Proper-vertex Ward-Takahashi identity in QED ($k_{\mu }{\Gamma }^{\mu }(p+k,p)=S^{-1}(p+k)-S^{-1}(p)$)

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.

In a gauge-preserving regularization and consistent renormalization convention, the proper-vertex Ward-Takahashi identity in QED equates the vertex and electron wave-function renormalization constants. With ${\psi }_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0={\mu }_R^{ϵ}{Z}_{1}e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2ϵ$. Gauge invariance restricts counterterms but does not forbid charge running.