  The relativistic three dimensional (3D) Lippmann-Schwinger-type equations for
the $ep$ scattering amplitude is derived based on unitarity condition in the
usual quantum electrodynamic (QED). The $ep$ scattering potential $V_{e'N',eN}$
consists of the leading one off mass shell photon exchange part and the
nonlocal multi-particle exchange potential. Unlike to the other
field-theoretical equations, both protons in the unitarity condition and in
$V_{e'N',eN}$ are on mass shell. Therefore in this approach are not required
the multi-variable input photon-nucleon vertexes with the off mass shell
nucleons.
  In the present formulation the standard leading one photon exchange potential
$V_{OPE}$ is generated by the canonical equal-time anti commutator between the
electron source and the interacted electron fields which are sandwiched by the
one nucleon asymptotic states. This anticommutator is calculated in the Coulomb
and Lorentz gauges, where only the transverse parts of the photon fields are
quantized. It is shown, that the leading one photon exchange
potentialn$V_{OPE}$ in the Coulomb and Lorentz gauges coincide. The complete
set of the next to leading order terms which are generated by the static
electric (Coulomb) interaction are exactly reproduced.
