(Semi)-Nonrelativistic Limits of the Dirac Equation with External Time-Dependent Electromagnetic Field

We perform a mathematical study of the limit of infinite velocity of light for the Dirac equation with given time-dependent electromagnetic potential. Our approach is based on the use of appropriate projection operators for the electron and the positron component of the spinor which are better suited than the widely used simple splitting into “upper (large)” and “lower (small) component”. The “semi-nonrelativistic limit” yields the approximation by the Pauli-equation for the electron component of the 4-spinor where first order corrections are kept. Like in the Foldy–Wouthuysen approach we use a rescaling of time to subtract the rest energy of the electron component and add it for the positron component which is assumed to be “small” initially. We give also rigorous results for the nonrelativistic limit to the Schrödinger equation. In this case we keep the symmetry of electron and positron components in the rescaling, thus avoiding smallness assumptions on the initial data, and obtain a decoupled pair of Schrödinger equations (with negative mass for the positron component). Convergence results for the relativistic current are included.