Ward identity

In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gauge symmetries of the theory, and which remains valid after renormalization.

The Ward–Takahashi identity of quantum electrodynamics was originally used by John Clive Ward and Yasushi Takahashi to relate the wave function renormalization of the electron to its vertex renormalization factor F₁(0), guaranteeing the cancellation of the ultraviolet divergence to all orders of perturbation theory. Later uses include the extension of the proof of Goldstone's theorem to all orders of perturbation theory.

The Ward–Takahashi identity is a quantum version of the classical Noether's theorem, and any symmetries in a quantum field theory can lead to an equation of motion for correlation functions. This generalized sense should be distinguished when reading literature, such as Michael Peskin and Daniel Schroeder's textbook, from the original sense of the Ward identity.

The Ward–Takahashi identity applies to correlation functions in momentum space, which do not necessarily have all their external momenta on-shell. Let

be a QED correlation function involving an external photon with momentum k (where ${\displaystyle \epsilon _{\mu }(k)}$ is the polarization vector of the photon and summation over ${\displaystyle \mu =0,\ldots ,3}$ is implied), n initial-state electrons with momenta ${\displaystyle p_{1}\cdots p_{n}}$, and n final-state electrons with momenta ${\displaystyle q_{1}\cdots q_{n}}$. Also define ${\displaystyle {\mathcal {M}}_{0}}$ to be the simpler amplitude that is obtained by removing the photon with momentum k from our original amplitude. Then the Ward–Takahashi identity reads

...
Wikipedia