Multiplets, Invariants, and Selection Rules

An exact symmetry organizes states and operators into representations, or multiplets. Correlators, amplitudes, matrix elements, and couplings must then be invariant tensors—or, equivalently, intertwiners—built from those representations. If the relevant tensor product contains no singlet, the observable vanishes. If it contains several singlets, symmetry reduces the answer to that many independent structures but does not determine their dynamical coefficients.

This gives a practical method: identify the exact group preserved by the full quantum setup, assign representations to every external object, construct the appropriate tensor product, and project it onto its invariant subspace. The method is exact for an unbroken unitary internal symmetry and an invariant state. Explicit breaking, a selected broken vacuum, antiunitary operations, boundaries, or anomalies change the group or the covariance equation and must be handled before applying the singlet test.

The finite-dimensional multiplets and finite or compact groups treated below admit the stated decomposition and averaging methods. Noncompact groups, infinite-dimensional multiplets, antiunitary actions, and spacetime actions require more careful versions of the invariant-tensor test.

Required background. What Is a Symmetry of a QFT? supplies the physical action, faithful quotient, and distinction between exact symmetry and an invariant presentation. Representations, Intertwiners, Invariants, and Tensor Decomposition supplies tensor products, dual representations, invariant subspaces, intertwiners, and the complete-reducibility conditions used below.

From a symmetry orbit to a multiplet

Let $G$ be an exact unitary internal symmetry, and denote the finite-dimensional representation spaces used for state and operator labels by $V_i$. The multiplets in this section carry genuine representations. If states instead transform projectively, use representations of a common central extension and verify that their multipliers combine consistently, as explained in Quantum Implementations, Projective Actions, and Central Extensions.

Let states $|\alpha,a\rangle$ with fixed physical labels $\alpha$ transform as

$$U(g)|\alpha,a\rangle =D^{(\alpha)}(g)_a{}^b|\alpha,b\rangle.$$

Their span is a representation of $G$. For an internal symmetry that preserves the Hamiltonian,

$$[U(g),H]=0,$$

so $U(g)$ maps an energy eigenstate to states with the same energy. All members of a realized irreducible multiplet therefore share the same energy. A truncation or boundary condition that omits some components is not invariant under the full representation; its spectrum must instead be organized under the subgroup it preserves. Reducible state spaces decompose into irreducible multiplets only under the hypotheses established by the mathematical prerequisite.

Local operators similarly transform in multiplets,

$$U(g)\mathcal O_a(x)U(g)^{-1} =D^{(\mathcal O)}(g)_a{}^b\mathcal O_b(x).$$

The matrices depend on the chosen basis, while the representation type, multiplicities, and dimension of the invariant subspace do not. Discrete unitary operations can also mix degenerate one-particle states; the basis and phase dependence of those mixing matrices is made explicit in Weinberg 1995, Vol. I, Appendix C to Chapter 2, pp. 100–104.

Correlators are invariant tensors

For an invariant vacuum, abbreviate $\boldsymbol a=(a_1,\ldots,a_n)$ and $\boldsymbol x=(x_1,\ldots,x_n)$. Then consider

$$\begin{aligned} G_{\boldsymbol a}(\boldsymbol x) &\equiv \langle\Omega|\mathrm T\{\mathcal X\}|\Omega\rangle,\\ \mathcal X &= \mathcal O^{(1)}_{a_1}(x_1)\cdots \mathcal O^{(n)}_{a_n}(x_n). \end{aligned}$$

Insert $U(g)^{-1}U(g)$ around every operator. The vacuum phases cancel, giving

$$\begin{aligned} G_{a_1\ldots a_n} ={}& D^{(1)}(g)_{a_1}{}^{b_1}\cdots D^{(n)}(g)_{a_n}{}^{b_n}\\ &\times G_{b_1\ldots b_n} \qquad\text{for every }g\in G. \end{aligned}$$

Thus the correlator is canonically an invariant covector,

$$\begin{aligned} G&\in \left(V_1^*\otimes\cdots\otimes V_n^*\right)^G \\ &\cong \operatorname{Hom}_G \left(V_1\otimes\cdots\otimes V_n,\mathbf 1\right). \end{aligned}$$

This proves the basic selection rule:

$$\begin{gathered} \operatorname{Hom}_G \left(V_1\otimes\cdots\otimes V_n,\mathbf 1\right)=0, \\[-2pt] \Downarrow \\[-2pt] G_{a_1\ldots a_n}=0. \end{gathered}$$

Ordinary global symmetries organize spectra into representations and impose Ward identities and selection rules in precisely this sense; see Gaiotto et al. 2015, § 1, pp. 1–2. The result is necessary, not sufficient: the presence of a singlet permits a structure but does not force its coefficient to be nonzero.

