 Solution

Use $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$. The stated derivative convention gives $[D_{\mu },D_{\nu }]=-ie{F}_{\mu \nu }$. Also $[{\gamma }^{\mu },{\gamma }^{\nu }]=-2i{\sigma }^{\mu \nu }$. Splitting products into symmetric and antisymmetric parts therefore yieldsMultiplying the first-order equation by $i{\gamma }^{\mu }{D}_{\mu }+m$ now givesThe antisymmetric derivative commutator is the electromagnetic field strength. It is the extra term absent when squaring an uncoupled Dirac operator with commuting derivatives, and it generates the spin–magnetic-field interaction in the nonrelativistic limit.