- Ward-Takahashi identity
In

quantum field theory , a**Ward-Takahashi identity**is an identity betweencorrelation function s that follows from the global or gauged symmetries of the theory, and which remains valid afterrenormalization .The Ward-Takahashi identity of

quantum electrodynamics was originally used byJohn Clive Ward andY. Takahashi to relate thewave function renormalization of theelectron to its vertex renormalization factor F_{1}(0), guaranteeing the cancellation of theultraviolet divergence to all orders in perturbation theory. Later uses include the extension of the proof ofGoldstone'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 asMichael Peskin andDaniel 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 particleson-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 thepolarization 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 electronvertex 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 S

_{gf}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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