---
title: First-Order Maxwell Operator Formalism
url: https://www.emergentmind.com/topics/first-order-maxwell-operator-formalism
type: topic
---

# First-Order Maxwell Operator Formalism

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 [1710.10148][2212.14845][2602.14410][1010.4728][2603.27475]. 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 \((\mathbf E,\mathbf H)=2\,\mathrm{Re}\,\Psi\) [1710.10148]. 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 [2212.14845]. In frequency-domain macroscopic QED, the dual field
\[
\mathcal E(\mathbf r,\omega)=\begin{bmatrix}\mathbf E(\mathbf r,\omega)\\ Z_0\,\mathbf H(\mathbf r,\omega)\end{bmatrix}
\]
obeys
\[
\mathcal M\,\mathcal E=\mathcal S,\qquad
\mathcal M=i\,\bar\nabla\times-k_0\,\bar\varepsilon,
\]
with \(\bar\nabla\times=J\otimes(\nabla\times)\) and \(\bar\varepsilon=\mathrm{diag}(\varepsilon,\mu)\) [2603.27475].

| Setting | Field multiplet | First-order equation |
|---|---|---|
| Linear media | \(\Psi=(E+iH)/2\) | \(i\,\partial_t\Psi=M\Psi-iJ\) |
| Covariant Palatini-like form | \((A,F)\) | \(\mathcal M(A,F)=(dA-F,\delta F)=0\) |
| Null-tetrad form | \((\phi_0,\phi_1,\phi_2)^T\) | \(\mathcal M\cdot\Psi=0\) |
| Macroscopic QED | \([\mathbf E,Z_0\mathbf H]^T\) | \(\mathcal M\,\mathcal E=\mathcal S\) |

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 \(A=A_\mu dx^\mu\), the Faraday 2-form
\[
F=dA=\tfrac12 F_{\mu\nu}\,dx^\mu\wedge dx^\nu,
\qquad
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,
\]
and its Hodge dual
\[
*F=\tfrac12 \epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}\,dx^\mu\wedge dx^\nu,
\]
with \(dF=0\) and \(d(*F)=-*J\). Gauge invariance is immediate from \(A\to A+d\Lambda\Rightarrow F\to F\) [2602.14410].

The NP decomposition introduces a null tetrad \(\{l,n,m,\bar m\}\) satisfying
\[
l\cdot l=n\cdot n=0,\qquad l\cdot n=-1,
\]
\[
m\cdot m=\bar m\cdot \bar m=0,\qquad m\cdot \bar m=+1,
\]
and
\[
g_{\mu\nu}=2\bigl(m_{(\mu}\bar m_{\nu)}-l_{(\mu}n_{\nu)}\bigr).
\]
The associated directional derivatives are
\[
D\equiv l^\mu\nabla_\mu,\qquad
\Delta\equiv n^\mu\nabla_\mu,\qquad
\delta\equiv m^\mu\nabla_\mu,\qquad
\bar\delta\equiv \bar m^\mu\nabla_\mu,
\]
and the electromagnetic field is encoded in the NP scalars
\[
\phi_0=F_{\mu\nu}l^\mu m^\nu,\qquad
\phi_1=\tfrac12 F_{\mu\nu}(l^\mu n^\nu+m^\mu \bar m^\nu),\qquad
\phi_2=F_{\mu\nu}\bar m^\mu n^\nu.
\]

In vacuum, \(dF=0\) and \(d(*F)=0\) yield four first-order PDEs:
\[
(D-2\rho)\phi_1-(\bar\delta-2\alpha)\phi_0+\kappa\,\phi_2=0,
\]
\[
(\delta-2\tau)\phi_1-(\Delta-2\gamma)\phi_0+\sigma\,\phi_2=0,
\]
\[
(D+2\epsilon-\rho)\phi_2-(\bar\delta+2\pi)\phi_1+\lambda\,\phi_0=0,
\]
\[
(\delta+2\beta-\tau)\phi_2-(\Delta+2\mu)\phi_1+\nu\,\phi_0=0.
\]
With \(\Psi=(\phi_0,\phi_1,\phi_2)^T\), these may be organized as a \(4\times 3\) differential operator \(\mathcal M\cdot \Psi=0\), or, after isolating a dynamical subsystem, as a square \(3\times 3\) operator [2602.14410].

