  We discuss two different methods of obtaining ``effective $2 \times 2$
Hamiltonians'' of the electromagnetic interaction which include relativistic
corrections. One is the standard Foldy--Wouthuysen transformation which we
compare with the Hamiltonian obtained from a direct reduction of the matrix
element of the interaction Hamiltonian between positive--energy solutions of
the free Dirac equation. The two approaches are applied to Compton scattering
by a proton for which a low--energy theorem exists. It is found that the
Foldy--Wouthuysen Hamiltonian yields the same result as a covariant
calculation. This is not true for the direct reduction method which will in
general lead to incorrect results even after restoring the gauge invariance
property of the Hamiltonian. Furthermore, it is shown that an identification of
the Z--diagrams of the usual Dirac representation with the contact graphs of
the Foldy--Wouthuysen representation is incorrect beyond the order of the
low--energy theorem.
