Relativistic treatment of the energy shifts caused by static electromagnetic effects on free electrons

Magnetic effects on free electron systems have been studied extensively in the context of spin-to-orbital angular momentum conversion. Starting from the Dirac equation, we derive a fully relativistic expression for the energy of free electrons in the presence of a spatiotemporally constant, weak electromagnetic field. The expectation value of the maximum energy shift, which is completely independent of the electron spin-polarization coefficients, is computed perturbatively to first order. This effect is orders of magnitude larger than that predicted by the quantum mechanical Zeeman shift. We then show, in the non-relativistic limit, how to discriminate between achiral and completely polarized states and discuss possible mesoscopic and macroscopic manifestations of electron spin states across many orders of magnitude in the physical world.