and identify the probability current multiplied by the charge (-e) as the current that couples to the EM field.

Removing the from the left and from the right and dotting into , we have the interaction Hamiltonian.

Note the difference between this interaction and the one we used in the non-relativistic case. The relativistic interaction has just one term, is linear in , and is naturally proportional to the coupling . There is

We may still assume that
is **transverse** and that
by choice of gauge.

Jim Branson 2013-04-22