The same formalism accommodates Lorentz-violating extensions while preserving gauge invariance because the actions are built from \(F\) and \(*F\). For the CPT-odd dimension-3 CFJ/MCS term, the field equation becomes
\[
d(*F)-2\,k_{AF}\wedge F=0,
\]
producing source-like NP terms linear in \(k_{AF}\cdot \phi\). For the CPT-even dimension-4 \(\kappa_F\) term,
\[
d[*F+2*(\kappa_F\cdot F)]=0,
\]
and the NP equations acquire additional linear couplings to directional derivatives of \(\phi\) and to \(\phi\) itself. Myers–Pospelov dimension-5 terms generate second derivatives along \(n^\mu\), while dimension-6 Podolsky/Lee–Wick operators produce up to third-order derivatives in the NP scalars [2602.14410].

The formal closure of the system relies on the NP commutators, Ricci identities, and decoupling identities such as
\[
(\Delta-2\gamma-\mu)(D-2\rho)\phi_2=(\delta-2\tau)(\bar\delta-2\pi)\phi_2+\cdots.
\]
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
\[
\mathcal H=\Bigl\{\Psi=(E,H):\mathbb R^3\to \mathbb C^6,\ \int \Psi^*(x)W(x)\Psi(x)\,dx<\infty\Bigr\}
\]
with scalar product
\[
\langle \Phi,\Psi\rangle_W=\int_{\mathbb R^3}\Phi^*(x)\,W(x)\,\Psi(x)\,dx.
\]
Under the assumptions \(W=W^*\) and \(cI\le W(x)\le CI\), one takes
\[
\mathrm{Dom}(M)=H(\mathrm{curl})\oplus H(\mathrm{curl}),
\]
and \(M=W^{-1}\mathrm{Rot}|_{\omega\ge 0}\) is self-adjoint. The corresponding unitary group \(U(t)=e^{-itM}\) yields
\[
\Psi(t)=U(t)\Psi(0)-i\int_0^t U(t-s)J(s)\,ds,
\]
and preserves the electromagnetic energy
\[
\mathcal E=\tfrac12\langle \Psi,\Psi\rangle_W.
\]
Because \(CMC=-M\), the spectrum is symmetric, with positive and negative branches and a zero eigenspace of longitudinal modes. The projector
\[
Q_+=P_+ + \tfrac12 P_0
\]
selects the unique complex nonnegative-frequency representative of a real field [1710.10148].

In an infinite cylinder \(\Pi=U\times \mathbb R\) with scalar coefficients \(\varepsilon(z)\) and \(\mu(z)\), the Maxwell operator is the self-adjoint first-order block matrix
\[
M\begin{pmatrix}E\\ H\end{pmatrix}
=
\begin{pmatrix}
0 & i\,\varepsilon^{-1}\mathrm{rot}\\
-\,i\,\mu^{-1}\mathrm{rot} & 0
\end{pmatrix}
\begin{pmatrix}E\\ H\end{pmatrix}
\]
in \(J=J(\varepsilon)\oplus J(\mu)\), with domain
\[
\mathrm{Dom}\,M=\Phi(\tau,\varepsilon)\oplus \Phi(\nu,\mu),
\]
subject to perfect-conductivity conditions
\[
E\times \nu|_{\partial\Pi}=0,\qquad \mu H\cdot \nu|_{\partial\Pi}=0,
\]
and divergence-free constraints \(\mathrm{div}(\varepsilon E)=0\), \(\mathrm{div}(\mu H)=0\) [2012.01034].

Because \(\varepsilon\) and \(\mu\) 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 \(\mathcal I_k^{el}\), \(\mathcal I_l^{mag}\), and \(\mathcal I_j^0\) carry \(3\times 3\), \(3\times 3\), and \(2\times 2\) 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, \(\sigma_{sc}(M)=\varnothing\), and the absolutely continuous spectrum is either \(\mathbb R\) or has a single gap around zero depending on the topology of the cross-section; for periodic coefficients, \(\sigma(M)=\sigma_{ac}(M)\), \(\sigma_p(M)=\sigma_{sc}(M)=\varnothing\), and \(\sigma(M)\) has a finite number of gaps [2012.01034].

