Application of Spinor Analysis to the Maxwell and Dirac Equations

Abstract

With the spinor analysis developed by B. van der Waerden which comprises all representations of the Lorentz group, even those not contained in ordinary tensor calculus, one is able to write all derivations and equations in an automatically covariant form. For the convenient translation into spinor language of the Maxwell equations, it becomes important to introduce three self-dual tensors, one representing the electromagnetic field, one corresponding to the Hertzian vector, and one representing a kind of current potential. These correspond to symmetric spinors of the 2nd rank. Many spinor equations thus become simpler than the corresponding tensorial equations, especially the expression for the stress energy tensor. From the 1st order Dirac equations in spinor form, as given by v.d. Waerden, we derived the 2nd order equation, which agrees with the Gordon-Klein form but for a correction term which again contains the self-dual field tensor. Further the expression for the current was derived, and its decomposition into conduction and polarization currents, and both Maxwell and Dirac equations were derived from a spinorial variation principle, analogous to the results of Gordon and Darwin. In addition to the divergence condition for the current three new invariant relations between the wave functions which are independent of the potentials were obtained (Chapter III, Eqs. (11), (12) and (13)).