A Maxwell Quadratic-Form Representation of the Parallel-Plate Casimir Trace from Codimension-Three Riesz Reduction

TL;DR

This paper develops a Maxwell-specific quadratic-form representation for the parallel-plate Casimir effect, extending previous scalar results to the physical electromagnetic case with explicit spectral analysis and energy density calculation.

Contribution

It introduces a Maxwell version of the Riesz quadratic-form representation for parallel plates, including spectral properties and energy density in finite volume.

Findings

Finite-volume Maxwell operator has a spectral gap and compact resolvent.

The trace of the Maxwell operator matches a scalar Dirichlet-Neumann channel sum.

Large-area energy density matches the known Casimir result of -π²ħc/(720a³).

Abstract

We formulate a Maxwell version of the codimension-three Riesz/Gaussian quadratic-form representation for perfectly conducting parallel plates. This paper is the Maxwell follow-up to the scalar codimension-three Riesz/Gaussian representation theorem presented earlier in arXiv:2605.06693(2026): the same transverse Riesz reduction and prescribed-covariance quadratic-form mechanism are carried over here to the physical parallel-plate Maxwell operator. The construction is carried out in finite lateral volume $\Omega_{L,a}=T_L^2\times[0,a]$, using the physical electric-field Hilbert space of divergence-free fields satisfying the perfect-conductor tangential condition $n\times E=0$, with the static normal zero mode removed. The Maxwell curl-curl operator is defined by its closed quadratic form, and an explicit Fourier-domain analysis proves the finite-volume spectral gap, compact resolvent, and heat-trace admissibility needed for the stochastic construction. For this reduced Maxwell operator $\mathcal L_{\mathrm{Mx}}$, the codimension-three Riesz integral gives the transversely reduced Riesz mediator $g\mathcal L_{\mathrm{Mx}}^{-1}$. A prescribed heat-regularized Gaussian source with covariance $(\hbar c/g)\mathcal L_{\mathrm{Mx}}^{3/2}e^{-\tau\mathcal L_{\mathrm{Mx}}}$ then has expected quadratic Green energy equal to the heat-regularized physical Maxwell trace. The finite-volume trace is shown to be spectrally equivalent to a scalar Dirichlet channel plus a scalar Neumann channel with its constant zero mode removed. Under the standard parallel-plate interaction finite-part prescription, the large-area energy density is $$ -\frac{\pi^2\hbar c}{720a^3}. $$ The result is a representation theorem for the Maxwell parallel-plate trace under a prescribed covariance in the flat-plate geometry considered here.