Dispersion relation, propagation features and canonical gauge structure of GNLED

We investigate the propagation properties and canonical gauge structure of Generalised Non-Linear Electrodynamics (GNLED) in the presence of a non-dynamical background field. Starting from the quadratic photon sector of the theory, we perform a field redefinition that converts the background-induced derivative coupling into a mass-like contribution characterized by the vector $V_{\mu }={\mathrm{\partial }}_{\mu }\alpha \mathrm{/}\alpha$ where $\alpha =\sqrt{{\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\mathcal{F}}\mid }_B}$. For a constant background, we derive the corresponding dispersion relation and show that, for a purely space-like $V_{\mu }$, the propagating mode obeys ${\omega }^2=\mathrm{\parallel }\vec{k}{\mathrm{\parallel }}^2+\mathrm{\parallel }\vec{V}{\mathrm{\parallel }}^2$, defining an effective mass scale $m_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{\mid }\vec{V}\mathrm{\mid }$ and a finite rest frequency. The resulting phase and group velocities exhibit the characteristic dispersive behaviour of a gapped mode, with subluminal group velocity and superluminal phase velocity. We then construct the propagator and show that the massive pole resides in the transverse sector, while the longitudinal singularity is associated with gauge fixing. A complete Hamiltonian analysis yields two first-class constraints and only two physical degrees of freedom, demonstrating that the effective mass does not introduce an additional polarisation. The associated gauge transformation is a deformed Abelian $U(1)$ symmetry. Finally, we show that the Hamiltonian is positive definite for purely space-like backgrounds.