Symmetry, currents, and Ward identities

A continuous symmetry becomes useful when it constrains correlation functions, not merely when a classical Lagrangian looks unchanged. The essential move is to replace its constant parameter by a smooth function. The coefficient of the parameter’s derivative identifies a current, while the same localized transformation inside a regulated functional integral produces a Ward identity whose contact terms say exactly how that current acts on inserted operators.

This lesson treats ordinary continuous internal symmetries in Lorentzian QFT. The main example is a complex scalar with global $U(1)$ symmetry. Spacetime symmetries, higher-form symmetries, and gauge-fixed redundancies require additional structures, so their identities should not be inferred by changing labels in the formulas below.

Required background. Classical fields, actions, and local dynamics supplies the bulk–boundary variation and elementary Noether reasoning. Vector fields and gauge redundancy separates a physical global symmetry from a redundancy of description.

Helpful background. Functional integrals and correlators supplies the regulated Lorentzian change-of-variables language, while Fermions, spin, and anticommutation explains why the order of fermionic insertions must remain fixed in a graded calculation.

A localized global transformation identifies the current

Let a continuous group act infinitesimally on fields $\Phi^I$ by

$$\delta_\epsilon\Phi^I =\epsilon^a\Delta_a\Phi^I,$$

where constant $\epsilon^a$ leaves the declared theory data invariant. This means more than invariance of one density: the state, boundary conditions, couplings, operator domain, and sectors under discussion must transform consistently. What is a symmetry of a QFT? develops that operational test.

Replace $\epsilon^a$ by a smooth compactly supported function $\alpha^a(x)$. Fix the current and explicit-breaking conventions by writing

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

The insertion $\mathcal B_a$ records explicit breaking at fixed couplings or sources; it vanishes for an exact symmetry. Compact support removes the surface term, so integration by parts gives

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

On a classical solution, an arbitrary compactly supported field variation has $\delta S=0$. Therefore

$$\partial_\mu j_a^\mu=-\mathcal B_a,$$

and an exact symmetry has $\partial_\mu j_a^\mu=0$ on shell. The sign is not a fact to memorize in isolation: it follows from the displayed definition of $j_a^\mu$. Reversing that definition reverses the current, charge, and every later contact sign together. The localized Noether construction is developed in Schwartz 2014, § 3.3, pp. 32–34 and Weinberg 1995, § 7.3, pp. 306–309.

Promoting the parameter to $\alpha(x)$ is a diagnostic variation. It does not turn a global symmetry into a gauge redundancy and does not introduce a new dynamical gauge field.

A current becomes a charge only with boundary control

For a future-oriented Cauchy surface $\Sigma$, define the candidate charge

$$Q_a[\Sigma] =\int_\Sigma\mathrm d\Sigma_\mu\,j_a^\mu.$$

Let $\Sigma_1$, $\Sigma_2$, and a side boundary $\mathscr B$ enclose a region $\Omega$. The divergence equation gives

$$\begin{aligned} Q_a[\Sigma_2]-Q_a[\Sigma_1] ={}&-\int_\Omega\mathrm d^d x\,\mathcal B_a \\ &-\int_{\mathscr B}\mathrm d\Sigma_\mu\,j_a^\mu. \end{aligned}$$

Even when $\mathcal B_a=0$, the bulk charge is surface-independent only if the side flux vanishes or is balanced by an explicitly included boundary system. In QFT, $j_a^\mu$ is an operator-valued distribution; its spatial integral also needs smearing, a limiting prescription, and an operator domain. A selected symmetry-breaking vacuum can support a conserved local current even when the global charge fails to exist on that Hilbert space.

Fix the active generator convention

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

Equivalently,

$$[Q_a,\mathcal O]=i\,\delta_a\mathcal O.$$

This relation fixes the contact signs below. It requires that the charge actually exists and generates the stated transformation on the operator in question; formal current conservation alone does not prove either fact. The full current-to-charge chain, including surface flux and domain qualifications, is given in Continuous symmetries, generators, and charges.

The quantum current may require composite-operator renormalization and mixing. An improvement

$$j_a^\mu\longmapsto j_a^\mu+\partial_\nu B_a^{[\nu\mu]}$$

has identically vanishing bulk divergence, but it can change local matrix elements and boundary charges. Thus a current should be specified by its normalization, insertion identity, and allowed improvements, not merely by a bare classical formula. See Quantum currents, improvements, and conservation.

