.

1. Conceptual scope and recurring structure

A common starting point is the contrast with the standard Maxwell description in which the field equations are written in second-order form for the potential $A_\mu$, schematically $\nabla_\mu F^{\mu\nu}=0$ with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. Several first-order programs instead enlarge the field space and let a linear differential operator act directly on a multiplet whose components already contain the electromagnetic degrees of freedom. In the Lyra-manifold DKP formulation, the field is a 10-component spin-1 multiplet $\psi$; in the $8\times 8$ Dirac-like formulation, the electromagnetic state is $\Psi=(0,\mathbf{E},0,\mathbf{B})^T$; in the optical Dirac formalism, it is a 6-component “optical spinor”; and in macroscopic QED it is the dual field $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$ (Casana et al., 2016, Li et al., 2023, Oue, 2019, Agarwal et al., 29 Mar 2026).

The motivation is not uniform, but several themes recur. First-order Lagrangians are described as making Hamiltonian dynamics more transparent, handling constrained systems more flexibly, and fitting naturally with Schwinger’s action principle (Casana et al., 2016). Other works use first-order operators to keep electric and magnetic variables on equal footing, to expose symmetry generators and boundary terms, or to replace constitutive intermediaries such as $\mathbf{D}$ and $\mathbf{H}$0 by direct operator maps from $\mathbf{H}$1 to $\mathbf{H}$2 (Gratus et al., 2019, Agarwal et al., 29 Mar 2026).

Strand	Basic variable	Representative operator
DKP spin-1 field	$\mathbf{H}$3	$\mathbf{H}$4 (Casana et al., 2016)
Dirac-like Maxwell	$\mathbf{H}$5	$\mathbf{H}$6 (Li et al., 2023)
Optical Dirac theory	$\mathbf{H}$7	$\mathbf{H}$8 (Oue, 2019)
CMCR constitutive theory	$\mathbf{H}$9	$A_\mu$0 (Gratus et al., 2019)
Macroscopic QED	$A_\mu$1	$A_\mu$2 (Agarwal et al., 29 Mar 2026)

This variety already shows that “first order” has several meanings. In some papers it refers to a field equation first order in spacetime derivatives; in others it refers to constitutive or current operators linear in fields and derivatives; in still others it refers to first-order perturbative expansions of boundary operators. A broader implication is that the phrase identifies an operator-centered reorganization of electromagnetism rather than one fixed set of equations.

2. First-order field equations and enlarged electromagnetic state spaces

In the massless Duffin–Kemmer–Petiau approach, the spin-1 electromagnetic sector is represented by a 10-component field whose Minkowskian Lagrangian reproduces Maxwell theory with its $A_\mu$3 local gauge symmetry. After minimal coupling to a Lyra manifold, the action becomes

$A_\mu$4

and the equation of motion is

$A_\mu$5

Projecting with the spin-1 operators $A_\mu$6 and $A_\mu$7 yields a Lyra-generalized Maxwell equation,

$A_\mu$8

so torsion enters explicitly through the trace $A_\mu$9 of the Lyra torsion tensor (Casana et al., 2016). The same formalism produces an energy-momentum tensor and a spin density tensor by the Schwinger variational principle.

A different line rewrites Maxwell’s equations exactly in Dirac form by using $\nabla_\mu F^{\mu\nu}=0$0 matrices. In that construction, the same gamma-matrix algebra supports both the electron and photon sectors; the electromagnetic field is encoded by

$\nabla_\mu F^{\mu\nu}=0$1

and Maxwell’s equations become

$\nabla_\mu F^{\mu\nu}=0$2

The formalism admits both a spin-$\nabla_\mu F^{\mu\nu}=0$3 operator and a spin-1 operator, but only the spin-1 operator preserves the constrained electromagnetic subspace $\nabla_\mu F^{\mu\nu}=0$4. Within the same operator framework, the oscillatory part of the Poynting vector is identified with photon Zitterbewegung (Li et al., 2023).

The optical Dirac formulation uses a 6-component spinor

$\nabla_\mu F^{\mu\nu}=0$5

so that

$\nabla_\mu F^{\mu\nu}=0$6

Because this is already Schrödinger-like, it admits an “optical density operator” $\nabla_\mu F^{\mu\nu}=0$7 and a thermal state

$\nabla_\mu F^{\mu\nu}=0$8

The thermal analysis shows equal weight for left- and right-handed circularly polarized transverse modes, nonvanishing electric-magnetic correlation at low temperature, and vanishing correlation at high temperature (Oue, 2019).

These formulations share a basic strategy: the electromagnetic field is not treated as one potential with constraints added later, but as a larger state on which a first-order operator acts directly. This suggests that the principal technical gain is structural: gauge, polarization, and constitutive constraints are encoded into the operator/state space rather than reconstructed from a second-order scalar or vector equation.

3. Operator geometry, constitutive response, and source construction

A particularly geometric version starts from a formally self-adjoint first-order linear differential operator

$\nabla_\mu F^{\mu\nu}=0$9

acting on $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$0-columns of complex-valued half-densities. In the special case $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$1, $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$2, and nondegenerate principal symbol, the determinant of the principal symbol takes the form

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$3

which defines a Lorentzian metric. The matrices $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$4 become Pauli matrices, and the covariant subprincipal symbol

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$5

can be written as

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$6

thereby identifying a real electromagnetic covector potential $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$7. Combining $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$8 with its adjugate produces a block Dirac operator equivalent, up to density rescaling, to the traditional Dirac operator with electromagnetic coupling (Fang et al., 2014).

A more radical constitutive reformulation dispenses with $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$9 and $\psi$0 altogether. The argument is that $\psi$1 and $\psi$2 are unmeasurable and function only as gauge potentials for charge and current, so the physically relevant constitutive content lies in direct operator equations relating $\psi$3 to $\psi$4. The resulting combined Maxwell and constitutive relations have the form

$\psi$5

with first-order linear differential operators constrained by charge conservation. In that framework, homogeneous axionic bulk couplings become admissible: $\psi$6 These operator terms permit bulk effects of homogeneous axionic media, including longitudinal-wave solutions and topological configurations that cannot be modeled globally with $\psi$7 and $\psi$8 (Gratus et al., 2019).

Electromagnetic current operators in phenomenological relativistic models provide a third operator-level construction. There the basic requirement is local gauge invariance in a Poincaré-invariant quantum mechanical model. A model Hamiltonian written in the Weyl representation is made gauge covariant by replacing momentum operators with gauge-covariant derivatives, and the current is obtained as the term first order in $\psi$9. For a two-body interaction, this produces explicit operator expressions for $8\times 8$0 in terms of derivatives of the interaction kernel with respect to momenta. The three-vector current comes directly from minimal substitution, while the charge density is then reconstructed from the dynamical boost generators (Polyzou, 2023).

These operator programs differ sharply in purpose, but they converge on one point: first-order structure is not limited to rewriting Maxwell’s propagation law. It also governs how geometry, gauge covariance, constitutive response, and many-body source operators are encoded.

4. Quantization, commutators, and open-system first-order operators

One usage of “operator approach” does not produce a first-order field equation, but instead promotes the first-order divergence $8\times 8$1 to a nontrivial operator. In covariant quantization with the gauge-fixed Lagrangian, canonical commutators imply

$8\times 8$2

To avoid conflict with imposing the Lorentz gauge as a strong operator identity, the extended Lorentz gauge sets

$8\times 8$3

with $8\times 8$4 a constant Lorentz scalar, then replaces $8\times 8$5 by an operator $8\times 8$6 such that

$8\times 8$7

The dynamics remain second order, but the gauge condition is recast as a first-order operator relation rather than a subsidiary condition on physical states (Morimoto, 2018).

Another construction places classical electromagnetism directly in a complex Hilbert space. The potentials $8\times 8$8 are treated as state vectors, position and wavenumber become Hermitian operators, and the wave commutator

$8\times 8$9

is derived from the Fourier kernel $\Psi=(0,\mathbf{E},0,\mathbf{B})^T$0. Time evolution is written in Schrödinger-like form,

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$1

while the fields are generated by first-order operators acting on potentials: $\Psi=(0,\mathbf{E},0,\mathbf{B})^T$2 The formalism parallels Dirac–von Neumann quantum mechanics but explicitly removes the Born rule and collapse postulate (Piasecki, 2024).