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 \(2+1\) dimensions, the Maxwell–Chern–Simons system
\[
\partial^\mu F_{\mu\nu}+p\,\epsilon_{\nu\mu\rho}F^{\mu\rho}=0
\]
admits a first-order matrix reformulation in terms of the six-component wavefunction
\[
\Psi_A(x)=\{\,\Psi_\nu(x)=pA_\nu(x),\ \Psi_{\rho\sigma}(x)=F_{\rho\sigma}(x)\,\}.
\]
The field equations become
\[
(\beta^\mu \partial_\mu+p\,P)\Psi(x)=0,
\]
or, equivalently,
\[
(\Gamma^\mu\partial_\mu+m\,S)\Psi(x)=0,\qquad m\equiv p,
\]
with \(S\) a projector onto the tensor subspace [1010.4728].

In the ordered basis \((A_1,A_2,A_3\equiv iA_0;F_{23},F_{31},F_{12})\), the \(6\times 6\) matrices \(\beta^\mu\) obey the Duffin–Kemmer–Petiau algebra
\[
\beta^\mu\beta^\nu\beta^\alpha+\beta^\alpha\beta^\nu\beta^\mu
=
g^{\mu\alpha}\beta^\nu+g^{\nu\alpha}\beta^\mu,
\qquad
g^{\mu\nu}=\mathrm{diag}(1,1,1).
\]
A Hermitianizing matrix exists,
\[
\eta=\beta^1\beta^1+\beta^2\beta^2-\beta^3\beta^3,
\]
satisfying \(\eta\beta^\mu=(\beta^\mu)^\dagger \eta\), so that the bilinear form \(\Psi^\dagger\eta\Psi\) is Lorentz invariant.

The formalism also supplies covariant projection operators. In momentum space, with \(\hat p=\beta^\mu P_\mu\) and \(A=i\hat p+p\,P\), one constructs
\[
\Pi(P)=N(A^2-P^2)(A-p),\qquad N=\frac1{2p^3},
\]
and then spin projectors from the covariant Pauli–Lubanski scalar
\[
W=\epsilon_{\mu\nu\rho}P^\mu J^{\nu\rho},\qquad J^{\mu\nu}=\beta^\mu\beta^\nu-\beta^\nu\beta^\mu,
\]
namely
\[
S(\pm1)=\tfrac12\Bigl(1\pm \frac{W}{|P|}\Bigr),\qquad
S(0)=1-\frac{W^2}{P^2},
\]
with \(\Pi(P,s)=\Pi(P)S(s)\).

After eliminating non-dynamical components by the Lorentz gauge, the theory reduces to a \(5\times 5\) Schrödinger Hamiltonian \(H_{\rm Sch}\). Its minimal polynomial implies the eigenvalues
\[
E=\pm \sqrt{\mathbf p^2+m^2}
\]
for the physical sector and
\[
E=0
\]
for the pure-gauge sector [1010.4728]. 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
\[
\mathcal L(A,P)= -\frac14 P_{\mu\nu}\bigl(P^{\mu\nu}-2F^{\mu\nu}\bigr),
\qquad
F_{\mu\nu}:=\partial_\mu A_\nu-\partial_\nu A_\mu,
\]
with independent \(A_\mu\) and antisymmetric auxiliary field \(P_{\mu\nu}\). The De Donder–Weyl polymomenta are
\[
\pi^{\mu\nu}=-\tfrac12 P^{\mu\nu},\qquad \rho^{\mu\nu,\alpha\beta}=0,
\]
and the generalized Dirac–Bergmann analysis yields the reduced Hamiltonian
\[
H^*(A,\pi)=-\pi_{\mu\nu}\pi^{\mu\nu}
\]
and reduced polysymplectic form
\[
\omega^*=dA_\mu\wedge d\pi^{\mu\nu}\wedge d^{\,n-1}x_\nu.
\]
The De Donder–Weyl equations then reproduce the first-order Maxwell system
\[
F_{\mu\nu}-2\partial_{[\mu}A_{\nu]}=0,\qquad
\partial_\mu F^{\mu\nu}=0,
\]
which is equivalently encoded by
\[
\mathcal M:(A_\mu,F_{\mu\nu})\longmapsto
\bigl(F_{\mu\nu}-2\partial_{[\mu}A_{\nu]},\ \partial_\mu F^{\mu\nu}\bigr)=0
\]
or, in differential-form notation,
\[
\mathcal M=(dA-F,\delta F)
\]
[2212.14845]. The same framework supports a covariant Hamilton–Jacobi equation
\[
\partial_\nu S^\nu(A,x)+H^*\!\bigl(A,\tfrac{\partial S}{\partial A}\bigr)=0,
\qquad
\pi^{\mu\nu}=\frac{\partial S^\nu}{\partial A_\mu},
\]
with the embedding condition \(\partial_\nu A_\mu=\partial S^\nu/\partial A_\mu\).