A regulated change of variables gives the Ward identity

Work at a finite regulator scale and use the Lorentzian weight $e^{iS}$. Let

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

be an ordered product of regulated local insertions. Initially assume that the transformation is one-to-one, preserves the integration domain and boundary data, and has unit Jacobian. Also assume that the insertions transform without derivatives of $\alpha$:

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

Invariance under a change of integration 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 vanishes as a distribution. Define

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

The local Ward–Takahashi identity is

$$\begin{aligned} \partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle ={}&-\left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle \\ &+i\sum_{k=1}^n \delta^{(d)}(x-x_k) \left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle. \end{aligned}$$

Three signs have distinct origins:

• the minus sign on $\mathcal B_a$ comes from $\partial_\mu j_a^\mu=-\mathcal B_a$;
• the factor $i$ multiplying the contacts comes from the Lorentzian weight $e^{iS}$; and
• the sign of each individual contact comes from the transformation $\delta_a\mathcal O_k$.

The identity is distributional. At $x\ne x_k$, the contacts vanish and one recovers the separated-point divergence equation. At $x=x_k$, differentiating a time-ordered product differentiates its step functions. For one bosonic insertion,

$$\partial_{x^0} \left\langle \mathrm T\{j_a^0(x)\mathcal O(y)\} \right\rangle \supset \delta(x^0-y^0) \left\langle[j_a^0(x),\mathcal O(y)]\right\rangle.$$

Integrating over space turns this into $\delta(x^0-y^0)\langle[Q_a,\mathcal O]\rangle =i\delta(x^0-y^0)\langle\delta_a\mathcal O\rangle$, exactly matching the functional derivation. With fermionic insertions, keep their displayed order; additional minus signs arise only from actual graded reordering. Coincident composite products can also require local counterterms or derivative contacts. The operator and functional derivations are compared in Schwartz 2014, § 14.8, pp. 278–282 and Weinberg 1995, § 10.4, pp. 442–448.

If $\Omega$ contains every insertion, integrating the local identity gives

$$\begin{aligned} \int_{\partial\Omega}\mathrm d\Sigma_\mu\, \left\langle\mathrm T\{j_a^\mu\mathcal X\}\right\rangle ={}&-\int_\Omega\mathrm d^d x\, \left\langle\mathrm T\{\mathcal B_a\mathcal X\}\right\rangle \\ &+i\sum_{k=1}^n \left\langle\mathrm T\{\mathcal X_{a,k}\}\right\rangle. \end{aligned}$$

For an exact symmetry, an invariant state, and zero boundary flux, this reduces to the global selection rule

$$\sum_{k=1}^n \left\langle\mathrm T\{\mathcal X_{a,k}\}\right\rangle=0.$$

A complex scalar fixes every sign

Consider

$$\mathcal L_0 =\partial_\mu\phi^\dagger\partial^\mu\phi -m^2\phi^\dagger\phi -\frac\lambda2(\phi^\dagger\phi)^2,$$

with the global transformation

$$\phi\longmapsto e^{i\alpha}\phi, \qquad \phi^\dagger\longmapsto e^{-i\alpha}\phi^\dagger.$$

The parameter-free infinitesimal variations are

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

Localizing $\alpha$ and keeping only first order gives

$$\delta_\alpha\mathcal L_0 =-j^\mu\partial_\mu\alpha,$$

where

$$j^\mu =i\left[ \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right].$$

The equations of motion give

$$\partial_\mu j^\mu =i\left( \phi^\dagger\Box\phi -(\Box\phi^\dagger)\phi \right)=0.$$

At equal time, with $\pi=\partial_0\phi^\dagger$ and $\pi^\dagger=\partial_0\phi$, the charge is

$$Q=i\int\mathrm d^{d-1}x\, \left(\phi^\dagger\pi^\dagger-\pi\phi\right).$$

The canonical commutators give

$$[Q,\phi]=-\phi, \qquad [Q,\phi^\dagger]=+\phi^\dagger.$$

This agrees with $\delta\mathcal O=-i[Q,\mathcal O]$: $-i[Q,\phi]=+i\phi$ and $-i[Q,\phi^\dagger]=-i\phi^\dagger$.

