Exact Foldy-Wouthuysen transformation for a Dirac spinor in torsion and other CPT and Lorentz violating backgrounds

We discuss the possibility to perform and use the exact Foldy-Wouthuysen transformation (EFWT) for the Dirac spinor coupled to different CPT and Lorentz violating terms. The classification of such terms is performed, selecting those of them which admit EFWT. For the particular example of an axial vector field, which can be associated with the completely antisymmetric torsion, we construct an explicit EFWT in the case when only a timelike component of this axial vector is present. In the cases when EFWT is not possible, one can still use the corresponding technique for deriving the perturbative Foldy-Wouthuysen transformation, as is illustrated in a particular example in the Appendix.