Macroscopic Quantum Electrodynamics (MQED)

• Macroscopic quantum electrodynamics (MQED) is a quantum-field theory that quantizes electromagnetic fields in linear, causal, dispersive, and absorbing media using dyadic Green tensors and bosonic noise operators.
• It bridges multiple formulations—canonical, noise-current, and integral—to accurately capture dispersion, absorption, and causal response functions as dictated by Kramers–Kronig relations.
• Recent advances include a first-order operator reformulation that unifies field quantization with input–output descriptions, enhancing both numerical modeling and analytical insight in open photonic systems.

Macroscopic quantum electrodynamics (MQED) is the quantum-field-theoretic framework for electromagnetic fields in arbitrary linear, causal, dispersive, and absorbing macroscopic media. In its standard form, MQED expresses field operators, light–matter couplings, dissipation, Lamb shifts, spontaneous-emission rates, and fluctuation-induced forces in terms of the classical dyadic Green tensor of Maxwell’s equations together with bosonic reservoir operators that encode material fluctuations and enforce the fluctuation–dissipation theorem (Liu et al., 5 Mar 2026, Philbin, 2010). Recent work has also reformulated MQED directly at the level of the first-order Maxwell operator acting on the dual field $[\mathbf{E},Z_0\mathbf{H}]^T$, retaining boundary terms and thereby producing a native quantum input–output description for open photonic systems (Agarwal et al., 29 Mar 2026).

1. Core formal structure

In MQED, the medium is specified by macroscopic response functions such as $\varepsilon(\mathbf{r},\omega)$ and, when required, $\mu(\mathbf{r},\omega)$ or more general magnetoelectric tensors. The central classical object is the dyadic Green tensor, defined for inhomogeneous magnetodielectrics by

$$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$

with outgoing-wave or retarded boundary conditions (Philbin, 2011, Liu et al., 5 Mar 2026).

The standard field quantization of absorbing media introduces bosonic noise operators. In a purely dielectric formulation, the electric-field operator can be written schematically as

$$\hat{\mathbf{E}}(\mathbf{r},\omega) = i\sqrt{\frac{\hbar}{\pi\varepsilon_0}\frac{\omega^2}{c^2}} \int d^3\mathbf{r}'\, \sqrt{\mathrm{Im}\,\varepsilon(\mathbf{r}',\omega)}\, \mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\!\cdot\!\hat{\mathbf{f}}(\mathbf{r}',\omega) +\mathrm{h.c.},$$

with

$$[\hat{f}_i(\mathbf{r},\omega),\hat{f}_j^\dagger(\mathbf{r}',\omega')] = \delta_{ij}\,\delta(\mathbf{r}-\mathbf{r}')\,\delta(\omega-\omega')$$

(Liu et al., 5 Mar 2026, Das et al., 2020). In magnetodielectric media the same structure is generalized to electric and magnetic noise sectors, and the field can equivalently be expressed via a noise current and the Green tensor (Philbin, 2010, Philbin, 2011).

A defining identity of MQED is that field fluctuations are controlled by the anti-Hermitian part of the Green tensor. In standard passive media,

$$\langle \hat{\mathbf{E}}(\mathbf{r},\omega)\hat{\mathbf{E}}^\dagger(\mathbf{r}',\omega')\rangle \propto \mathrm{Im}\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\,\delta(\omega-\omega'),$$

and corresponding noise-current correlators are proportional to $\mathrm{Im}\,\varepsilon$ or $\mathrm{Im}\,\mu$, implementing the fluctuation–dissipation theorem (Philbin, 2011, Oue, 25 Feb 2026). This is why MQED places radiative observables and mechanical observables under a single formal object: the field correlation function determined by $\mathbf{G}$ (Oue, 25 Feb 2026).

2. Canonical, reservoir, and integral formulations

A long-standing obstacle for canonical quantization in macroscopic media is that dispersion and absorption spoil naive mode expansions. Canonical MQED resolves this by coupling the electromagnetic field to continua of harmonic-oscillator reservoir fields representing dissipative material channels. In the canonical magnetodielectric theory, the Hamiltonian diagonalizes into bosonic operators,

$\varepsilon(\mathbf{r},\omega)$0

and the phenomenological Green-tensor prescriptions emerge from that canonical construction rather than being postulated independently (Philbin, 2010). Philbin’s canonical framework similarly derives the Casimir energy density and stress tensor for arbitrary inhomogeneous magnetodielectrics directly from a field theory with an action, Hamiltonian, commutators, and Noether stress tensor (Philbin, 2011).

Microscopic and mesoscopic reservoir models provide complementary realizations of the same logic. The Huttner–Barnett dielectric model treats the medium polarization as a harmonic field coupled to a bath; integrating out the damped polaritons yields an exact displacement-field propagator in the presence of a dispersive and absorbing dielectric half-space, and the resulting noise-current commutator matches phenomenological MQED (Eberlein et al., 2012). A related Hopfield-type integral formulation for finite dispersive dielectric objects expresses the electromagnetic field operators directly as retarded integrals over the polarization density operator, reducing the Heisenberg dynamics to a closed integral equation for the polarization operator and enabling direct reuse of classical integral-equation solvers in open, absorbing environments (Forestiere et al., 2022).

