Localized Transformations and Ward–Takahashi Identities

A Ward–Takahashi identity is the insertion-level form of a continuous symmetry. One replaces the constant symmetry parameter by a smooth function, performs the resulting transformation as a regulated change of integration variables, and uses the arbitrariness of that function. The divergence of the current is then related to delta-supported variations of every inserted operator.

This is stronger than the separated-point statement $\partial_\mu j^\mu=0$. The contact terms are not defects in conservation: they are precisely how the current acts on local operators. The derivation also exposes its own limits. It requires a well-defined regulator, a controlled measure Jacobian, invariant integration data, and an honest account of time ordering and composite insertions.

The derivation concerns ordinary continuous global symmetries. The localized parameter is a diagnostic change of variables, not a promotion of the symmetry to a gauge redundancy.

Required background. Quantum Currents, Improvements, and Conservation supplies the normalized current insertion and explicit-breaking operator. Changes of Variables and Regulated Jacobians supplies the regulated measure and Jacobian argument. The Generating Functional supplies Lorentzian time-ordered correlators with sources.

Regulated localized variation

Use the Lorentzian weight $e^{iS}$ and fix

$$U(\epsilon)=e^{-i\epsilon^aQ_a}, \qquad \delta_a\mathcal O=-i[Q_a,\mathcal O].$$

Work first at a finite regulator scale $\Lambda$, and suppress the subscript $\Lambda$ in intermediate formulas. Let

$$\mathcal X =\mathcal O_1(x_1)\cdots\mathcal O_n(x_n)$$

be a product of regulated local insertions. Localize the infinitesimal transformation:

$$\delta_\alpha\Phi^I(x) =\alpha^a(x)\Delta_a\Phi^I(x),$$

where $\alpha^a(x)$ is smooth and compactly supported. Fix the current convention by writing the action variation as

$$\delta_\alpha S =-\int\mathrm d^d x\, j_a^\mu\partial_\mu\alpha^a +\int\mathrm d^d x\, \alpha^a\mathcal B_a.$$

The insertion $\mathcal B_a$ records explicit breaking. For an exact symmetry it vanishes. After integrating by parts, compact support gives

$$\delta_\alpha S =\int\mathrm d^d x\, \alpha^a \left(\partial_\mu j_a^\mu+\mathcal B_a\right).$$

For now assume each insertion transforms without derivatives of the localized parameter:

$$\delta_\alpha\mathcal O_k(x_k) =\alpha^a(x_k)\delta_a\mathcal O_k(x_k).$$

Derivative operators or noncovariant composite representatives can also produce derivatives of $\alpha$; those terms become additional derivative contact distributions and must not be discarded.

Deriving the local identity

Assume that the regulated transformation is one-to-one, preserves the integration domain and boundary conditions, and has unit Jacobian. Invariance of an integral under a change of variables gives

$$0 =\left\langle\delta_\alpha\mathcal X\right\rangle +i\left\langle \mathrm T\{\mathcal X\,\delta_\alpha S\} \right\rangle.$$

Because $\alpha^a(x)$ is arbitrary, its coefficient must vanish as a distribution. The result is

$$\begin{aligned} \partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle ={}&-\left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle \\ &+\mathcal C_a(x;\mathcal X), \end{aligned}$$

where the contact distribution is

$$\begin{aligned} \mathcal C_a(x;\mathcal X) ={}&i\sum_{k=1}^n \delta^{(d)}(x-x_k) \\ &\quad\times \left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle, \end{aligned}$$

where

$$\mathcal X_{a,k} =\mathcal O_1(x_1)\cdots \big(\delta_a\mathcal O_k(x_k)\big) \cdots\mathcal O_n(x_n).$$

The factor of $i$ follows from the Lorentzian weight $e^{iS}$ and the site’s active-generator convention. Integrating across a thin time slab containing $x_k$, with spatial smearing and no side flux, recovers $[Q_a,\mathcal O_k]=i\delta_a\mathcal O_k$, so the contact sign agrees with the charge action.

Schwartz derives the invariant-measure change of variables and its delta-supported insertion terms in Schwartz 2014, § 14.8.1, pp. 278–279. In the QED current/Dirac-field example, Weinberg gives an independent operator derivation from time ordering, current conservation, and equal-time commutators in Weinberg 1995, Vol. I, § 10.4, pp. 442–448.

