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.
Contents
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.
Ward–Takahashi 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
where −e is the charge of the electron. Note that if
Ward identity
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
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.
Derivation in the path integral formulation
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
expresses the invariance of the functional measure where S is the action and
for some "current" J (as a functional of the fields φ) 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 S is the gauge invariant action and Sgf 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 λ is some functional of the fields φ, we have an anomalous Ward–Takahashi identity. This happens when we have a chiral anomaly, for example.