Effective-metric formulation of Casimir energies in nonlinear scalar and electromagnetic theories

Abstract

We study the Casimir effect in nonlinear field theories through the effective geometries that \mbox{govern} their linearized fluctuations. Previous analyses of Lorentz-violating scalar fields showed that a constant kinetic background modifies the parallel-plate Casimir energy by a rescaling of the plate separation and an overall determinant factor. We show that this structure is not merely a consequence of diagonalizing the reduced Green function. It follows from a common Schur-complement structure: after Fourier reduction parallel to the plates, the same reduced quadratic form controls the spectral denominator of the reduced Green function and the numerator generated by the energy-density insertion. This observation allows the Lorentz-violating scalar result to be used as an effective-metric prescription for regular fluctuation sectors arising from the linearization of nonlinear theories around constant backgrounds. In nonlinear scalar theories, the effective tensor is the Hessian of the Lagrangian evaluated on a constant-gradient background. In nonlinear electrodynamics $\mathcal{L}(\mathcal{F})$, a constant magnetic background splits the fluctuations into an ordinary Maxwell branch and an extraordinary optical branch. For this electromagnetic sector, we compute the parallel-plate Casimir energy both by direct mode summation and by applying the effective-metric formula branch by branch, finding exact agreement. The resulting energy depends on the orientation of the magnetic background relative to the plates, providing a concrete anisotropic Casimir response in a regular nonlinear electromagnetic sector.