Rigorous Derivation of the Pauli Equation With Time-dependent Electromagnetic Field

In this work we discuss relativistic corrections for the description of charge carriers in aquantum mechanical framework. The fundamental equation is the Dirac equationwhich takes into account also the electron's spin. However, this equation intrinsicallyalso incorporates positrons which play no role in applications in solid state physics. Wegive a rigorous derivation of the Pauli equation describing electrons in a first orderapproximation of the Dirac equation in the limit of infinite velocity of light. We dealwith time-dependent electromagnetic potentials where no rigorous results have beengiven before. Our approach is based on the use of appropriate projection operators forthe electron and the positron component of the spinor which are better suited than thewidely used simple splitting into ‘upper (large)’ and ‘lower (small) component’. We alsosystematically derive corrections at second order in 1/c where we essentially recover theresults of the Foldy-Wouthuysen approach. However, due to the non-static problem,differences occur in the term which couples the electric field with the spin