We pursue an investigation of logarithmic electrodynamics, for which the field energy of a point-like charge is finite, as happens in the case of the usual Born–Infeld electrodynamics. We also show that, contrary to the latter, logarithmic electrodynamics exhibits the feature of birefringence. Next, we analyze the lowest-order modifications for both logarithmic electrodynamics and for its non-commutative version, within the framework of the gauge-invariant path-dependent variables formalism. The calculation shows a long-range correction ( \(1/r^5\) -type) to the Coulomb potential for logarithmic electrodynamics. Interestingly enough, for its non-commutative version, the interaction energy is ultraviolet finite. We highlight the role played by the new quantum of length in our analysis.

The propagation of electromagnetic waves in a Lorentz-symmetry violating scenario is investigated in connection with non-linear (photon self-interacting) terms induced by quantum effects. It turns out that the photon field acquires an interesting polarization state and, from our calculations of phase and group velocities, we contemplate different scenarios with physically realizable magnetic fields and identify situations where non-linearity effects dominate over Lorentz-symmetry breaking ones and vice versa.

Background fermion condensates in a landscape dominated by global supersymmetry are reassessed in connection with a scenario where Lorentz symmetry is violated in the bosonic sector (actually, the photon sector) by a CPT-even \(k_F\) term. An effective photonic action is discussed that originates from the supersymmetric background fermion condensates. Also, the photino mass emerges in terms of a particular condensate contrary to what happens in the case of \( k_{AF}\)-violation. Finally, the interparticle potential induced by the effective photonic action is investigated and a confining profile is identified.