Ward–Takahashi Identities and Schwinger–Dyson Equations

The classical Noether theorem says a continuous symmetry gives a conserved current. In the quantum theory the corresponding statements are relations among correlation functions: Ward–Takahashi identities (for symmetries) and Schwinger–Dyson equations (the quantum equations of motion). Both follow from a single principle — invariance of the functional integral under a change of integration variable — and both are exact, holding to all orders in perturbation theory. Their failure, when the integration measure is not invariant, is an anomaly.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The master principle: field redefinitions leave $Z$ invariant

A change of integration variable cannot change the value of an integral. In the path integral, shift $\varphi (x)\to \varphi (x)+ϵ(x)$ by an infinitesimal amount. Since $\int \mathcal{D}\varphi$ is unchanged (assuming the measure is invariant — the crucial proviso), the integrand's variation must integrate to zero:

$$0=\int \mathcal{D}\varphi \frac{\delta }{\delta \varphi (x)}[(\cdots )e^{iS[\varphi ]}].$$

Everything below is a special case of this one identity. Which identity results depends on what the shift $ϵ(x)$ represents: an equation of motion (Schwinger–Dyson) or a symmetry transformation (Ward–Takahashi).

Schwinger–Dyson equations

Take the shift to be arbitrary. The resulting identity is the quantum equation of motion: the classical Euler–Lagrange equation holds as an operator insertion inside correlators, up to contact terms:

$$⟨\frac{\delta S}{\delta \varphi (x)}\varphi (y_1)\cdots \varphi (y_n)⟩=i\sum _k⟨\varphi (y_1)\cdots \widehat{\varphi (y_k)}\cdots \varphi (y_n)⟩{\delta }^4(x-y_k).$$

The left side is the classical EOM ($⟨(\square +m^2)\varphi \cdots ⟩$ for the free scalar); the right side is a sum of contact terms where the shifted point collides with an external point. These Schwinger–Dyson equations are the exact, all-orders field equations of the quantum theory — the diagrammatic recursion relations among Green's functions.

Ward–Takahashi identities (Theorem)

Now take the shift to be a symmetry transformation with a spacetime-dependent parameter, $\delta \varphi =ϵ(x){\delta }_{\text{sym}}\varphi$. If the symmetry is exact, the action changes only through the derivative of $ϵ$, by $\int ({\partial }_{\mu }ϵ)j^{\mu }$ with $j^{\mu }$ the Noether current. Invariance of $Z$ then yields the Ward–Takahashi identity — the quantum statement of current conservation inside correlators:

$$\boxed{{\partial }_{\mu }⟨j^{\mu }(x)\varphi (y_1)\cdots \varphi (y_n)⟩=-i\sum _k⟨\varphi (y_1)\cdots {\delta }_{\text{sym}}\varphi (y_k)\cdots \varphi (y_n)⟩{\delta }^4(x-y_k).}$$

Away from the insertion points ($x\ne y_k$) this is just ${\partial }_{\mu }⟨j^{\mu }\cdots ⟩=0$; the contact terms on the right encode how the current "acts" on the fields it passes through — the quantum image of the charge-generates-symmetry relation $[Q,\varphi ]=-\delta \varphi$.

The QED Ward identity

For the $U(1)$ current of QED, the Ward–Takahashi identity relates the photon–fermion vertex ${\Gamma }^{\mu }$ to the electron self-energy $\Sigma$, and in momentum space reduces to the Ward identity on the amputated vertex:

$$q_{\mu }{\Gamma }^{\mu }(p+q,p)=S_F^{-1}(p+q)-S_F^{-1}(p)\stackrel{q\to 0}{\to }{\Gamma }^{\mu }=\frac{\partial }{\partial p_{\mu }}{S}_F^{-1}(p).$$

Three consequences that make QED consistent and calculable:

• Transversality of the photon self-energy, $q_{\mu }{\Pi }^{\mu \nu }(q)=0$, which forbids a photon mass term and protects the photon's masslessness under renormalization.
• $Z_1=Z_2$: the vertex and electron-field renormalization constants are equal, so the renormalized charge is universal (the same $e$ for every charged species) — this is why charge is conserved and quantized identically across the theory.
• Gauge independence of physical amplitudes: the $q_{\mu }{q}_{\nu }$ pieces of the photon propagator drop out against $q_{\mu }{\mathcal{M}}^{\mu }=0$, the check invoked in tree-level examples.

In non-abelian gauge theories the analogous (more intricate) relations are the Slavnov–Taylor identities, derived from BRST symmetry.

When the measure is not invariant: anomalies

Every identity above assumed the integration measure $\mathcal{D}\varphi$ is invariant under the transformation. For chiral transformations of fermions, this assumption fails: the measure $\mathcal{D}\bar{\psi }\mathcal{D}\psi$ picks up a non-trivial Jacobian. The classical current conservation is then violated by a calculable c-number — the anomaly:

$${\partial }_{\mu }{j}^{\mu 5}=\frac{e^2}{16{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }\ne 0.$$

This is the Fujikawa derivation of the chiral anomaly: the Ward identity for the axial current is anomalous precisely because the fermion measure is not chirally invariant. Whether a would-be symmetry is exact or anomalous is thus a question about the measure, answered cleanly only in the path integral.

Summary

Where this leads

• The anomaly in full: the chiral anomaly and anomaly cancellation.
• Non-abelian version: BRST symmetry and Slavnov–Taylor identities.
• Why $Z_1=Z_2$ matters: renormalization and counterterms.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 7.4, 9.6.
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 16–17, 22.
• Srednicki, Quantum Field Theory, Ch. 22, 67, 75.