Dual-Field First-Order Maxwell Operator

• Dual-field first-order Maxwell operator formalism recasts standard Maxwell equations into a coupled first-order system for paired fields (e.g. E and B) based on divergences and curls.
• It provides explicit retarded integral formulas to reconstruct fields directly from source data, unifying classical, material, and quantum electrodynamics.
• The operator-based approach preserves boundary contributions and reciprocity, leading to enhanced computational methods and elimination of nonphysical modes.

Searching arXiv for the specified papers and closely related work to ground the article. arXiv search query: (Heras et al., 2020) Maxwell operator formalism (Agarwal et al., 29 Mar 2026, Kudryavtsev et al., 2012) The dual-field first-order Maxwell operator formalism denotes a class of reformulations of classical and macroscopic quantum electrodynamics in which electromagnetism is expressed as a coupled first-order system for paired fields rather than as a second-order equation for a single field. In one lineage, the formalism appears as a retarded two-field generalization of the Helmholtz theorem, where two vector fields are uniquely reconstructed from their divergences and coupled curls, with Maxwell’s equations emerging as the special case $\mathbf F_1=\mathbf E$, $\mathbf F_2=\mathbf B$ (Heras et al., 2020). In another lineage, the electromagnetic state is organized as a dual field such as $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$, and Maxwell’s equations are written as a first-order operator equation whose Green operator propagates both bulk sources and boundary data (Agarwal et al., 29 Mar 2026). Closely related potential-based formulations replace the standard curl-plus-constraint structure by a first-order hyperbolic system for two vector and two scalar potentials, eliminating zero-speed nonphysical modes and making the equations directly compatible with upwind and shock-capturing methods (Kudryavtsev et al., 2012). Taken together, these developments define a common operator viewpoint: the physically relevant data are the coupled first-order relations among dual electromagnetic variables, not an electric-field-only second-order reduction.

1. Retarded dual-field generalization of Helmholtz theory

The most explicit classical dual-field formulation is the theorem for two retarded vector fields $\mathbf F_1(\mathbf r,t)$ and $\mathbf F_2(\mathbf r,t)$, proved in "Helmholtz's theorem for two retarded fields and its application to Maxwell's equations" (Heras et al., 2020). It extends the static Helmholtz theorem from one vector field to two time-dependent retarded fields whose curls are coupled by time derivatives. The prescribed data are

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$

and

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$

The theorem states that if $$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$ vanish at $t=0$, if $$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$ go to zero faster than $\mathbf F_2=\mathbf B$0 as $\mathbf F_2=\mathbf B$1, and if $\mathbf F_2=\mathbf B$2 as $\mathbf F_2=\mathbf B$3, then the fields are uniquely determined by these divergences and coupled curls (Heras et al., 2020). The source functions must satisfy the compatibility relations

$\mathbf F_2=\mathbf B$4

This construction is tailored to the time-dependent Maxwell system. In static theory, Helmholtz reconstruction depends on $\mathbf F_2=\mathbf B$5 and $\mathbf F_2=\mathbf B$6. In the dynamical case, however, Maxwell’s equations do not separately prescribe $\mathbf F_2=\mathbf B$7 and $\mathbf F_2=\mathbf B$8 as independent source equations. Instead they furnish the coupled combinations

$\mathbf F_2=\mathbf B$9

This is the precise sense in which the Maxwell system acquires a dual-field first-order structure (Heras et al., 2020).

A plausible implication is that the formalism does not merely generalize a decomposition theorem; it isolates the exact first-order data that are natural for relativistic field propagation. The pair $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$0 is the dynamical object, and the coupled curls play the role that the ordinary curl plays in static Helmholtz theory.

2. Explicit decomposition and uniqueness structure

The theorem provides an explicit retarded reconstruction formula. The fields are written as

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

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$2

with retarded potentials

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$3

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$4

Here $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$5, and $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$6 denotes evaluation at the retarded time $\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$7 (Heras et al., 2020).

The proof relies on two structural ingredients. First, the auxiliary potentials satisfy Lorenz-type conditions,

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

Second, they satisfy wave equations,

$\mathcal{E}=[\mathbf{E},Z_0\mathbf{H}]^T$9

with

$\mathbf F_1(\mathbf r,t)$0

The key identity is

$\mathbf F_1(\mathbf r,t)$1

for scalar or vector sources $\mathbf F_1(\mathbf r,t)$2 (Heras et al., 2020).

The uniqueness step is especially important. If another pair shares the same $\mathbf F_1(\mathbf r,t)$3, then the difference fields satisfy a homogeneous coupled system implying $\mathbf F_1(\mathbf r,t)$4 and $\mathbf F_1(\mathbf r,t)$5. The Appendix proves a uniqueness theorem for the homogeneous vector wave equation: if a vector field $\mathbf F_1(\mathbf r,t)$6 satisfies $\mathbf F_1(\mathbf r,t)$7, vanishes together with $\mathbf F_1(\mathbf r,t)$8 at $\mathbf F_1(\mathbf r,t)$9, and obeys suitable boundary conditions on $\mathbf F_2(\mathbf r,t)$0 at the boundary, then $\mathbf F_2(\mathbf r,t)$1 (Heras et al., 2020). This is presented as a direct uniqueness proof for wave equations and underwrites the legitimacy of the dual-field retarded Helmholtz theorem.

