The Ward identity is a specialization of the Ward–Takahashi identity to S-matrix elements, which describe physically possible scattering processes and thus have all their external particles on-shell. Again let be the amplitude for some QED process involving an external photon with momentum , where is the polarization vector of the photon. Then the Ward identity reads:
Physically, what this identity means is the longitudinal polarization of the photon which arises in the ξ gauge is unphysical and disappears from the S-matrix.
Examples of its use include constraining the tensor structure of the vacuum polarization and of the electron vertex function in QED.
In the path integral formulation, the Ward–Takahashi identities are a reflection of the invariance of the functional measure under a gauge transformation. More precisely, if represents a gauge transformation by (and this applies even in the case where the physical symmetry of the system is global or even nonexistent; we are only worried about the invariance of the functional measure here), then
expresses the invariance of the functional measure where is the action and is a functional of the fields. If the gauge transformation corresponds to a global symmetry of the theory, then,
for some "current" J (as a functional of the fields ) after integrating by parts and assuming that the surface terms can be neglected.
Then, the Ward–Takahashi identities become
This is the QFT analog of the Noether continuity equation .
If the gauge transformation corresponds to an actual gauge symmetry then
where is the gauge invariant action and is a non-gauge-invariant gauge fixing term.
But note that even if there is not a global symmetry (i.e. the symmetry is broken), we still have a Ward–Takahashi identity describing the rate of charge nonconservation.
If the functional measure is not gauge invariant, but happens to satisfy
where is some functional of the fields , we have an anomalous Ward–Takahashi identity, for example when the fields have a chiral anomaly.