First order Maxwell operator formalism for macroscopic quantum electrodynamics

Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both mathbf{E} and mathbf{H} and keeps those boundary terms, naturally leading to a quantum input-output formalism. We recast Maxwell's equations as an operator equation for the dual field mathit{E}=\[mathbf{E},mathbf{H}\]^T, whose first-order Green operator g propagates the electromagnetic state between surfaces. Symmetries of the Maxwell operator under energy and reciprocal inner products yield the propagation formula, Lorentz reciprocity, and a generalized optical theorem, with minimal vector calculus. Quantizing via a Heisenberg-Langevin approach for absorptive, dispersive media yields two independent quantum noise sources: bulk Langevin operators from material absorption and input-output field operators at the boundary. Expressing the interior field in terms of these operators and the Green propagator yields an exact closed commutation relation \[{mathit{E}},{mathit{E}}^dagger\]propto Im g, consistent with the fluctuation-dissipation theorem. This identity holds even when dielectrics extend to the boundary, as in waveguide input-output problems, and enables quantum input-output descriptions of complex photonic structures where the Green's function is obtained numerically, extending the framework beyond cavities and waveguides.