First-Order Maxwell Operator Formalism

• First-order Maxwell operator formalism redefines Maxwell’s equations as a first-order system using varied field multiplets to enhance analysis of gauge invariance and spectral properties.
• Different representations—including Palatini-like, Newman–Penrose, and DKP-type—enable precise treatment of boundary conditions, energy conservation, and asymptotic behavior.
• Applications range from macroscopic QED and cylindrical waveguides to curved spacetime, offering tractable evolution equations and explicit spectral decompositions.

First-order Maxwell operator formalism denotes a family of representations in which Maxwell theory is written as a first-order system on an enlarged field space rather than exclusively as a second-order wave equation. In these representations the fundamental variable may be a complex electromagnetic six-vector, a pair $(A,F)$ of potential and field strength, a null-frame multiplet $(\phi_0,\phi_1,\phi_2)^T$, or a Duffin–Kemmer–Petiau-type wavefunction, and the operator may be realized as a self-adjoint generator, a covariant constraint map, or a resolvent-based propagator (Nittis et al., 2017, Pietrzyk et al., 2022, Xiao et al., 16 Feb 2026, Kruglov, 2010, Agarwal et al., 29 Mar 2026). The common objective is to retain the first-order differential structure of Maxwell’s equations while making gauge invariance, spectral theory, asymptotics, conserved quantities, boundary data, or quantization more transparent.

1. Core operator reformulations

A central pattern is the replacement of the real-field Maxwell system by a complex first-order evolution equation. In linear, dispersionless media with

$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$

the dynamical equations can be written as

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$

and, after complexification and restriction to nonnegative frequencies,

$$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$

with $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$ (Nittis et al., 2017). In the covariant Palatini-like formulation, the first-order operator is instead the map

$\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$

so that $dA-F=0$ encodes the definition of $F$ and $\delta F=0$ encodes Maxwell’s equations (Pietrzyk et al., 2022). In frequency-domain macroscopic QED, the dual field

$(\phi_0,\phi_1,\phi_2)^T$0

obeys

$(\phi_0,\phi_1,\phi_2)^T$1

with $(\phi_0,\phi_1,\phi_2)^T$2 and $(\phi_0,\phi_1,\phi_2)^T$3 (Agarwal et al., 29 Mar 2026).

Setting	Field multiplet	First-order equation
Linear media	$(\phi_0,\phi_1,\phi_2)^T$4	$(\phi_0,\phi_1,\phi_2)^T$5
Covariant Palatini-like form	$(\phi_0,\phi_1,\phi_2)^T$6	$(\phi_0,\phi_1,\phi_2)^T$7
Null-tetrad form	$(\phi_0,\phi_1,\phi_2)^T$8	$(\phi_0,\phi_1,\phi_2)^T$9
Macroscopic QED	$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$0	$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$1

These constructions differ in state space, boundary conditions, and interpretation, but they all preserve the first-order character of the field equations. This suggests that the phrase “first-order Maxwell operator” refers not to a unique canonical matrix but to a structural class of formulations adapted to different analytical tasks.

2. Newman–Penrose null-tetrad operator

A differential-form and Newman–Penrose (NP) formulation expresses Maxwell theory in terms of the gauge potential $$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$2, the Faraday 2-form

$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$3

and its Hodge dual

$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$4

with $$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$5 and $$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$6. Gauge invariance is immediate from $$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$7 (Xiao et al., 16 Feb 2026).

The NP decomposition introduces a null tetrad $$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$8 satisfying

$$W(x)=\begin{pmatrix}\varepsilon&\chi\ \chi^*&\mu\end{pmatrix}(x),$$9

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$0

and

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$1

The associated directional derivatives are

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$2

and the electromagnetic field is encoded in the NP scalars

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$3

In vacuum, $W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$4 and $W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$5 yield four first-order PDEs: $W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$6

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$7

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$8

$W \partial_t\Psi_{\mathrm{phys}}=\mathrm{Rot}\,\Psi_{\mathrm{phys}}-J_{\mathrm{phys}}, \qquad \mathrm{Rot}:=\begin{pmatrix}0&i\nabla\times\-i\nabla\times&0\end{pmatrix},$9

With $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$0, these may be organized as a $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$1 differential operator $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$2, or, after isolating a dynamical subsystem, as a square $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$3 operator (Xiao et al., 16 Feb 2026).

The same formalism accommodates Lorentz-violating extensions while preserving gauge invariance because the actions are built from $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$4 and $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$5. For the CPT-odd dimension-3 CFJ/MCS term, the field equation becomes

$$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$6

producing source-like NP terms linear in $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$7. For the CPT-even dimension-4 $$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$8 term,

$$i\,\partial_t\Psi(t)=M\Psi(t)-i\,J(t),\qquad M=W^{-1}\,\mathrm{Rot}\big|_{\omega\ge 0},$$9

and the NP equations acquire additional linear couplings to directional derivatives of $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$0 and to $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$1 itself. Myers–Pospelov dimension-5 terms generate second derivatives along $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$2, while dimension-6 Podolsky/Lee–Wick operators produce up to third-order derivatives in the NP scalars (Xiao et al., 16 Feb 2026).