At $x\neq x_k$, the contact sum vanishes and the identity reduces inside such correlators to the separated-point equation

$$\begin{aligned} &\partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle \\ ={}&-\left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle, \qquad x\neq x_k. \end{aligned}$$

At coincidence, it is incorrect to move $\partial_0$ through the time-ordering symbol as though the step functions were constant. Their derivatives generate the same equal-time contact terms. Contact Terms, Equal-Time Commutators, and Schwinger Terms develops that operator-side calculation.

If a symmetry-preserving renormalized limit exists, the same identity holds with normalized renormalized insertions $[j_a^\mu]_R$, $[\mathcal B_a]_R$, and renormalized operator variations. The regulated identity alone does not prove that this limit exists.

The integrated identity and selection rules

Integrate the local identity over a region $\Omega$ containing every insertion. If the current flux through its boundary vanishes, then

$$\begin{aligned} &\sum_{k=1}^n \left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle \\ ={}&-i\int_\Omega\mathrm d^d x\, \\ &\quad\times \left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle. \end{aligned}$$

For an exact symmetry, $\mathcal B_a=0$, and the sum of insertion variations vanishes. This is the global Ward identity. For operators in charge eigenrepresentations it becomes a selection rule: a correlator can be nonzero only if the product contains an invariant component.

This conclusion assumes that the state or path-integral boundary conditions are themselves invariant. A selected symmetry-breaking state, a physical boundary, or nonzero asymptotic flux changes the integrated statement even when the bulk current is locally conserved.

Momentum contraction is still off shell

For an exact symmetry define

$$\begin{aligned} G_a^\mu(x;\{x_i\}) &=\left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle, \\ C_{a,k}(\{x_i\}) &=\left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle. \end{aligned}$$

With the site Fourier convention

$$\widetilde G_a^\mu(p;\{x_i\}) =\int\mathrm d^d x\, e^{+ip\cdot x}G_a^\mu(x;\{x_i\}),$$

integration by parts gives $\widetilde{\partial_\mu G^\mu}=-ip_\mu\widetilde G^\mu$. The local identity therefore becomes

$$p_\mu\widetilde G_a^\mu(p;\{x_i\}) =-\sum_{k=1}^n e^{+ip\cdot x_k}C_{a,k}(\{x_i\}).$$

This is a momentum-contraction identity for correlation functions at generic momenta. It is off shell: external propagators are still present, and no mass-shell or polarization condition has been imposed. To obtain an on-shell Ward identity for scattering amplitudes, one must establish the relevant pole structure, amputate with LSZ, and show that the contact contributions behave appropriately. Schwartz separates these steps explicitly in Schwartz 2014, §§ 14.8.2–14.8.3, pp. 279–282.

Nonlinear currents, derivative couplings, or insertions of additional currents can generate extra local or “seagull” terms in a momentum-space identity. The simple contraction formula applies only when the preceding transformation law exhausts the localized variation.

Threaded complex-scalar identity

For the exact complex-scalar $U(1)$,

$$j^\mu =i\left( \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right),$$

and the parameter-free variations are

$$\delta\phi=+i\phi, \qquad \delta\phi^\dagger=-i\phi^\dagger.$$

Let

$$G(y,z) =\left\langle \mathrm T\{\phi(y)\phi^\dagger(z)\} \right\rangle.$$

Also define the current-inserted correlator

$$G_j^\mu(x;y,z) =\left\langle \mathrm T\{j^\mu(x)\phi(y)\phi^\dagger(z)\} \right\rangle.$$

The exact local identity is

$$\begin{aligned} \partial_\mu G_j^\mu(x;y,z) ={}&\big[ \delta^{(d)}(x-z) \\ &-\delta^{(d)}(x-y) \big]G(y,z). \end{aligned}$$

The minus sign at a $\phi$ insertion agrees with $[Q,\phi]=-\phi$; the plus sign at a $\phi^\dagger$ insertion agrees with $[Q,\phi^\dagger]=+\phi^\dagger$. After Fourier transforming only $x$,

$$\begin{aligned} p_\mu\widetilde G_j^\mu(p;y,z) =i\big(&e^{+ip\cdot z} \\ &-e^{+ip\cdot y}\big)G(y,z). \end{aligned}$$

