Quantum transport theory with vector interaction

We derive the relativistic quantum kinetic equation for massless fermions with vector and axial vector interaction using the Wigner function formalism. The vector and axial vector currents are self-consistently treated with corresponding constraint equations. The kinetic equations are derived and the condition for equilibrium is discussed up to the first order of $\hbar$. In addition to the vorticity and shear contributions, the divergence of the local particle number density is also found to contribute to the kinetic equations.