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, An Introduction to Quantum Field Theory (see references), from the original sense of the Ward identity.
The Ward–Takahashi identity
be a QED correlation function involving an external photon with momentum k (where is the polarization vector of the photon and summation over =0,...,3 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. Note that if has its external electrons on-shell, then the amplitudes on the right-hand side of this identity each have one external particle off-shell, and therefore they do not contribute to S-matrix elements.
The 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 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.
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 represents a gauge transformation by ε (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
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.
- Y. Takahashi, Nuovo Cimento, Ser 10, 6 (1957) 370.
- J.C. Ward, Phys. Rev. 78, (1950) 182
- For a pedagogical derivation, see section 7.4 of Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 0-201-50397-2.