- Ward-Takahashi identity
quantum field theory, a Ward-Takahashi identity is an identity between correlation functions that follows from the global or gauged symmetries of the theory, and which remains valid after renormalization.
The Ward-Takahashi identity of
quantum electrodynamicswas originally used by John Clive Wardand Y. Takahashito relate the wave function renormalizationof the electronto its vertex renormalization factor F1(0), guaranteeing the cancellation of the ultraviolet divergenceto all orders in perturbation theory. Later uses include the extension of the proof of Goldstone's theoremto 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 Peskinand Daniel Schroeder's textbook, "An Introduction to Quantum Field Theory" (see references), from the original sense of the Ward identity.
The Ward-Takahashi identity
The Ward-Takahashi identity applies to correlation functions in momentum space, which do not necessarily have all their external momenta
be a QED correlation function involving an external photon with momentum k (where is the
polarizationvector of the photon), "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
The Ward identity
The Ward identity is a specialization of the Ward-Takahashi identity to
S-matrixelements, 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 k, where is the polarizationvector 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 gaugeis unphysical and disappears from the S-matrix.
Examples of its use include constraining the tensor structure of the
vacuum polarizationand of the electron vertex functionin QED.
Derivation in the path integral formulation
See also: 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
expresses the invariance of the functional measure where S is the action and is a functional of the fields. If the gauge transformation corresponds to a "global" symmetry of the theory, then,
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,
where S is the gauge invariant action and Sgf is a nongauge 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.
* [http://prola.aps.org/abstract/PR/v78/i2/p182_1 J.C. Ward, "Phys. Rev." 78, (1950) 182]
*For a pedagogical derivation, see section 7.4 of cite book
author=Michael E. Peskin and Daniel V. Schroeder
title=An Introduction to Quantum Field Theory
id = ISBN 0-201-50397-2
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