This suggests that the formalism is not simply representational. It is an existence-and-uniqueness theorem for a first-order electromagnetic data problem, where the primitive objects are paired fields and their coupled differential invariants.

3. Specialization to Maxwell’s equations

When the theorem is applied to Maxwell’s equations in SI units with

$\mathbf F_2(\mathbf r,t)$2

the prescribed data become

$\mathbf F_2(\mathbf r,t)$3

$\mathbf F_2(\mathbf r,t)$4

The corollary then yields directly

$\mathbf F_2(\mathbf r,t)$5

$\mathbf F_2(\mathbf r,t)$6

With

$\mathbf F_2(\mathbf r,t)$7

one obtains the standard retarded-field formulas

$\mathbf F_2(\mathbf r,t)$8

directly from the theorem (Heras et al., 2020).

The same framework also treats material sources. With polarization $\mathbf F_2(\mathbf r,t)$9 and magnetization $$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$0, the potentials become

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$1

For magnetic monopoles in Gaussian units, the theorem yields the dual-potential form

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$2

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$3

with corresponding retarded scalar and vector potentials built from $$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$4 (Heras et al., 2020).

A common misconception is that the dual-field perspective merely rephrases the familiar potential formalism. The explicit corollary shows a stronger claim: one can obtain the retarded electric and magnetic fields directly from source data, and the potential representation then appears as a consequence rather than as the starting point (Heras et al., 2020).

4. Operator formulation for the dual electromagnetic state

A more explicitly operator-theoretic version of the first-order formalism is developed in "First order Maxwell operator formalism for macroscopic quantum electrodynamics" (Agarwal et al., 29 Mar 2026). There the electromagnetic state is written as the dual field

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$5

with the corresponding dual source vector

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$6

Using $$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$7 time dependence, the frequency-domain Maxwell equations are

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$8

These are combined by introducing the dual curl operator

$$\nabla\cdot\mathbf F_1=D_1,\qquad \nabla\cdot\mathbf F_2=D_2,$$9

and the block-diagonal dual material tensor

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$0

The first-order Maxwell system takes the form

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$1

with $$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$2 (Agarwal et al., 29 Mar 2026).

Multiplying by $$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$3, the paper defines the Maxwell “Hamiltonian”

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$4

and then the Maxwell operator

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$5

The field equation becomes

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$6

Its Green operator is

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$7

with retarded Green operator $$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$8 and kernel

$$-\nabla\times\mathbf F_1-\frac{\partial \mathbf F_2}{\partial t}=\mathbf C_2,\qquad \nabla\times\mathbf F_2-\frac{1}{c^2}\frac{\partial \mathbf F_1}{\partial t}=\mathbf C_1.$$9

(Agarwal et al., 29 Mar 2026).

This operator packaging retains both $$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$0 and $$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$1 on equal footing. It contrasts with the standard second-order Helmholtz formulation, which propagates only $$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$2 and typically reconstructs the magnetic field afterward. The paper explicitly derives the second-order operator

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$3

and the relation

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$4

showing that the first-order formalism is not different physics, but a more complete operator organization of the same system (Agarwal et al., 29 Mar 2026).

5. Boundary terms, reciprocity, and generalized optical structure

A defining feature of the operator formalism is that it keeps the boundary term generated by integration by parts. The propagation formula is

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$5

where

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$6

The volume term propagates bulk sources, and the surface term propagates tangential boundary data (Agarwal et al., 29 Mar 2026).

The energy inner product is defined by

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$7

Under this inner product,

$$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$8

so $$\mathbf F_1,\mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$9 is formally self-adjoint up to the boundary form. Combining this with the material tensor gives

$t=0$0

The paper identifies this identity as underlying Poynting’s theorem, the generalized optical theorem, and the relation between bulk absorption and boundary flux (Agarwal et al., 29 Mar 2026).

The reciprocal inner product is

$t=0$1

and the Green kernel obeys

$t=0$2

From this follow Lorentz reciprocity and the symmetry of the Green operator in reciprocal media; the paper states explicitly that reciprocity does not require losslessness, since loss is allowed as long as the medium is reciprocal (Agarwal et al., 29 Mar 2026).

The generalized optical theorem is written as

$t=0$3

or in operator form,

$t=0$4

The anti-Hermitian part of the Green operator is therefore decomposed into bulk dissipation and boundary radiation flux (Agarwal et al., 29 Mar 2026).

This boundary-aware treatment marks a substantive difference from formulations that discard open-system surface terms. In the dual-field operator formalism, those terms are not auxiliary; they are part of the exact propagator structure.

