The influence of a Lorentz-violating fixed background on fermions is considered by means of a torsion-free nonminimal coupling. The nonrelativistic regime is assessed and the Lorentz-violating Hamiltonian is determined. The effect of this Hamiltonian on the hydrogen spectrum is determined to first-order evaluation (in the absence and in the presence of an external magnetic field), revealing that there appear some energy shifts that modify the fine structure of the spectrum. In the case the nonminimal coupling is torsionlike, no first order correction shows up in the absence of an external field; in the presence of an external field, a secondary Zeeman effect is implied. Such effects are then used to set up stringent bounds on the parameters of the model.

The gauge-invariant Chern-Simons-type Lorentz- and $\mathrm{CPT}$-breaking term is here reassessed and a spin-projector method is adopted to account for the breaking (vector) parameter. Issues such as causality, unitarity, spontaneous gauge-symmetry breaking, and vortex formation are investigated, and consistency conditions on the external vector are identified.

Taking as a starting point a Lorentz and CPT noninvariant Chern-Simons-like model defined in $1+3$ dimensions, we proceed to realize its dimensional reduction to $D=1+2.\mathrm{}$ One then obtains a new planar model, composed by the Maxwell-Chern-Simons (MCS) sector, a Klein-Gordon massless scalar field, and a coupling term that mixes the gauge field to the external vector $v^{\mu }.$ In spite of breaking Lorentz invariance in the particle frame, this model may preserve CPT symmetry for a single particular choice of $v^{\mu }.$ Analyzing the dispersion relations, one verifies that the reduced model exhibits stability, but the causality can be jeopardized by some modes. The unitarity of the gauge sector is assured without any restriction, while the scalar sector is unitary only in the spacelike case.

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.