Ward–Takahashi identity
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Transcript of Ward–Takahashi identity
Ward–Takahashi identityFrom Wikipedia, the free encyclopedia
See also: Isserlis' theorem
In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gaugedsymmetries 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 wavefunction renormalization of the electron to its vertex renormalization factor F1(0), guaranteeing the cancellation of the ultraviolet divergence to allorders 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 toan 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.
Contents1 The Ward–Takahashi identity2 The Ward identity3 Derivation in the path integral formulation4 References
The Ward–Takahashi identityThe 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), n initial-state electrons with momenta , and n final-state electrons with momenta . Also define to be the simpler amplitudethat 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 thisidentity each have one external particle off-shell, and therefore they do not contribute to S-matrix elements.
The Ward identityThe Ward identity is a specialization of the Ward–Takahashi identity to S-matrix elements, which describe physically possible scatteringprocesses and thus have all their external particles on-shell. Again let be the amplitude for some QED processinvolving 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 fromthe 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 formulationSee 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 gaugetransformation. More precisely, if represents a gauge transformation by ε (and this applies even in the case where the physical symmetry ofthe 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 transformationcorresponds 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 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 ofcharge 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.
ReferencesY. Takahashi, Nuovo Cimento, Ser 10, 6 (1957) 370.J.C. Ward, Phys. Rev. 78, (1950) 182 (http://prola.aps.org/abstract/PR/v78/i2/p182_1)For a pedagogical derivation, see section 7.4 of Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to QuantumField Theory. Westview Press. ISBN 0-201-50397-2.
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