Ward–Takahashi 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 (QED) was originally used by John Clive Ward and Yasushi Takahashi to relate the wave function renormalization of the electron to its vertex renormalization factor, 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. More generally, a Ward–Takahashi identity is the quantum version of classical current conservation associated to a continuous symmetry by Noether's theorem. Such symmetries in quantum field theory (almost) always give rise to these generalized Ward–Takahashi identities which impose the symmetry on the level of the quantum mechanical amplitudes. This generalized sense should be distinguished when reading literature, such as Michael Peskin and Daniel Schroeder's textbook, from the original Ward–Takahashi identity. The detailed discussion below concerns QED, an abelian theory to which the Ward–Takahashi identity applies. The equivalent identities for non-abelian theories such as quantum chromodynamics (QCD) are the Slavnov–Taylor identities. 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 is the polarization vector of the photon and summation over is implied), n initial-state electrons with momenta , and n final-state electrons with momenta . Also define 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 where e is the charge of the electron and is negative in sign.