In curved spacetime, the source-free Maxwell equations
\[
\nabla_\nu F^{\mu\nu}=0,\qquad F_{[\mu\nu,\rho]}=0
\]
together with the Lorenz gauge \(\nabla_\mu A^\mu=0\) imply
\[
\Box_g A_\mu-R_\mu{}^\nu A_\nu=0.
\]
For a vacuum gravitational wave in TT gauge, \(R_{\mu\nu}=0\), so \(\Box_g A_\mu=0\). Expanding
\[
g_{\mu\nu}=\eta_{\mu\nu}+\epsilon h_{\mu\nu},\qquad
A_\mu=A_\mu^{(0)}+\epsilon A_\mu^{(1)},
\]
one obtains the first-order perturbation equation
\[
L_\mu{}^\nu A_\nu^{(1)}+\delta L_\mu{}^\nu A_\nu^{(0)}=0,
\]
with
\[
L_\mu{}^\nu=\delta_\mu^\nu \eta^{\alpha\beta}\partial_\alpha\partial_\beta,
\]
\[
\delta L_\mu{}^\nu=
-\,h^{\lambda\alpha}\delta_\mu^\nu \partial_\lambda\partial_\alpha
+\bigl[h_\mu{}^\lambda{}_{,\nu}+h_\mu{}^\nu{}_{,\lambda}-h_\lambda{}^\nu{}_{,\mu}\bigr]\partial_\lambda.
\]
The paper also shows that the \(F\)-tensor and \(A\)-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 [2605.28663].

Explicit first-order fields and stress tensor are available:
\[
E_\mu=F_{\mu\nu}U^\nu,\qquad
B_\mu=-\tfrac12 \epsilon_{\mu\alpha\beta\gamma}F^{\alpha\beta}U^\gamma,
\]
and
\[
T_{\mu\nu}=\frac1{4\pi}\Bigl[g^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma}-\frac14 g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\Bigr].
\]
For a plane EM wave interacting with a TT gravitational wave, the maximum modulus of the coupling coefficient is on the order of \(10^2\), and a typical astrophysical strain \(h_0\sim 10^{-21}\) generates a first-order EM response on the order of \(10^{-19}\) relative to the incident field amplitude [2605.28663]. 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 \(\mathbf E\) and \(\mathbf H\) as components of a dual field,
\[
\mathcal E(\mathbf r,\omega)=
\begin{bmatrix}
\mathbf E(\mathbf r,\omega)\\
Z_0\,\mathbf H(\mathbf r,\omega)
\end{bmatrix},
\qquad
\mathcal S(\mathbf r,\omega)=
i\begin{bmatrix}
Z_0\,\mathbf J_E(\mathbf r,\omega)\\
\mathbf J_M(\mathbf r,\omega)
\end{bmatrix}.
\]
The operator equation is
\[
\bigl(i\,\bar\nabla\times-k_0\bar\varepsilon(\mathbf r,\omega)\bigr)\mathcal E(\mathbf r,\omega)=\mathcal S(\mathbf r,\omega),
\]
with
\[
\bar\nabla\times=
\begin{pmatrix}
0 & \nabla\times\\
-\nabla\times & 0
\end{pmatrix},
\qquad
\bar\varepsilon(\mathbf r,\omega)=
\begin{pmatrix}
\varepsilon(\mathbf r,\omega)&0\\
0&\mu(\mathbf r,\omega)
\end{pmatrix}.
\]
The retarded Green operator
\[
G^{(+)}(\omega)=\lim_{\eta\to 0^+}(\mathcal M-i\eta)^{-1}
\]
has kernel \(g^{(+)}\) satisfying
\[
\mathcal M_{\mathbf r}\,g^{(+)}(\mathbf r,\mathbf r')=\mathbb I_6\,\delta^{(3)}(\mathbf r-\mathbf r'),
\]
and the interior field obeys the Stratton–Chu representation
\[
\mathcal E(\mathbf r)=\int_V d^3r'\,g^{(+)}(\mathbf r,\mathbf r')\mathcal S(\mathbf r')
-i\oint_S dS\, g^{(+)}(\mathbf r,\mathbf s)\,(\bar n\times \mathcal E)(\mathbf s).
\]
The boundary term is intrinsic rather than discarded, and \(g^{(+)}\) propagates both volume sources and tangential surface data [2603.27475].

