The non-minimal coupling of fermions to a background responsible for the breaking of Lorentz symmetry is introduced in Dirac's equation; the non-relativistic regime is contemplated, and the Pauli's equation is used to show how an Aharonov-Casher phase may appear as a natural consequence of the Lorentz violation, once the particle is placed in a region where there is an electric field. Different ways of implementing the Lorentz breaking are presented and, in each case, we show how to relate the Aharonov-Casher phase to the particular components of the background vector or tensor that realises the violation of Lorentz symmetry. * Electronic address: belich@cce.ufes.br † Electronic address: tcsoares@cbpf.br ‡ Electronic address: manojr@ufma.br § Electronic address: helayel@cbpf.br

The gauge-invariant Chern-Simons-type Lorentz-and CPT-breaking term is here reassessed and a spinprojector method is adopted to account for the breaking ͑vector͒ parameter. Issues such as causality, unitarity, spontaneous gauge-symmetry breaking, and vortex formation are investigated, and consistency conditions on the external vector are identified.

The influence of a Lorentz-violating fixed background on fermions is considered by means of a torsion-free non-minimal coupling. The non-relativistic 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 of external magnetic field), revealing that there appear some energy shifts that modify the fine structure of the spectrum. In the case the non-minimal coupling is torsion-like, 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.

Taking as starting point a Lorentz and CPT non-invariant Chern-Simons-like model defined in 1+3 dimensions, we proceed realizing its dimensional reduction to D = 1 + 2. 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 µ . In spite of breaking Lorentz invariance in the particle frame, this model may preserve the CPT symmetry for a single particular choice of v µ . 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 space-like case.