The formal closure of the system relies on the NP commutators, Ricci identities, and decoupling identities such as

$(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$3

In this setting the first-order Maxwell operator is simultaneously a compact encoding of the field equations and an asymptotic tool for radiation and polarization analysis.

3. Self-adjoint generators in media and cylindrical waveguides

In the Schrödinger formalism for electromagnetism, the Maxwell operator acts on a weighted Hilbert space

$(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$4

with scalar product

$(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$5

Under the assumptions $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$6 and $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$7, one takes

$(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$8

and $(\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi$9 is self-adjoint. The corresponding unitary group $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$0 yields

$\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$1

and preserves the electromagnetic energy

$\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$2

Because $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$3, the spectrum is symmetric, with positive and negative branches and a zero eigenspace of longitudinal modes. The projector

$\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$4

selects the unique complex nonnegative-frequency representative of a real field (Nittis et al., 2017).

In an infinite cylinder $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$5 with scalar coefficients $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$6 and $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$7, the Maxwell operator is the self-adjoint first-order block matrix

$\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$8

in $\mathcal M:(A,F)\longmapsto (dA-F,\delta F),$9, with domain

$dA-F=0$0

subject to perfect-conductivity conditions

$dA-F=0$1

and divergence-free constraints $dA-F=0$2, $dA-F=0$3 (Filonov, 2020).

Because $dA-F=0$4 and $dA-F=0$5 depend only on the longitudinal variable, separation of variables reduces the three-dimensional operator to a countable family of one-dimensional first-order matrix systems. The invariant subspaces $dA-F=0$6, $dA-F=0$7, and $dA-F=0$8 carry $dA-F=0$9, $F$0, and $F$1 blocks, respectively. Under stabilization at infinity, each block square is unitarily equivalent to a direct sum of scalar Schrödinger operators with short-range potential; under periodicity, the squares become Hill operators. The resulting spectral description is explicit: for stabilizing coefficients, $F$2, and the absolutely continuous spectrum is either $F$3 or has a single gap around zero depending on the topology of the cross-section; for periodic coefficients, $F$4, $F$5, and $F$6 has a finite number of gaps (Filonov, 2020).

Together these results place the first-order Maxwell operator within functional analysis and spectral theory: self-adjointness is not merely formal, but the basis for well-posed evolution, conserved quadratic observables, and reduction to tractable one-dimensional blocks.

4. Duffin–Kemmer–Petiau-type formulation of Maxwell–Chern–Simons theory

In $F$7 dimensions, the Maxwell–Chern–Simons system

$F$8

admits a first-order matrix reformulation in terms of the six-component wavefunction

$F$9

The field equations become

$\delta F=0$0

or, equivalently,

$\delta F=0$1

with $\delta F=0$2 a projector onto the tensor subspace (Kruglov, 2010).

In the ordered basis $\delta F=0$3, the $\delta F=0$4 matrices $\delta F=0$5 obey the Duffin–Kemmer–Petiau algebra

$\delta F=0$6

A Hermitianizing matrix exists,

$\delta F=0$7

satisfying $\delta F=0$8, so that the bilinear form $\delta F=0$9 is Lorentz invariant.

The formalism also supplies covariant projection operators. In momentum space, with $(\phi_0,\phi_1,\phi_2)^T$00 and $(\phi_0,\phi_1,\phi_2)^T$01, one constructs

$(\phi_0,\phi_1,\phi_2)^T$02

and then spin projectors from the covariant Pauli–Lubanski scalar

$(\phi_0,\phi_1,\phi_2)^T$03

namely

$(\phi_0,\phi_1,\phi_2)^T$04

with $(\phi_0,\phi_1,\phi_2)^T$05.

After eliminating non-dynamical components by the Lorentz gauge, the theory reduces to a $(\phi_0,\phi_1,\phi_2)^T$06 Schrödinger Hamiltonian $(\phi_0,\phi_1,\phi_2)^T$07. Its minimal polynomial implies the eigenvalues

$(\phi_0,\phi_1,\phi_2)^T$08

for the physical sector and

$(\phi_0,\phi_1,\phi_2)^T$09

for the pure-gauge sector (Kruglov, 2010). In this case the first-order Maxwell operator is not a self-adjoint curl generator but a relativistic wave operator of DKP type, adapted to topologically massive spin-1 dynamics.

5. Covariant Hamiltonian and perturbative curved-spacetime operators

A covariant Hamiltonian treatment begins from the Palatini-like Lagrangian

$(\phi_0,\phi_1,\phi_2)^T$10

with independent $(\phi_0,\phi_1,\phi_2)^T$11 and antisymmetric auxiliary field $(\phi_0,\phi_1,\phi_2)^T$12. The De Donder–Weyl polymomenta are

$(\phi_0,\phi_1,\phi_2)^T$13

and the generalized Dirac–Bergmann analysis yields the reduced Hamiltonian

$(\phi_0,\phi_1,\phi_2)^T$14

and reduced polysymplectic form

$(\phi_0,\phi_1,\phi_2)^T$15

The De Donder–Weyl equations then reproduce the first-order Maxwell system

$(\phi_0,\phi_1,\phi_2)^T$16

which is equivalently encoded by

$(\phi_0,\phi_1,\phi_2)^T$17

or, in differential-form notation,

$(\phi_0,\phi_1,\phi_2)^T$18

(Pietrzyk et al., 2022). The same framework supports a covariant Hamilton–Jacobi equation

$(\phi_0,\phi_1,\phi_2)^T$19

with the embedding condition $(\phi_0,\phi_1,\phi_2)^T$20.