Two bilinear structures organize the formalism. Under the energy inner product,
\[
\langle \mathcal E_1|\mathcal E_2\rangle=\int_V d^3r\,\mathcal E_1^\dagger(\mathbf r)\mathcal E_2(\mathbf r),
\]
one derives the generalized optical theorem,
\[
g-g^\dagger
=
2ik_0\int_V g^\dagger \bar\varepsilon_I g
-i\oint_S g^\dagger(\bar n\times)g.
\]
Under the reciprocal inner product
\[
\langle \mathcal E_1|\mathcal E_2\rangle_R
=
\int_V d^3r\,\mathcal E_1^T(\mathbf r)\Pi\,\mathcal E_2(\mathbf r),
\qquad
\Pi=\mathrm{diag}(I_3,-I_3),
\]
one obtains Lorentz reciprocity,
\[
g(\mathbf r_1,\mathbf r_2)=\Pi\,g^T(\mathbf r_2,\mathbf r_1)\,\Pi,
\]
for reciprocal media.

Quantization proceeds through a Heisenberg–Langevin treatment. The constitutive relation is
\[
\hat{\mathcal P}(\mathbf r,\omega)=
\bar\chi(\mathbf r,\omega)\hat{\mathcal E}(\mathbf r,\omega)+\hat{\mathcal P}_N(\mathbf r,\omega),
\]
with bulk noise commutator
\[
[\hat{\mathcal P}_{N,i}(\mathbf r,\omega),\hat{\mathcal P}_{N,j}^\dagger(\mathbf r',\omega')]
=
\frac{\hbar}{\pi\varepsilon_0}
(\bar\varepsilon_I)_{ij}(\mathbf r,\omega)
\delta^{(3)}(\mathbf r-\mathbf r')\delta(\omega-\omega').
\]
The operator-valued field solution is
\[
\hat{\mathcal E}(\mathbf r,\omega)
=
k_0\int_V d^3r'\,g(\mathbf r,\mathbf r';\omega)\hat{\mathcal P}_N(\mathbf r',\omega)
-i\oint_S dS\,g(\mathbf r,\mathbf s;\omega)\,(\bar n\times)\hat{\mathcal E}_{\mathrm{in}}(\mathbf s,\omega),
\]
and the total field satisfies the exact closed commutation relation
\[
[\hat{\mathcal E}_i(\mathbf r,\omega),\hat{\mathcal E}_j^\dagger(\mathbf r',\omega')]
=
\frac{\hbar k_0}{\pi\varepsilon_0}\,\mathrm{Im}\,g_{ij}(\mathbf r,\mathbf r';\omega)\,\delta(\omega-\omega').
\]
The same framework yields an exact quantum transfer relation between input and output surfaces,
\[
\hat\Psi_2(\mathbf s_2,\omega)
=
\oint_{S_1}T_{21}(\mathbf s_2,\mathbf s_1;\omega)\hat\Psi_1(\mathbf s_1,\omega)\,dS_1
+\hat\Psi_{N,21}(\mathbf s_2,\omega),
\]
with
\[
T_{21}(\mathbf s_2,\mathbf s_1;\omega)=-\,i\,(\bar n_2\times)\,g(\mathbf s_2,\mathbf s_1;\omega).
\]
This formalism extends first-order Maxwell operators from classical propagation to open-system quantization in absorptive and dispersive photonic structures [2603.27475].

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.

Source: https://www.emergentmind.com/topics/first-order-maxwell-operator-formalism