Casimir effect near spontaneously Lorentz-breaking magnetic vacua in Plebański nonlinear electrodynamics

Abstract

We study the Casimir response of electromagnetic fluctuations near magnetic vacua that spontaneously break Lorentz symmetry in gauge-invariant nonlinear electrodynamics. The theory is formulated in the Plebański first-order representation, with a single-invariant Hamiltonian potential $\widehat V(P)$ taken as the fundamental nonlinear object. This formulation is particularly useful because the nontrivial vacua are obtained as stationary points of the effective Hamiltonian, rather than as extrema of $\widehat V(P)$ itself. In the magnetic branch, the symmetry-breaking condition is governed by $S_m(P) \equiv \widehat V_P(P)+2P\widehat V_{PP}(P)$, whose vanishing also signals the degeneracy of the Hamiltonian constraint structure and the loss of rank of the longitudinal magnetic response. We first linearize around a regular purely magnetic background $\bar P$, with $S_m(\bar P)\neq0$, and obtain an ordinary Maxwell-like branch together with an extraordinary anisotropic branch controlled by $α(\bar P)=\widehat V_P(\bar P)/S_m(\bar P)$. We then compute the regularized parallel-plate Casimir energy for magnetic backgrounds perpendicular and parallel to the plates. As the regular background approaches the exact Lorentz-breaking vacuum $P_\star$, where $S_m(P_\star)=0$, the extraordinary branch becomes singular and, in the parallel configuration, the Casimir energy diverges within the regular-sector description. A direct analysis on the degenerate surface shows, however, that the extraordinary branch does not survive as an independent physical propagating mode for generic momenta. The divergence is therefore not a prediction of an infinite vacuum energy at the exact Lorentz-breaking state, but a diagnostic of the noncommutativity between quantizing the regular theory and imposing $S_m(P_\star)=0$ before quantization.