Matrix elements and couplings use the same test

Suppose $\mathcal O$ transforms in $V_{\mathcal O}$, while the initial and final state multiplets are $V_{\mathrm{in}}$ and $V_{\mathrm{out}}$. Its matrix elements define an intertwiner

$$\mathcal M: V_{\mathcal O}\otimes V_{\mathrm{in}} \longrightarrow V_{\mathrm{out}}.$$

If

$$\operatorname{Hom}_G \left( V_{\mathcal O}\otimes V_{\mathrm{in}}, V_{\mathrm{out}} \right)=0,$$

the transition is forbidden. If this space has dimension $m$, symmetry allows $m$ independent tensor structures. Kinematics and dynamics determine the corresponding reduced functions or coefficients.

A coupling tensor obeys the same rule. A term

$$\lambda^{a_1\ldots a_n} \mathcal O^{(1)}_{a_1}\cdots \mathcal O^{(n)}_{a_n}$$

is invariant only when $\lambda$ is an invariant tensor in the appropriate dual product. Even then, locality, statistics, Hermiticity, dimensions, anomaly freedom, and renormalization decide whether the coupling is physically admissible or generated.

A reproducible singlet-projector method

Finite groups and compact groups with normalized Haar measure admit the averaging projector used here.

Let $R(g)$ be the tensor-product representation relevant to the observable. For a finite group,

$$P_{\mathrm{inv}} =\frac{1}{|G|} \sum_{g\in G}R(g),$$

while for a compact group,

$$P_{\mathrm{inv}} =\int_G \mathrm d\mu(g)\,R(g), \qquad \int_G\mathrm d\mu(g)=1.$$

Left invariance of the sum or Haar measure gives

$$R(h)P_{\mathrm{inv}}=P_{\mathrm{inv}}, \qquad P_{\mathrm{inv}}^2=P_{\mathrm{inv}}.$$

Therefore its image is exactly the invariant subspace. The number of independent singlets is

$$m_{\mathbf 1} =\operatorname{tr}P_{\mathrm{inv}}.$$

For finite $G$, this supplies the efficient character check

$$m_{\mathbf1} =\frac{1}{|G|} \sum_{g\in G} \chi_R(g).$$

For compact $G$, the corresponding formula is

$$m_{\mathbf1} =\int_G\mathrm d\mu(g) \chi_R(g).$$

If $R=W_1\otimes\cdots\otimes W_n$, then $\chi_R=\prod_i\chi_{W_i}$, with each $W_i$ chosen as the representation or dual representation required by the observable. In particular, the correlator above uses the dual product. A representation and its full dual have the same trivial multiplicity, but constructing the actual invariant covector still requires the dual representation. The underlying definitions of representations, intertwiners, duals, and tensor products are given in Etingof 2020, §§ 4.3 and 11.1 (MIT OpenCourseWare PDF).

Abelian charge conservation from projection

For $U(1)$, an object of integer charge $Q$ transforms by $e^{iQ\alpha}$. Projecting that one-dimensional representation gives

$$\frac{1}{2\pi} \int_0^{2\pi} \mathrm d\alpha\,e^{iQ\alpha} =\delta_{Q,0}.$$

Hence a correlator, matrix element, or coupling can be nonzero only when the signed sum of all charges is zero. For the subgroup $\mathbb Z_N$, the same calculation becomes

$$\frac{1}{N} \sum_{k=0}^{N-1} e^{2\pi i kQ/N} = \begin{cases} 1,&Q=0\pmod N,\\ 0,&Q\ne0\pmod N. \end{cases}$$

Continuous charge conservation and conservation modulo $N$ are therefore two instances of the same singlet projector.

Threaded complex-scalar application

Take the charge-one scalar of the preceding pages,

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

With exact $U(1)$ and an invariant vacuum, the operator

$$\mathcal O_{n,m} =\phi^n(\phi^\dagger)^m$$

has charge $Q=n-m$. Thus $\langle\mathcal O_{n,m}\rangle$ vanishes unless $n=m$. More generally, the signed charges of all insertions in a correlator must sum to zero. The two-point function

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

is allowed, while $\langle\phi(x)\phi(y)\rangle$ is forbidden by the full $U(1)$.