For a product containing $n_+$ fields $\phi$ and $n_-$ fields $\phi^\dagger$, the integrated exact identity gives

$$(n_+-n_-) \left\langle\mathrm T\{\mathcal X\}\right\rangle=0.$$

Now restore the controlled deformation

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2.$$

In the spurion-covariant renormalized convention used on the preceding page,

$$[\mathcal B]_R =iN\left( h[\phi^N]_R -h^*[(\phi^\dagger)^N]_R \right).$$

The local identity acquires the insertion $-\langle\mathrm T\{[\mathcal B]_R(x)\mathcal X\}\rangle$. The continuous $U(1)$ selection rule is therefore broken. The exact residual $\mathbb Z_N$ can still impose charge conservation modulo $N$, but that finite-group rule is not obtained by differentiating an infinitesimal current identity. Treating $h$ as a transforming source instead organizes a covariant family; its functional form belongs to Current Sources and Generating Functionals.

Failure tests and boundaries of the formula

Measure Jacobian. Unit Jacobian is a hypothesis at the regulated level. A nontrivial local Jacobian adds another term to the identity; Regulated Jacobians and Measure Variation develops that anomalous case.

Symmetry-breaking regulator. If local symmetry-restoring counterterms exist as the regulator is removed, the renormalized Ward identity must include them. If no compatible restoration exists, the obstruction is not erased by a formal unregulated change of variables.

Composite and derivative insertions. Their localized transformations can contain derivatives of the parameter, operator mixing, and coincident-product counterterms. Symmetry-Protected Operators, Currents, and Improvement develops the detailed renormalized identities.

Time ordering and coincident support. The displayed equation is distributional. Equal-time commutators, Schwinger terms, and derivative contacts cannot be reconstructed by checking only $x\neq x_k$.

Boundaries and noninvariant states. Compact support avoided a spacetime boundary in the derivation. Removing that restriction requires the actual flux and boundary variation; a selected noninvariant state can also invalidate the global selection rule.

Gauge-fixed identities. A gauge redundancy is not an ordinary physical global symmetry. Its gauge-fixed functional identity is the Slavnov–Taylor and Zinn-Justin identity, with ghosts, antifield sources, and BRST structure that are absent here.

Check your understanding

Starting from the general local identity, reproduce the two contact signs in the complex-scalar two-point function. Fourier transform only the current position using $e^{+ip\cdot x}$, and then integrate over $x$ to recover the charge-neutrality rule.

Check

At the $\phi(y)$ insertion, $i\delta(x-y)\delta\phi(y) =i\delta(x-y)(i\phi(y)) =-\delta(x-y)\phi(y)$. At the $\phi^\dagger(z)$ insertion, $i\delta(x-z)\delta\phi^\dagger(z) =i\delta(x-z)(-i\phi^\dagger(z)) =+\delta(x-z)\phi^\dagger(z)$. This gives the displayed position-space identity.

Because $\int\mathrm d^d x\,e^{+ipx}\partial_\mu G^\mu =-ip_\mu\widetilde G^\mu$, its Fourier transform is

$$p_\mu\widetilde G_j^\mu =i\left( e^{+ip\cdot z}-e^{+ip\cdot y} \right)G.$$

Integrating the position-space identity makes the two contacts cancel. More generally, each $\phi$ contributes $+i$ to the infinitesimal variation and each $\phi^\dagger$ contributes $-i$, so invariance gives $(n_+-n_-)\langle\mathrm T\mathcal X\rangle=0$.

What to carry forward

The Ward–Takahashi identity contains three distinct pieces: a current divergence, any explicit breaking insertion, and the contact-supported transformations of local operators. Its momentum-space form remains an off-shell correlator identity until pole and LSZ hypotheses are supplied.

The next page, Contact Terms, Equal-Time Commutators, and Schwinger Terms, resolves the contact distributions from the operator side. A later structural treatment, Ward Identities and Anomalous Obstructions, explains how such identities and their obstructions are organized in causal perturbation theory.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI