Ward–Takahashi identity

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 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 F₁(0), 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.

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,^[3] 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 and summation over $\mu$=0,...,3 is implied), 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

$k_{\mu} \mathcal{M}^{\mu}(k; p_1 \cdots p_n; q_1 \cdots q_n) = e \sum_i \left[ \mathcal{M}_0(p_1 \cdots p_n; q_1 \cdots (q_i-k) \cdots q_n) \right.$

$\left. - \mathcal{M}_0(p_1 \cdots (p_i+k) \cdots p_n; q_1 \cdots q_n) \right]$

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 ${\displaystyle \epsilon _{\mu }(k)}$ 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

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

$\int \delta_\epsilon \left(\mathcal{F} e^{iS}\right) \mathcal{D}\phi = 0$

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,

$\delta_\epsilon S=\int \left(\partial_\mu \epsilon\right) J^\mu \mathrm{d}^dx = -\int \epsilon \partial_\mu J^\mu \mathrm{d}^dx$

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

$\langle \delta_\epsilon \mathcal{F}\rangle - i \int \epsilon \langle \mathcal{F} \partial_\mu J^\mu \rangle \mathrm{d}^dx = 0$

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 then

$\int \delta_\epsilon \left( \mathcal{F} e^{i\left(S+S_{gf}\right)}\right) \mathcal{D}\phi = 0$

where S is the gauge invariant action and S_gf 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 of charge nonconservation.

If the functional measure is not gauge invariant, but happens to satisfy

$\int \delta_\epsilon \left(\mathcal{F} e^{iS}\right) \mathcal{D}\phi = \int \epsilon \lambda \mathcal{F} e^{iS} \mathrm{d}^dx$

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

1. Ward, John Clive (1950). "An Identity in Quantum Electrodynamics". Physical Review. 78: 182. doi:10.1103/PhysRev.78.182.
2. Takahashi, Yasushi (1957). "On the generalized ward identity". Il Nuovo Cimento. 6 (2): 371–375. doi:10.1007/BF02832514.
3. Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. Westview Press. Section 7.4 ("The Ward-Takahashi identity"). ISBN 0-201-50397-2.