= Proper-vertex Ward-Takahashi identity in QED
{title2=$k_\mu\Gamma^\mu(p+k,p)=S^{-1}(p+k)-S^{-1}(p)$}
Define the proper electromagnetic vertex with the <electric charge> removed, and define $S^{-1}$ without the propagator's overall factor $i$. The displayed off-shell identity follows by localizing the vector symmetry in a time-ordered correlation function, including the contact terms at the charged-field insertions, and amputating. Its on-shell form gives the <Ward identity> for physical <photon> amplitudes. At zero <photon> momentum it identifies the vertex with the momentum derivative of the inverse <fermion> propagator.
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