Now define

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

and

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

The contact at $\phi(y)$ is negative because $i\,\delta\phi=i(i\phi)=-\phi$. The contact at $\phi^\dagger(z)$ is positive because $i\,\delta\phi^\dagger=i(-i\phi^\dagger)=+\phi^\dagger$. Therefore

$$\partial_\mu G_j^\mu(x;y,z) =\left[ \delta^{(d)}(x-z) -\delta^{(d)}(x-y) \right]G(y,z).$$

Away from $y$ and $z$, the divergence is zero. At the insertions, the two delta distributions encode their opposite charge actions. Integrating over $x$ with no boundary flux makes the contacts cancel, as required for a neutral two-point function.

Using the Fourier convention

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

integration by parts gives

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

This is an off-shell correlator identity. It becomes an on-shell amplitude identity only after pole isolation and LSZ amputation; the external-photon replacement test on the preceding page is its simpler scattering counterpart.

Explicit breaking adds a bulk insertion

Add

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\ge2.$$

The localized variation now contains

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

so

$$\partial_\mu j^\mu=-\mathcal B.$$

The two-point Ward identity becomes

$$\begin{aligned} \partial_\mu G_j^\mu(x;y,z) ={}&-\left\langle \mathrm T\{\mathcal B(x)\phi(y)\phi^\dagger(z)\} \right\rangle \\ &+\left[ \delta^{(d)}(x-z)-\delta^{(d)}(x-y) \right]G(y,z). \end{aligned}$$

The bulk term is different from the insertion contacts: it can remain nonzero when $x$ is separated from $y$ and $z$. At fixed nonzero $h$, the continuous $U(1)$ is explicitly broken, although the interaction retains a finite $\mathbb Z_N$ symmetry. That finite selection rule is not obtained by differentiating an infinitesimal current.

Jacobians, boundaries, and anomalies require separate tests

The simple Ward identity relied on a unit regulated Jacobian. To expose that hypothesis, define a regulated measure contribution $\mathcal A_{a,\Lambda}$ by

$$\delta_\alpha\ln\mathcal J_\Lambda =-i\int\mathrm d^d x\, \alpha^a\mathcal A_{a,\Lambda}.$$

With this convention, the regulated identity becomes

$$\begin{aligned} \partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle ={}&-\left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle \\ &+\left\langle \mathrm T\{\mathcal A_{a,\Lambda}(x)\mathcal X\} \right\rangle \\ &+i\sum_k\delta^{(d)}(x-x_k) \left\langle\mathrm T\{\mathcal X_{a,k}\}\right\rangle. \end{aligned}$$

A nontrivial Jacobian is not automatically an anomaly. It may depend on the regulator or be shifted by an allowed local counterterm. A genuine anomaly is the obstruction that remains after the regulator, renormalization prescription, backgrounds, boundary conditions, and allowed local counterterms are fixed. Fujikawa’s regulated measure calculation is the standard local mechanism Fujikawa 1979, pp. 1195–1198, while the counterterm test is summarized in Bilal 2008, § 6.2, pp. 42–43, PDF.

The physical conclusion also depends on which transformation is at issue. An uncanceled anomaly in a dynamical gauge redundancy obstructs the proposed quantum gauge theory. An anomaly of an exact global symmetry is instead an ’t Hooft anomaly: without additional inflow, it obstructs making the background-coupled generating functional invariant under all admitted background gauge transformations, and its class must be matched under renormalization-group flow. Neither conclusion follows from seeing a nonunit formal determinant alone. Use What is an anomaly? for the full obstruction test.

Physical boundaries introduce a different modification. Removing the compact support assumption produces current flux, possible variation of the boundary action or boundary conditions, and possibly boundary degrees of freedom. These terms are classical or quantum boundary balance terms; they are not automatically anomalies.

Finally, a gauge redundancy is not an ordinary global symmetry acting on distinct physical states. After gauge fixing, its quantum identities are organized by BRST symmetry and the Slavnov–Taylor or Zinn–Justin equation, including ghosts and gauge-fixing sources absent from this lesson. Gauge transformations with nontrivial boundary action may instead define physical boundary symmetries, so the allowed transformation group must be stated before applying either language.

Common pitfalls

Writing $\partial_\mu j^\mu=0$ inside every time-ordered product. It holds away from insertions for an exact nonanomalous symmetry. Delta-function contacts at the insertions are required for the charge to act on them.