A fully boundary-aware open-system version appears in macroscopic QED. There the dual field is

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$3

the first-order Maxwell operator is

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$4

and Maxwell’s equations become

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$5

Its Green operator $\Psi=(0,\mathbf{E},0,\mathbf{B})^T$6, with kernel $\Psi=(0,\mathbf{E},0,\mathbf{B})^T$7, propagates the electromagnetic state between surfaces and retains the boundary term

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$8

Quantization then yields two independent noise sources: a bulk Langevin noise polarization and an input-output boundary field. Their combined commutator gives the exact closed identity

$\Psi=(0,\mathbf{E},0,\mathbf{B})^T$9

consistent with the fluctuation-dissipation theorem even when dielectrics extend to the boundary (Agarwal et al., 29 Mar 2026).

A persistent theme in these works is that first-order operator formulations are especially well suited to commutators, boundary traces, and Hamiltonian or Green-operator identities. This suggests that their main advantage is often algebraic and structural rather than merely notational.

5. Dual charges, symmetry algebras, and supersymmetric interactions

For $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$0-form gauge theories, a first-order formulation can be built as BF theory with a potential,

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$1

Eliminating $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$2 recovers the second-order Maxwell-like action $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$3, but the first-order parent theory has two gauge-type symmetries: ordinary gauge transformations of $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$4 and translational shifts of $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$5. When the translational symmetry is reducible, with zero modes $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$6, its charges survive the reduction and become the magnetic or dual charges of the $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$7-form theory. The resulting electric and magnetic charges obey a centrally extended algebra,

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$8

and the existence condition for nontrivial dual charges is $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$9 (Geiller et al., 2021). In this setting, the first-order operator viewpoint makes electric and magnetic charges descendants of the same topological parent structure.

A different symmetry-based extension appears in linearized non-minimal supergravity in superspace. There the first-order variable is an $\mathbf{D}$0-invariant superfield

$\mathbf{D}$1

supplemented by connection-like auxiliary superfields. After integrating out the auxiliaries, one obtains a single-superfield action equivalent to the conventional second-order description, and this first-order framework is then used to construct the supersymmetric cubic electromagnetic vertex $\mathbf{D}$2. Trivial symmetries between the two $\mathbf{D}$3 multiplets imply that the cubic vertex must depend on the vector multiplet only through its gauge-invariant field strength

$\mathbf{D}$4

not the bare prepotential $\mathbf{D}$5, and the cubic interaction generates nontrivial deformations of the gauge transformations (Buchbinder et al., 2021).

These examples show that first-order operator approaches are not confined to rewriting free Maxwell equations. They also organize symmetry reduction, central extensions, and interaction vertices in ways that are difficult to see from second-order formulations alone.

6. Perturbative, scattering, and computational extensions

A broader usage of the phrase appears in black-hole perturbation theory. For electromagnetic perturbations of Schwarzschild, the relevant scalar master variables satisfy

$\mathbf{D}$6

with

$\mathbf{D}$7

The retarded Green function of $\mathbf{D}$8 turns the source problem into a convolution, and for point charges the radial delta support collapses the two-dimensional integral to a one-dimensional path integral along the worldline. The same operator is then reused at second order with quadratic effective sources, which yields mode mixing in the electromagnetic sector (Aly et al., 2024). Here “first order” refers not to a Dirac-like photon equation, but to the first-order perturbative level and its associated Green-operator inversion.

In computational electromagnetics, uncertain-shape scattering is treated by linearizing the boundary integral operators with respect to small random perturbations of a nominal surface. For PEC and dielectric scatterers, the shape derivative satisfies deterministic operator equations on the nominal boundary, while the randomness enters through the right-hand side and its covariance. Tensorizing the second-moment equation produces operator systems such as

$\mathbf{D}$9

which are then solved by the first-order sparse boundary element method through the combination technique. The resulting second-moment approximation has the error bound

$\mathbf{H}$00

and the method directly computes statistical moments without Monte Carlo sampling (Escapil-Inchauspé et al., 2023).

Taken together, these perturbative and numerical works show that the expression “first-order electromagnetic operator approach” also covers first-order linearization of electromagnetic operators with respect to geometry or perturbative amplitude. A common misconception is therefore that the term always means a photon Dirac equation. The literature instead supports at least four distinct meanings: first-order field equations, first-order constitutive or current operators, first-order Green-operator formalisms with explicit boundary traces, and first-order perturbative operator expansions (Li et al., 2023, Gratus et al., 2019, Agarwal et al., 29 Mar 2026, Escapil-Inchauspé et al., 2023).