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QED with external field: Hamiltonian treatment for anisotropic medium formed by the Lorentz-non-invariant vacuum

Nonlinear electrodynamics, QED included, is considered against the Lorentz-noninvariant external field background, treated as an anisotropic medium. Hamiltonian formalism is applied to electromagnetic excitations over the background, and entities of electrodynamics of media, such as field inductions and intensities, are made sense of in terms of canonical variables. Both conserved and nonconserved generators of space-time translations and rotations are defined on the phase space, and their Hamiltonian equations of motion and Dirac bracket relations, different from the Poincaré algebra, are established. Nonsymmetric, but--in return--gauge-invariant, energy-momentum tensor suggests a canonical momentum density other than the Poynting vector. A photon magnetic moment is found to govern the evolution of the photon angular momentum. It is determined by the antisymmetric part of the energy-momentum tensor.