Constitutive Origin of Hamiltonian Degeneracy in Nonlinear Electrodynamics with Spontaneous Lorentz Symmetry Breaking

TL;DR

This paper reveals the structural origin of Hamiltonian degeneracy in nonlinear electrodynamics with spontaneous Lorentz symmetry breaking, linking it to the constitutive relations and the effective Hamiltonian's properties.

Contribution

It demonstrates that the Hamiltonian degeneracy arises from the constitutive relations and the complementary energy structure in Plebański nonlinear electrodynamics.

Findings

Magnetic stationary points correspond to surfaces where the linearized map loses rank.

The effective Hamiltonian is the complementary energy related to magnetic response.

The magnetic constitutive Jacobian appears as a local block in the Poisson-bracket matrix.

Abstract

In Pleba\'nski nonlinear electrodynamics with spontaneous Lorentz symmetry breaking, nontrivial magnetic backgrounds are selected by stationary points of an effective Hamiltonian. Previous branchwise Hamiltonian analyses showed that this same stationarity requirement coincides with the vanishing of the determinant of the Poisson-bracket matrix among the second-class constraints, but the structural origin of this coincidence was not manifest. We show that it follows from the constitutive origin of the theory. The structural potential \(V(P,Q)\) generates the electromagnetic constitutive relations, while the effective Hamiltonian for magnetic vacua is the complementary energy associated with the magnetic response at fixed \(\Dvec\). Moreover, because the first-order constitutive relation enters the Dirac constraint structure, the magnetic constitutive Jacobian appears as a local block of the Poisson-bracket matrix among the second-class constraints. This complementary-energy structure implies that every nontrivial magnetic stationary point lies on a surface where the linearized map \(\delta\Hvec\mapsto\delta\Bvec\), at fixed \(\Dvec\), loses rank. We use this interpretation to formulate the reduced linearized theory at the vacuum, discuss the removal of the radial mode in the vacuum-restricted theory, and clarify why electric and mixed stationary branches are obstructed in single-invariant models.