The interaction Hamiltonian for the Dirac equation can be deduced in several ways.
The simplest for now is to just use the same interaction term that we had for electromagnetism
and identify the probability current multiplied by the charge (-e) as the current that couples to the EM field.
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 no longer an term with a different power of
This will make our perturbation series also a series in powers of
We may still assume that
is transverse and that
by choice of gauge.