Lagrangian approach to the semi-relativistic electron dynamics in the   mean-field approximation
  We derive a mean-field model that is based on a two-component Pauli-like equation and incorporates quantum, spin, and relativistic effects up to second order in $1/c$. Using a Lagrangian approach, we obtain the self-consistent charge and current densities that act as sources in the Maxwell equations. A physical interpretation is provided for the second-order corrections to the sources. The Maxwell equations are also expanded to the same order. The resulting self-consistent model constitutes a suitable semi-relativistic approximation to the full Dirac-Maxwell equations. 
Anant Dixit, Yannick Hinschberger, Jens Zamanian, Giovanni Manfredi
quant-ph