6. Quantum extension and numerical-potential variants

In macroscopic quantum electrodynamics, the same first-order framework is quantized by a Heisenberg-Langevin treatment rather than Hamiltonian diagonalization (Agarwal et al., 29 Mar 2026). The dual polarization is

$t=0$5

and the constitutive relation becomes

$t=0$6

where $t=0$7 is the Langevin noise polarization. The paper distinguishes two independent quantum noise sources: bulk Langevin noise generated by absorption in the material and boundary input noise represented by incoming fields on the boundary (Agarwal et al., 29 Mar 2026).

The interior field splits as

$t=0$8

with

$t=0$9

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$0

The exact closed commutation relation is

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$1

and is stated to hold even when dielectrics extend all the way to the boundary, including waveguide input-output problems (Agarwal et al., 29 Mar 2026). The surface-to-surface propagation law,

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$2

with

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$3

extends the formalism to quantum input-output theory for complex photonic structures (Agarwal et al., 29 Mar 2026).

A numerically oriented variant appears in "Two-Potential Formalism for Numerical Solution of the Maxwell Equations" (Kudryavtsev et al., 2012). There the fields are represented by two vector potentials $$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$4 and two scalar potentials $$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$5. In the source-free vacuum case,

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$6

and also

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$7

Equating these produces

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$8

$$\nabla \mathbf F_1,\nabla \mathbf F_2,\partial\mathbf F_1/\partial t,\partial\mathbf F_2/\partial t$$9

closed by Lorentz gauge conditions

$\mathbf F_2=\mathbf B$00

(Kudryavtsev et al., 2012).

The state vector is

$\mathbf F_2=\mathbf B$01

and the characteristic matrix has only

$\mathbf F_2=\mathbf B$02

with multiplicity four, where $\mathbf F_2=\mathbf B$03. By contrast, the standard first-order field formulation has eigenvalues

$\mathbf F_2=\mathbf B$04

and the zero eigenvalues correspond to nonphysical modes associated with the divergence constraints (Kudryavtsev et al., 2012). The potential formalism therefore yields a hyperbolic system containing only evolutionary equations and no differential constraints. The paper specifically identifies compatibility with upwind differences, shock-capturing methods, and high-order WENO schemes; it uses a fifth-order WENO spatial discretization and a fourth-order Runge-Kutta-Gill time integrator (Kudryavtsev et al., 2012).

These two directions—operator quantization and hyperbolic potential evolution—are distinct in purpose but structurally aligned. Both retain a doubled electromagnetic description, both preserve the first-order coupling between electric and magnetic sectors, and both avoid reducing the theory to an electric-field-only second-order problem.

7. Conceptual scope, related first-order enlargements, and interpretive boundaries

The shared conceptual core of the dual-field first-order Maxwell operator formalism is the replacement of a single-field second-order description by an enlarged first-order state space. In the retarded Helmholtz theorem, the state is $\mathbf F_2=\mathbf B$05, determined by divergences and coupled curls (Heras et al., 2020). In macroscopic QED, the state is $\mathbf F_2=\mathbf B$06, acted on by the Maxwell operator $\mathbf F_2=\mathbf B$07 (Agarwal et al., 29 Mar 2026). In the two-potential hyperbolic method, the state is $\mathbf F_2=\mathbf B$08, whose first-order equations eliminate the usual divergence constraints as separate conditions (Kudryavtsev et al., 2012).

A related but not identical enlargement appears in "Maxwell - Chern - Simons topologically massive gauge fields in the first-order formalism" (Kruglov, 2010). That paper does not formulate a separate “dual-field first-order Maxwell operator formalism” in the modern sense, but it rewrites the $\mathbf F_2=\mathbf B$09-dimensional Maxwell-Chern-Simons theory as a first-order relativistic wave equation for the six-component multiplet

$\mathbf F_2=\mathbf B$10

satisfying

$\mathbf F_2=\mathbf B$11

The matrices $\mathbf F_2=\mathbf B$12 obey the Duffin-Kemmer-Petiau algebra

$\mathbf F_2=\mathbf B$13

and the construction is explicitly described as a first-order operator formalism in an enlarged space containing both the potential sector and the tensor sector (Kruglov, 2010). This is structurally similar to dual-field approaches because one component carries the field, the other its curvature, and the dynamics are encoded in a matrix-valued first-order operator.

Several interpretive boundaries follow from the source material. First, the dual-field formalism is not a claim that potentials are unnecessary in every setting; in (Heras et al., 2020), direct field reconstruction coexists with an induced retarded-potential structure. Second, the first-order operator viewpoint is not a different electrodynamics; (Agarwal et al., 29 Mar 2026) derives the second-order formulation from the first-order one and states that the first-order formalism is a more complete operator packaging. Third, the numerical two-potential formulation is not presented as a universal source treatment; its sourceful extension is described as mostly formal unless the source decomposition is already known (Kudryavtsev et al., 2012).

Within these limits, the formalism provides a unified way to think about electromagnetism as a first-order coupled system. The basic observables may be retarded fields, impedance-normalized dual fields, or doubled potentials, but in each case the decisive structure is the same: electromagnetic propagation is encoded by a coupled first-order operator acting on a doubled state space, with boundary terms, reciprocity, and causal propagation emerging as intrinsic features rather than secondary corrections.