These formulations are mathematically distinct but physically aligned. They all encode the same three structural requirements: causal response functions obeying Kramers–Kronig relations, bosonic reservoir variables representing loss channels, and a Green-operator solution of the macroscopic Maxwell problem. This suggests that “canonical MQED,” “noise-current MQED,” and “integral-equation MQED” are best regarded as different realizations of one response-theoretic framework rather than competing theories (Philbin, 2010, Forestiere et al., 2022).

3. First-order Maxwell operator MQED

A recent reformulation recasts MQED directly as a first-order operator theory for the dual electromagnetic field

$\varepsilon(\mathbf{r},\omega)$1

with

$\varepsilon(\mathbf{r},\omega)$2

(Agarwal et al., 29 Mar 2026). Here both $\varepsilon(\mathbf{r},\omega)$3 and $\varepsilon(\mathbf{r},\omega)$4 are kept on equal footing, unlike the usual second-order electric-field formulation.

The first-order formalism is organized by two adjoint structures. Under the energy inner product, the departure from self-adjointness splits exactly into a bulk absorption term proportional to $\varepsilon(\mathbf{r},\omega)$5 and a surface-flux term. Under the reciprocal bilinear pairing, reciprocal media satisfy a Maxwell-operator symmetry that yields Lorentz reciprocity and the Green-kernel symmetry

$\varepsilon(\mathbf{r},\omega)$6

(Agarwal et al., 29 Mar 2026). The same framework produces a generalized optical theorem,

$\varepsilon(\mathbf{r},\omega)$7

whose anti-Hermitian part partitions dissipation into bulk absorption and radiative flux through the boundary (Agarwal et al., 29 Mar 2026).

Quantization proceeds through a Heisenberg–Langevin construction with two independent noise sectors: bulk Langevin operators from material absorption and input–output field operators on the boundary. The interior field operator becomes

$\varepsilon(\mathbf{r},\omega)$8

and the exact closed commutator is

$\varepsilon(\mathbf{r},\omega)$9

This identity remains valid even when structured dielectrics extend to the boundary, including waveguide input–output configurations (Agarwal et al., 29 Mar 2026).

The formal consequence is significant: propagation, reciprocity, power balance, commutation relations, and input–output transfer are all encoded in the same first-order Green operator. A plausible implication is that numerically computed first-order resolvents can serve simultaneously as classical propagators and as the kernels of quantum fluctuation theory in complex open devices.

4. Boundaries, gauges, and open-system structure

Open-system MQED is subtle because boundary terms are not a dispensable technicality. In the usual second-order, “volume-only” Langevin noise formalism, finite lossy objects embedded in vacuum are handled cleanly only by taking the limit $\mu(\mathbf{r},\omega)$0 at the end of the calculation. If one sets $\mu(\mathbf{r},\omega)$1 strictly in vacuum, the field in those regions would vanish because scattering modes are not separated from medium-assisted modes (Ciattoni, 2024). A modified Langevin noise formalism resolves this by decomposing the field as

$\mu(\mathbf{r},\omega)$2

with one bosonic sector for medium-assisted polaritons and a second bosonic sector for scattering polaritons (Ciattoni, 2024). The key integral identity contains both a volume-loss term and a surface term built from the far-field amplitude of the Green tensor, and the scattering contribution balances that surface term exactly (Ciattoni, 2024).

Gauge issues at boundaries exhibit a related structure. Near a polarizable surface, the generalized Coulomb gauge

$\mu(\mathbf{r},\omega)$3

is technically natural, whereas the true Coulomb gauge demands

$\mu(\mathbf{r},\omega)$4

everywhere (Zietal et al., 2019). The explicit gauge transformation between them introduces an operator-valued scalar potential generated by fluctuating surface charge density, and the true-Coulomb-gauge Hamiltonian acquires an extra interaction term

$\mu(\mathbf{r},\omega)$5

(Zietal et al., 2019). Nevertheless, the total electrostatic interaction energy is gauge invariant, and the paper shows that only gauge-dependent quantities such as $\mu(\mathbf{r},\omega)$6 are altered by the presence of boundaries; commutators of physical fields such as $\mu(\mathbf{r},\omega)$7 are unchanged (Zietal et al., 2019).

The first-order Maxwell-operator theory makes the same point in a different language. By keeping the tangential dual trace $\mu(\mathbf{r},\omega)$8 explicitly, it generates surface-to-surface transfer kernels

$\mu(\mathbf{r},\omega)$9

and in the lossless case these satisfy a pseudo-unitarity condition preserving the surface symplectic metric (Agarwal et al., 29 Mar 2026). This suggests that boundary input channels in MQED are not an auxiliary construction; they are part of the exact bookkeeping required by open Maxwell dynamics.