In curved spacetime, the source-free Maxwell equations

$(\phi_0,\phi_1,\phi_2)^T$21

together with the Lorenz gauge $(\phi_0,\phi_1,\phi_2)^T$22 imply

$(\phi_0,\phi_1,\phi_2)^T$23

For a vacuum gravitational wave in TT gauge, $(\phi_0,\phi_1,\phi_2)^T$24, so $(\phi_0,\phi_1,\phi_2)^T$25. Expanding

$(\phi_0,\phi_1,\phi_2)^T$26

one obtains the first-order perturbation equation

$(\phi_0,\phi_1,\phi_2)^T$27

with

$(\phi_0,\phi_1,\phi_2)^T$28

$(\phi_0,\phi_1,\phi_2)^T$29

The paper also shows that the $(\phi_0,\phi_1,\phi_2)^T$30-tensor and $(\phi_0,\phi_1,\phi_2)^T$31-potential formulations are mathematically equivalent at first order, and that the perturbative equation is gauge-invariant under residual gauge transformations compatible with the first-order Lorenz-gauge constraint (Lou et al., 27 May 2026).

Explicit first-order fields and stress tensor are available: $(\phi_0,\phi_1,\phi_2)^T$32 and

$(\phi_0,\phi_1,\phi_2)^T$33

For a plane EM wave interacting with a TT gravitational wave, the maximum modulus of the coupling coefficient is on the order of $(\phi_0,\phi_1,\phi_2)^T$34, and a typical astrophysical strain $(\phi_0,\phi_1,\phi_2)^T$35 generates a first-order EM response on the order of $(\phi_0,\phi_1,\phi_2)^T$36 relative to the incident field amplitude (Lou et al., 27 May 2026). Here the first-order Maxwell operator functions as a perturbative transport operator on electromagnetic degrees of freedom in a weakly curved background.

6. First-order Green operators and macroscopic quantum electrodynamics

In macroscopic QED, the first-order operator formalism is formulated directly in the frequency domain and retains both $(\phi_0,\phi_1,\phi_2)^T$37 and $(\phi_0,\phi_1,\phi_2)^T$38 as components of a dual field,

$(\phi_0,\phi_1,\phi_2)^T$39

The operator equation is

$(\phi_0,\phi_1,\phi_2)^T$40

with

$(\phi_0,\phi_1,\phi_2)^T$41

The retarded Green operator

$(\phi_0,\phi_1,\phi_2)^T$42

has kernel $(\phi_0,\phi_1,\phi_2)^T$43 satisfying

$(\phi_0,\phi_1,\phi_2)^T$44

and the interior field obeys the Stratton–Chu representation

$(\phi_0,\phi_1,\phi_2)^T$45

The boundary term is intrinsic rather than discarded, and $(\phi_0,\phi_1,\phi_2)^T$46 propagates both volume sources and tangential surface data (Agarwal et al., 29 Mar 2026).

Two bilinear structures organize the formalism. Under the energy inner product,

$(\phi_0,\phi_1,\phi_2)^T$47

one derives the generalized optical theorem,

$(\phi_0,\phi_1,\phi_2)^T$48

Under the reciprocal inner product

$(\phi_0,\phi_1,\phi_2)^T$49

one obtains Lorentz reciprocity,

$(\phi_0,\phi_1,\phi_2)^T$50

for reciprocal media.

Quantization proceeds through a Heisenberg–Langevin treatment. The constitutive relation is

$(\phi_0,\phi_1,\phi_2)^T$51

with bulk noise commutator

$(\phi_0,\phi_1,\phi_2)^T$52

The operator-valued field solution is

$(\phi_0,\phi_1,\phi_2)^T$53

and the total field satisfies the exact closed commutation relation

$(\phi_0,\phi_1,\phi_2)^T$54

The same framework yields an exact quantum transfer relation between input and output surfaces,

$(\phi_0,\phi_1,\phi_2)^T$55

with

$(\phi_0,\phi_1,\phi_2)^T$56

This formalism extends first-order Maxwell operators from classical propagation to open-system quantization in absorptive and dispersive photonic structures (Agarwal et al., 29 Mar 2026).

Taken together, these developments suggest that first-order Maxwell operator formalisms are best understood as a broad operator-theoretic framework in which Maxwell’s equations are embedded into enlarged state spaces so that gauge structure, spectral decomposition, asymptotic propagation, reciprocity, and quantization become directly accessible within a single first-order language.