Now add

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

Assigning $h$ charge $-N$ makes the family spurionically covariant, but fixed nonzero $h$ leaves only the faithful $\mathbb Z_N$ symmetry. The exact fixed-theory selection rule is therefore

$$Q=0\pmod N.$$

In particular, $\phi^N$ and an $N$-field correlator become symmetry-allowed, though not guaranteed to be nonzero.

Perturbation theory around the $U(1)$-symmetric theory gives a useful cross-check. At order $h^r(h^*)^s$, the reference-theory correlator contains $r$ insertions of $\phi^N$ and $s$ insertions of $(\phi^\dagger)^N$. An observable of charge $Q$ can contribute only if

$$Q+Nr-Ns=0.$$

This equation implies $Q=0\pmod N$, reproducing the residual-group rule independently. For example, $\langle\phi^N\rangle$ can first receive a contribution proportional to $h^*$, for which $Q=N$, $r=0$, and $s=1$.

A finite non-Abelian check

The two-dimensional real irreducible representation of $S_3\simeq D_3$ can be written using $z=x+iy$. Let its generators act as

$$z\longmapsto\omega z, \qquad z\longmapsto z^*, \qquad \omega=e^{2\pi i/3}.$$

No nonzero linear invariant exists, because multiplication by $\omega$ fixes no nonzero vector. Two invariant polynomials are

$$|z|^2=x^2+y^2, \qquad \operatorname{Re}z^3=x^3-3xy^2.$$

Both survive the rotation because $\omega^3=1$, and both survive reflection by complex conjugation. The first is a rank-two invariant tensor; the second is a symmetric rank-three invariant tensor. This direct generator check shows why a non-Abelian rule is a singlet-multiplicity problem rather than a single additive-charge equation.

Where the simple rule stops

Explicit breaking. Use the stabilizer of the fixed couplings or sources, not the larger spurionic group. The scalar example changes equality of charge to congruence modulo $N$.

Spontaneous breaking. A chosen vacuum may preserve only $H\subset G$. Full-$G$ covariance then relates observables in different vacua; correlators within one selected vacuum obey only the unbroken-$H$ rule.

Anomalies. If a classical transformation is not an exact quantum symmetry, its naive selection rule is replaced by the anomalous Ward identity or by the rule for the surviving subgroup. By contrast, a ’t Hooft anomaly can obstruct gauging an otherwise exact global symmetry without erasing its flat-space selection rules.

Boundaries and defects. A boundary condition may preserve only a subgroup or absorb charge. Apply the invariant-tensor test to the complete bulk-boundary observable, including boundary and defect insertions.

Antiunitary or spacetime transformations. Antiunitary operations conjugate coefficients, and spacetime transformations move insertion points. The simple fixed-tensor equation must be replaced by the covariance law developed in Internal, Spacetime, Discrete, and Antiunitary Symmetries.

Infinite-dimensional or nonsemisimple settings. Do not assume a direct-sum decomposition. The robust statement is the existence or absence of an invariant functional or intertwiner.

Check your understanding

For exact $U(1)$, decide whether each correlator is symmetry-allowed:

$$\langle\phi\phi^\dagger\rangle, \qquad \langle\phi^2\phi^\dagger\rangle, \qquad \langle\phi^N\rangle.$$

Repeat after fixed nonzero $h$ leaves $\mathbb Z_N$, taking $N>2$.

Check

Under $U(1)$ the total charges are $0$, $1$, and $N$, respectively, so only $\langle\phi\phi^\dagger\rangle$ is allowed. Under $\mathbb Z_N$, charges need only vanish modulo $N$: the first and third are allowed, while the second remains forbidden for $N>2$. “Allowed” means symmetry does not force the correlator to vanish; it is not a prediction that the correlator is dynamically nonzero.

What to carry forward

The operational rule is: type the observable as an intertwiner, project its representation space onto singlets, and count the surviving invariant tensors. Zero singlets forbids the observable; $m$ singlets leaves $m$ dynamical structures.

The next treatments separate distinct tasks:

• Continuous Symmetries, Generators, and Charges turns a continuous multiplet action into infinitesimal generators and conserved-charge statements.
• DHR Sectors and Gauge Reconstruction owns rigorous superselection-sector reconstruction.
• Sector Selection, Localization, and Transportability develops the operator-algebraic setting in which sector selection has a different, more technical meaning.

References

• Etingof, Pavel. Lie Groups and Lie Algebras I. MIT OpenCourseWare 18.745, Fall 2020. Official PDF
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI