Ward–Takahashi identity

Remarks

The identity follows from current conservation, $\partial_\mu j^\mu = 0$, by Noether's theorem for the global phase symmetry. Inside time-ordered products, $\partial_\mu$ acting on $j^\mu(x)$ produces only contact terms where $x$ meets a charged field. These contact terms give the right-hand side.

Tree-level check: with $\Gamma^\mu = \gamma^\mu$ and $S^{-1}(p) = -i(\gamma^\mu p_\mu - m)$, both sides equal $-i\gamma^\mu k_\mu$.

Consequences: setting $k \to 0$ gives $\Gamma^\mu(p, p) = i\,\partial S^{-1}(p)/\partial p_\mu$, the Ward identity. It implies that the vertex and the electron field strength renormalization constants are equal, $Z_1 = Z_2$. So the electric charge renormalizes only through the photon field, and all charged particles keep equal charges after renormalization. For on-shell amplitudes with an external photon of momentum $k$, written $\mathcal{M}^\mu \epsilon_\mu$, the identity gives $k_\mu \mathcal{M}^\mu = 0$. This is why the unphysical photon polarizations decouple.

Going deeper

For a student

The identity is the quantum form of gauge invariance in QED, and it is a Schwinger–Dyson equation for a symmetry variation. It is used to check calculations and to show that QED is renormalizable with a gauge-invariant set of counterterms. In non-abelian gauge theories, the analogous identities are the Slavnov–Taylor identities, which involve the ghost fields.