Lorentz-violating extension of Wigner function formalism and chiral kinetic theory

The quantum kinetic equation for the gauge-invariant Wigner function, constructed from spinor fields that obey the Dirac equation modified by $CPT$ and Lorentz symmetry-violating terms, is presented. The equations for the components of the Wigner function in the Clifford algebra basis are accomplished. Focusing on the massless case, an extended semiclassical chiral kinetic theory in the presence of external electromagnetic fields is developed. We calculate the chiral currents and establish the anomalous magnetic and separation effects in a Lorentz-violating background. The chiral anomaly within the context of extended quantum electrodynamics is elucidated. Finally, we derive the semiclassical Lorentz-violating extended chiral transport equation.