From Selection Rules to Ward Identities

Selection rules and Ward identities are two expressions of the same symmetry logic at different levels of description.

In ordinary quantum mechanics, a symmetry can force a matrix element to vanish:

$$\langle f|O|i\rangle=0.$$

In quantum field theory, the same symmetry often appears as an identity among correlation functions:

$$\partial_\mu \langle T\,j^\mu(x)\mathcal O_1(x_1)\cdots\mathcal O_n(x_n) \rangle = \text{contact terms}.$$

The current $j^\mu$ is the local field-theory representative of the conserved generator. The contact terms say how the inserted operators transform. This page explains that bridge. It does not derive full gauge-theory Ward identities, BRST identities, or Slavnov–Taylor identities.

The Matrix-Element Version

Let $U$ be a unitary symmetry. Suppose the states transform as

$$U|i\rangle=u_i|i\rangle, \qquad U|f\rangle=u_f|f\rangle,$$

and the operator transforms as

$$UOU^\dagger=u_OO.$$

Then

$$\begin{aligned} \langle f|O|i\rangle &= \langle f|U^\dagger UOU^\dagger U|i\rangle \\ &= u_f^*u_Ou_i \langle f|O|i\rangle. \end{aligned}$$

A nonzero matrix element therefore requires

$$u_f^*u_Ou_i=1.$$

If this condition fails, the amplitude vanishes. This is the compact symmetry argument behind many selection rules: parity rules, charge rules, angular-momentum rules, and many point-group rules.

Infinitesimal Form

For a continuous symmetry generated by $G$,

$$U(\alpha)=e^{-i\alpha G/\hbar}.$$

If

$$G|i\rangle=g_i|i\rangle, \qquad G|f\rangle=g_f|f\rangle,$$

then

$$\langle f|[G,O]|i\rangle = (g_f-g_i)\langle f|O|i\rangle.$$

If the operator has definite charge under the symmetry,

$$[G,O]=\hbar q_OO,$$

then

$$(g_f-g_i-\hbar q_O) \langle f|O|i\rangle=0.$$

Thus a nonzero matrix element requires

$$g_f=g_i+\hbar q_O.$$

This is the infinitesimal version of a selection rule: the operator must carry exactly the quantum number needed to connect the two states.

From Global Generators to Local Insertions

For a nonanomalous continuous internal symmetry of a local field action, a Noether current and its charge satisfy:

$$\partial_\mu j^\mu=0, \qquad Q(t)=\int d^3x\,j^0(t,\mathbf x).$$

The charge $Q$ generates the symmetry on fields and local operators. The new feature is locality. Instead of asking only how $Q$ acts on a state, one asks what happens when the current is inserted at a spacetime point in a correlation function.

Away from operator insertions, current conservation suggests

$$\partial_\mu \langle T\,j^\mu(x)\mathcal X\rangle =0,$$

where

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

At the insertion points $x=x_i$, the current can act on the operators themselves. Those localized contributions are the contact terms.

Ward Identity and Its Contact Terms

For a bosonic internal current in $d$ spacetime dimensions, suppose the equal-time action is local:

$$[j_a^0(t,\mathbf x),\mathcal O_i(t,\mathbf y)] =\delta^{(d-1)}(\mathbf x-\mathbf y)[Q_a,\mathcal O_i(t,\mathbf y)].$$

With ordinary Minkowski time ordering, differentiating its step functions gives

$$\partial_\mu\langle Tj_a^\mu(x)\mathcal X\rangle =\sum_i\delta^{(d)}(x-x_i) \times\langle T\mathcal O_1\cdots[Q_a,\mathcal O_i]\cdots\mathcal O_n\rangle.$$

Here $\partial_\mu j_a^\mu=0$ is assumed away from insertions. If $\delta_a\mathcal O=(i/\hbar)[Q_a,\mathcal O]$, the right-hand side is $-i\hbar$ times the same sum with $\delta_a\mathcal O_i$. This fixes the convention instead of hiding the coefficient in a schematic equality. Derivative contact terms, anomalous divergences, and renormalized composite insertions need additional care when the displayed equal-time assumption does not apply.

Global Ward Identities as Selection Rules

If the operators have definite charge under a global $U(1)$ symmetry,

$$\delta\mathcal O_i = iq_i\alpha\,\mathcal O_i,$$

then invariance of the vacuum and the action implies

$$0 = \delta \langle \mathcal O_1\cdots\mathcal O_n \rangle = i\alpha \left( \sum_{i=1}^n q_i \right) \langle \mathcal O_1\cdots\mathcal O_n \rangle.$$

Therefore

$$\left( \sum_{i=1}^n q_i \right) \langle \mathcal O_1\cdots\mathcal O_n \rangle =0.$$

A nonzero correlator must have total charge zero, unless the vacuum or boundary conditions violate the assumptions. This is a field-theoretic selection rule. It is the many-operator cousin of the matrix-element condition $g_f=g_i+\hbar q_O$.

For example, if $\phi$ carries charge $+1$, then in a charge-invariant vacuum

$$\langle \phi(x)\phi(y)\rangle=0,$$

while

$$\langle \phi(x)\phi^\dagger(y)\rangle$$

is not forbidden by charge conservation.

Angular Momentum and Tensor Selection Rules

Rotational selection rules in quantum mechanics are usually expressed through tensor operators and the Wigner–Eckart theorem. The theorem says that rotational covariance fixes the angular dependence of matrix elements up to reduced matrix elements.

Field theory keeps the same representation-theoretic idea, but applies it to local fields, local composite operators, currents, and stress-energy tensors. A rotational or Lorentz Ward identity constrains how correlators transform under rotations, boosts, or spacetime translations.

