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

On the field theoretical formulation of the electron-proton scattering in the Coulomb and Lorentz gauges