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[1] and Yasushi Takahashi[2] 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,[3] from the original Ward–Takahashi identity.
The detailed discussion below concerns QED, an abelian theory to which the Ward–Takahashi identity applies.
It is closely related to the BRST operator and plays a central role in providing a geometric description of the consistent quantization of gauge theories.
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
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 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.
be the amplitude for some QED process involving an external photon with momentum
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.
(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
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
Gauge-fixing terms are required so as to be able to perform second quantization of a classical gauge theory.
The path-integral (Lagrangian) formulation of quantum field theory does not entirely avoid the need for gauge-fixing, as there is still a need to compute the asymptotic states of the scattering matrix (e.g in the interaction picture.)
In short, gauge-fixing is required, but it breaks the overall gauge invariance of the theory.
The Ward–Takahashi identities then describe exactly how all of the different fields are tied to one-another, under an infinitessimal gauge transformation.
When the functional measure is not gauge invariant, but happens to satisfy with
This example is prominent in the sigma model theory of nuclear forces.
This theory has not one, but two distinct global symmetries: the vector
The breaking is due to quantization, that is, due to the anomalous Ward–Takahashi identity (rather than to a Higgs-style Mexican-hat potential, which results in an entirely different kind of symmetry breaking).
The divergence of the axial current relates the pion-nucleon interaction to pion decay, fixing
In this way, the chiral anomaly provides the canonical description of the pion-nuclean interaction.