5. Applications and computational practice

Because MQED reduces quantum observables to Green tensors, it has become a unifying computational language across nanophotonics, dispersion forces, and open quantum systems. In canonical magnetodielectrics, thermal and zero-point field correlations generate the Casimir energy density and stress tensor for arbitrary inhomogeneous media, and the resulting expressions reproduce the standard Lifshitz pressure while remaining fully quantum-field-theoretic inside media (Philbin, 2011).

For molecular dispersion interactions, MQED expresses the van der Waals potential of two polarizable particles directly as a frequency integral over polarizabilities and the Green tensor. In a fullerene dimer, this produces analytic distance-, orientation-, and anisotropy-dependent formulas such as

$$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$0

and comparison with DFT shows that the MQED dipole model reproduces the long-range tail while missing the short-range Pauli wall and higher-multipole structure (Das et al., 2020).

In open quantum dynamics, MQED-QD operationalizes the standard Green-tensor workflow for exciton transport in arbitrary dielectric and plasmonic environments. Its pipeline is explicit: construct $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$1 analytically or from electromagnetic solvers, map it to coherent couplings

$$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$2

and collective decay rates

$$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$3

then propagate the resulting Lindblad dynamics (Liu et al., 5 Mar 2026). The reported silver-nanorod example shows that long-range couplings mediated by surface plasmon polaritons enhance mean-square displacement and participation ratio relative to planar geometries (Liu et al., 5 Mar 2026).

In quantum nanophotonics, MQED also supports exact basis reductions. “Emitter-centered modes” provide an exact, complete, and minimal basis for multi-emitter problems, with one bright continuum per emitter and decoupled dark modes, all constructed directly from $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$4 (Feist et al., 2020). When the Purcell spectrum is Lorentzian, the same Green-tensor information can be mapped without free parameters to a pseudomode Lindblad model, yielding analytically equivalent dynamics to the original mQED wavefunction approach (Wang et al., 2021). These developments show that MQED is not only a formal framework for quantization; it is also a bridge between full-wave numerics and reduced open-system models.

6. Assumptions, extensions, and recurrent controversies

Most MQED constructions assume linearity, causality, passivity, and usually spatial locality. Reciprocity is often assumed, but it is not fundamental. A conductivity-tensor formulation treats the most general linear, absorbing media, including nonlocal and Onsager-violating responses, and bases quantization on $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$5 rather than on reciprocal constitutive tensors (Buhmann et al., 2011). For local bianisotropic media, a canonical mode-expansion theory exists for inhomogeneous magneto-electric response consistent with Kramers–Kronig and Onsager relations, with constitutive couplings encoded in a $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$6 susceptibility block and explicit polariton eigenmodes (Judge et al., 2012). In that setting, duality invariance becomes continuous precisely when nonreciprocal magnetoelectric responses are allowed (Buhmann et al., 2011).

Time dependence introduces a sharper complication. Simply replacing $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$7 by $$\nabla\times\!\Big[\mu^{-1}(\mathbf{r},\omega)\,\nabla\times\mathbf{G}(\mathbf{r},\mathbf{r}',\omega)\Big] -\frac{\omega^2}{c^2}\,\varepsilon(\mathbf{r},\omega)\,\mathbf{G}(\mathbf{r},\mathbf{r}',\omega) =\mathbf{I}\,\delta(\mathbf{r}-\mathbf{r}'),$$8 in standard MQED produces nonphysical polarization currents; for a time-dependent Drude model, a step in carrier density causes the noise polarization and its current to become singular (Horsley et al., 2024). The consistent extension instead modulates reservoir dynamics, not the field–reservoir coupling, yielding a causal two-time susceptibility and finite nonequilibrium noise-current correlators with additional “temporal reflection” correlations (Horsley et al., 2024). Active media require an analogous modification of the noise sector: gain channels contribute through a creation-like term in the noise current, and consistent MQED requires that all poles of the Green function remain in the lower half-plane (Oue, 25 Feb 2026).

Two recurring controversies are thereby clarified. First, boundary effects do not imply that physical field commutators are altered by the presence of bodies; gauge-dependent and field-decomposition-dependent quantities are the ones that change (Zietal et al., 2019). Second, macroscopic quantization in open or finite systems cannot be reduced to bulk loss alone; surface flux, scattering channels, or boundary input operators must be retained if the formalism is to remain exact (Ciattoni, 2024, Agarwal et al., 29 Mar 2026).

Recent work on dispersion forces pushes this further by allowing the internal spectra of the interacting objects themselves to respond self-consistently to electromagnetic backaction. Within mQED, a self-consistent dressing of the polarizabilities can lead to substantial, long-ranged modifications of effective van der Waals interactions through repeated photon-mediated scattering processes, exposing a limitation of perturbative dispersion theories with fixed spectra (Fiedler, 4 May 2026). This suggests that MQED is increasingly being used not merely to quantize fields in prescribed media, but to treat self-consistent matter–field dressing at the level of macroscopic response.