Rigorous derivation of the Pauli equation with time-dependent electromagnetic field

In this work we discuss relativistic corrections for the description of cha rge carriers in a quantum mechanical framework. The fundamental equation is the Dirac equation which takes into account also the electron's spin. Howe ver, this equation intrinsically also incorporates positrons which play no role in applications in solid state physics. We give a rigorous derivation of the Pauli equation describing electrons in a first order approximation o f the Dirac equation in the limit of infinite velocity of light. We deal wi th time-dependent electromagnetic potentials where no rigorous results have been given before. Our approach is based on the use of appropriate project ion operators for the electron and the positron component of the spinor whi ch are better suited than the widely used simple splitting into 'upper (lar ge)' and 'lower (small) component'. We also systematically derive correctio ns at second order in lie where we essentially recover the results of the F oldy-Wouthuysen approach. However, due to the non-static problem, differenc es occur in the term which couples the electric field with the spin.