The direct analogy is:

$$\begin{array}{rcl} \text{QM} &:& \text{operator transforms in a representation,} \\ && \text{so only compatible matrix elements survive,} \\ \text{QFT} &:& \text{local operators transform in representations,} \\ && \text{so only compatible correlator structures survive.} \end{array}$$

Ward identities impose rotational covariance on entire correlation functions, extending the same constraint that fixes angular-momentum matrix elements.

A Ward–Takahashi Prototype

The name Ward–Takahashi identity is often used for quantum identities associated with continuous symmetries, especially in theories with charged fields. A typical structural example involves a conserved $U(1)$ current inserted into a two-point function:

$$\partial_\mu \langle T\,j^\mu(x)\psi(y)\psi^\dagger(z) \rangle = \text{contact term at }x=y + \text{contact term at }x=z.$$

The two contact terms encode the fact that $\psi$ and $\psi^\dagger$ carry opposite charges. After Fourier transformation and amputation, such identities can become relations between vertices and propagators. In a familiar QED convention, the schematic momentum-space form is

$$q_\mu\Gamma^\mu(p+q,p) \sim S^{-1}(p+q)-S^{-1}(p),$$

where $S$ is a fermion propagator and $\Gamma^\mu$ is a current or photon vertex. This equation is not needed for ordinary selection rules, but it shows how the same symmetry principle constrains quantum corrections in field theory.

Contact Terms Are the Point

It is tempting to read

$$\partial_\mu j^\mu=0$$

inside a correlator and conclude that the divergence always vanishes. That misses the central feature.

In a time-ordered product, differentiating can also act on the ordering symbols. In a path-integral derivation, changing variables can transform the inserted operators. In both viewpoints, the current is conserved away from the insertions but produces localized terms where it meets an operator.

Those contact terms are not a failure of conservation. They are the local form of the symmetry action:

$$Q \quad \text{acts on} \quad \mathcal O_i.$$

This is why Ward identities are stronger than a bare conservation equation. They know how each operator in the correlator transforms.

From Zeros to Relations

Selection rules often say that an amplitude is zero. Ward identities can do that, but they often do more. They can relate different amplitudes, different tensor structures, or different renormalized quantities.

Examples include:

• charge conservation forcing charged correlators with nonzero total charge to vanish;
• current conservation making longitudinal current insertions determined by contact terms;
• rotational or Lorentz covariance restricting tensor structures;
• global internal symmetry relating correlators of operators in the same multiplet;
• gauge-theory identities relating propagators, vertices, and counterterms.

The first item looks most like an ordinary selection rule. The later items show why field theory uses identities rather than only lists of forbidden transitions.

Assumptions and Failure Modes

Ward identities require assumptions, just as selection rules do.

An exact identity may fail or change when:

• the symmetry is explicitly broken;
• the regulator or quantum measure has an anomaly;
• boundary conditions allow charge flux;
• the vacuum is not invariant under the symmetry;
• sources or background fields transform nontrivially;
• the operator inserted in the correlator is not properly renormalized.

For spontaneous symmetry breaking, the current may still be conserved while the vacuum is not invariant. The corresponding Ward identities then imply gapless modes under suitable hypotheses; small explicit breaking can instead produce light pseudo-Goldstone modes, rather than simply saying that all charged correlators vanish.

For gauge theories, the most useful identities involve gauge fixing, constraints, ghosts, BRST symmetry, or physical gauge-invariant operators. Those refinements belong to quantum field theory proper.

References

• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• Y. Takahashi, “On the Generalized Ward Identity,” Nuovo Cimento 6, 371-375 (1957).
• J. C. Ward, “An Identity in Quantum Electrodynamics,” Physical Review 78, 182 (1950).
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., Oxford University Press, 2002.

Exercises

1. Charge selection rule from a commutator.

Suppose $G|i\rangle=g_i|i\rangle$, $G|f\rangle=g_f|f\rangle$, and $[G,O]=\hbar q_OO$. Show that $\langle f|O|i\rangle$ vanishes unless $g_f=g_i+\hbar q_O$.

Solution

Compute the matrix element of the commutator:

$$\langle f|[G,O]|i\rangle = (g_f-g_i) \langle f|O|i\rangle.$$

Using $[G,O]=\hbar q_OO$ also gives

$$\langle f|[G,O]|i\rangle = \hbar q_O \langle f|O|i\rangle.$$

Equating the two expressions,

$$(g_f-g_i-\hbar q_O) \langle f|O|i\rangle=0.$$

If $g_f-g_i-\hbar q_O\ne0$, the matrix element must vanish.

2. Correlator selection rule.

Three local operators have global $U(1)$ charges $q_1=1$, $q_2=1$, and $q_3=-1$. In a charge-invariant vacuum, what does symmetry say about $\langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle$?

Solution

The total charge is

$$q_1+q_2+q_3 = 1+1-1 =1.$$

The global Ward identity gives

$$(q_1+q_2+q_3) \langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle =0.$$

Since the total charge is nonzero, the correlator must vanish under the stated assumptions.

3. Why do contact terms appear in a current Ward identity?

Solution

The current is conserved away from operator insertions. At an insertion point, however, the charge associated with the current acts on the inserted operator. In a time-ordered derivation this appears when the derivative acts on time-ordering step functions. In a path-integral derivation it appears because the inserted operators transform under the change of variables. The contact terms therefore encode the local action of the symmetry generator.

4. Why is a Ward identity usually stronger than a selection rule?

Solution

A selection rule often says that a particular matrix element is zero because the symmetry labels do not match. A Ward identity can imply such zeros, but it can also relate different correlation functions, constrain tensor structures, and connect vertices with propagators. It is an identity among functions or amplitudes, not just a list of forbidden transitions.