Inferring a conserved charge from a conserved local current. Boundary flux, infrared behavior, smearing limits, and the operator domain can prevent $Q$ from being surface-independent or well defined.

Changing one sign convention in isolation. Reversing the sign of the current requires corresponding changes to $Q$, its commutator action, and all contact terms. Check the full chain rather than one formula.

Calling every Jacobian an anomaly. A regulated Jacobian is an input to the quantum identity. Only a remainder that cannot be removed by compatible local counterterms represents an anomaly.

Using a global Ward identity for gauge redundancy. Gauge-fixed theories require their BRST and Slavnov–Taylor structure. A gauge choice is not a physical global symmetry.

Treating a bare composite current as an exact operator. Same-point products need a renormalized definition, and derivative or current insertions can add local contact terms beyond the simplest formula.

Exercises

1. Localize the phase rotation of the complex-scalar Lagrangian and derive the current. Then use the equations of motion to verify its divergence.

   Solution

   Use $\delta_\alpha\phi=i\alpha\phi$ and $\delta_\alpha\phi^\dagger=-i\alpha\phi^\dagger$. The potential depends only on $\phi^\dagger\phi$, so its variation vanishes. Differentiating the localized fields produces

   $$\delta_\alpha\mathcal L_0 =i(\partial_\mu\alpha) \left[ (\partial^\mu\phi^\dagger)\phi -\phi^\dagger\partial^\mu\phi \right].$$

   Comparing with $\delta_\alpha\mathcal L_0=-j^\mu\partial_\mu\alpha$ gives

   $$j^\mu=i\left[ \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right].$$

   Its divergence is $i[\phi^\dagger\Box\phi-(\Box\phi^\dagger)\phi]$. The two equations of motion contain the same real factor $m^2+\lambda\phi^\dagger\phi$, so those terms cancel and the divergence is zero. With the $h\phi^N+h^*(\phi^\dagger)^N$ deformation, the uncancelled terms instead give $\partial_\mu j^\mu=-\mathcal B$.

2. Derive the two contact signs in the current-inserted complex-scalar two-point function, first from the general Ward identity and then from the equal-time charge action.

   Solution

   At the $\phi(y)$ insertion, $i\delta(x-y)\delta\phi=i\delta(x-y)(i\phi) =-\delta(x-y)\phi$. At the $\phi^\dagger(z)$ insertion, $i\delta(x-z)\delta\phi^\dagger=i\delta(x-z)(-i\phi^\dagger) =+\delta(x-z)\phi^\dagger$. Hence

   $$\partial_\mu G_j^\mu =\left[\delta^{(d)}(x-z)-\delta^{(d)}(x-y)\right]G.$$

   On the operator side, differentiating the time-ordering step functions produces equal-time commutators. Spatial integration gives $[Q,\phi]=-\phi$ and $[Q,\phi^\dagger]=+\phi^\dagger$, reproducing the same negative and positive contacts. Their integral cancels because the two-point product is neutral.

3. Suppose a regulated calculation produces a local measure term but the current also has nonzero flux through a physical boundary. What must be checked before either effect is called an anomaly?

   Solution

   First write the complete regulated identity, including insertion contacts, explicit breaking, the measure term, current flux, variation of boundary data, and any boundary degrees of freedom. Then determine whether the measure term changes under allowed local counterterms or a different compatible regulator. A removable term is scheme dependence, not an anomaly. A classical boundary flux is a balance term, not an anomaly by itself. Only an unremovable failure of the desired quantum transformation law for the complete bulk–boundary system defines an anomaly. Its interpretation then depends on whether the transformation is a physical global symmetry or a dynamical gauge redundancy.

Where to go next

You are ready to proceed when you can derive a current from a localized variation, obtain the time-ordered identity without losing its contacts, recover those contacts from equal-time commutators, and state separately any explicit-breaking, boundary, or measure contribution.

After also completing the free-fermion branch, continue to Perturbative expansion and Feynman rules. There, Ward identities become practical checks on gauge-dependent intermediate rules and physical amplitudes.

References

• Bilal, Adel. “Lectures on Anomalies.” 2008. arXiv:0802.0634. arXiv.
• Fujikawa, Kazuo. “Path-Integral Measure for Gauge-Invariant Fermion Theories.” Physical Review Letters 42, no. 18 (1979): 1195–1198. DOI.
• 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.