Ward-Takahashi identity


Ward-Takahashi identity

In 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 electrodynamics was originally used by John Clive Ward and Y. 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 in 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

The Ward-Takahashi identity applies to correlation functions in momentum space, which do not necessarily have all their external momenta on-shell. Let

::mathcal{M}(k; p_1 cdots p_n; q_1 cdots q_n) = epsilon_{mu}(k) mathcal{M}^{mu}(k; p_1 cdots p_n; q_1 cdots q_n)

be a QED correlation function involving an external photon with momentum k (where epsilon_{mu}(k) is the polarization vector of the photon), "n" initial-state electrons with momenta p_1 cdots p_n, and "n" final-state electrons with momenta q_1 cdots q_n. Also define mathcal{M}_0 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

::k_{mu} mathcal{M}^{mu}(k; p_1 cdots p_n; q_1 cdots q_n) = -e sum_i left [ mathcal{M}_0(p_1 cdots p_n; q_1 cdots (q_i-k) cdots q_n) ight. :::::::::::::: left. - mathcal{M}_0(p_1 cdots (p_i+k) cdots p_n; q_1 cdots q_n) ight]

where "−e" is the charge of the electron. Note that if mathcal{M} 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 mathcal{M}(k) = epsilon_{mu}(k) mathcal{M}^{mu}(k) be the amplitude for some QED process involving an external photon with momentum k, where epsilon_{mu}(k) is the polarization vector of the photon. Then the Ward identity reads:

:: k_{mu} mathcal{M}^{mu}(k) = 0

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

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 delta_epsilon 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

:int delta_epsilon left(mathcal{F} e^{iS} ight) mathcal{D}phi = 0

expresses the invariance of the functional measure where S is the action and mathcal{F} is a functional of the fields. If the gauge transformation corresponds to a "global" symmetry of the theory, then,

:delta_epsilon S=int left(partial_mu epsilon ight) J^mu mathrm{d}^dx = -int epsilon partial_mu J^mu mathrm{d}^dx

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

:langle delta_epsilon mathcal{F} angle - i int epsilon langle mathcal{F} partial_mu J^mu angle mathrm{d}^dx = 0

This is the QFT analog of the Noether continuity equation partial_mu J^mu=0.

If the gauge transformation corresponds to an actual gauge symmetry,

:int delta_epsilon left( mathcal{F} e^{ileft(S+S_{gf} ight)} ight) mathcal{D}phi = 0

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

:int delta_epsilon left(mathcal{F} e^{iS} ight) mathcal{D}phi = int epsilon lambda mathcal{F} e^{iS} mathrm{d}^dx

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.

References

*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
publisher=Westview Press
year=1995
id = ISBN 0-201-50397-2


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