Symmetry Groups and Representations

A symmetry group tells you which transformations can be composed. A representation tells you how those transformations act on a vector space, especially a Hilbert space of quantum states. In quantum mechanics, this distinction is essential: the same abstract group can act on ordinary vectors, wavefunctions, spinors, tensor operators, multiparticle states, or degenerate eigenspaces in different ways.

For the mathematical definitions used below, see Groups, Group Actions, Representations, and Unitary Representations.

Group Action on States

A group $G$ is a collection of transformations with composition, identity, and inverses. A quantum symmetry assigns to each group element $g\in G$ an operation on physical states.

For ordinary unitary symmetries one writes

$$g \longmapsto U(g),$$

where $U(g)$ is a unitary operator on a Hilbert space $\mathcal H$. The representation property is

$$U(g_1g_2) = U(g_1)U(g_2), \qquad U(e)=I.$$

This equation says that group composition is mirrored by operator composition. If one performs $g_2$ and then $g_1$, the Hilbert-space operator is $U(g_1)U(g_2)$.

Quantum mechanics adds two important refinements:

• antiunitary symmetries, such as time reversal, are antilinear rather than unitary;
• because pure states are rays, some symmetries are represented projectively, so composition may hold up to a phase.

Those refinements do not remove the central idea: symmetry becomes useful when its composition law is represented on the spaces where quantum states and operators live.

Which space transforms?

Naming the group is not enough. One must say what space carries the representation.

For example, rotations can act on:

• ordinary vectors in $\mathbb R^3$;
• scalar wavefunctions $\psi(\mathbf r)$;
• spherical harmonics of fixed $\ell$;
• spin-$1/2$ spinors in $\mathbb C^2$;
• spin-$j$ multiplets of dimension $2j+1$;
• tensor operators such as dipoles and quadrupoles.

All of these involve rotation symmetry, but they are not the same representation. The group may be $SO(3)$ or its double cover $SU(2)$, while the representation space changes with the physical degree of freedom.

This is why “the rotation group” is not a complete answer to a symmetry problem. A calculation needs the group, the representation space, the action on that space, and the conventions for phases and bases.

Reducible and Irreducible Representations

A subspace $W\subset\mathcal H$ is invariant under a representation if

$$U(g)W\subset W$$

for every $g\in G$. A representation is reducible if it has a nonzero proper closed invariant subspace (closedness is automatic in finite dimensions). It is irreducible if it has no such subspace.

Irreducible representations, or irreps, are the elementary symmetry types. They are the pieces into which many quantum symmetry problems decompose. Every finite-dimensional unitary representation splits into orthogonal irreducible subspaces. Compact-group continuous unitary representations also decompose into finite-dimensional irreducible types. In the finite-dimensional case, write

$$\mathcal H = \bigoplus_\lambda (M_\lambda\otimes V_\lambda),$$

where $V_\lambda$ carries the irrep type $\lambda$ and $M_\lambda$ counts its independent copies. The group acts as $I_{M_\lambda}\otimes U_\lambda(g)$.

The multiplicity label is different from a basis label within $V_\lambda$. A rotation can mix the magnetic labels in one multiplet without mixing its separate copies.

Multiplets

A multiplet is a set of states that transform together under a symmetry. In representation language, a multiplet is usually a basis for an irreducible representation, or for a physically selected copy of one.

For rotations, the spin-$j$ multiplet has basis states

$$\lvert j,m\rangle, \qquad m=-j,-j+1,\ldots,j,$$

and dimension

$$2j+1.$$

The label $j$ identifies the irrep of $SU(2)$ or, for integer $j$, an ordinary representation of $SO(3)$. The label $m$ chooses a basis vector within that irrep after selecting a quantization axis.

This is the conceptual meaning of angular momentum labels. They are not merely names for eigenvalues; they say how states transform.

Quantum Numbers as Representation Labels

A quantum number can have several roles. Some quantum numbers label eigenvalues of commuting observables. Some label irreducible representations. Some label basis vectors inside an irrep. Some separate repeated copies of the same symmetry type.

For angular momentum:

Label	Meaning
$j$	irrep label for a spin or angular-momentum multiplet
$m$	basis label inside the $j$ multiplet
additional labels such as $n$ or $\alpha$	distinguish different copies or radial/dynamical structure

For translations on the line, momentum $p$ labels a one-dimensional irreducible representation of the translation group:

$$T(a)\lvert p\rangle = e^{-iap/\hbar} \lvert p\rangle.$$

On the infinite line, $|p\rangle$ is a generalized eigenvector, not a normalizable Hilbert-space vector, and the decomposition involves an integral over $p$. Periodic boundary conditions make the allowed momenta discrete; a lattice instead identifies crystal momentum modulo reciprocal lattice vectors.

The practical rule is: ask whether a label tells you how the state transforms under a symmetry, or whether it comes from dynamics beyond the symmetry alone.

Symmetry and Hamiltonians

If a Hamiltonian is invariant under a unitary representation of $G$,

$$U(g)HU(g)^\dagger = H$$

for every $g\in G$, then the Hamiltonian commutes with the symmetry action:

$$HU(g) = U(g)H.$$

Therefore each energy eigenspace is invariant under the representation. If

$$H\lvert E,a\rangle = E\lvert E,a\rangle,$$

then

$$H\,U(g)\lvert E,a\rangle = E\,U(g)\lvert E,a\rangle.$$

This is why degeneracies and multiplets often appear together. A nontrivial irrep may force an energy eigenspace to contain several states that transform into one another.

There are cautions:

• ordinary complex unitary irreps of abelian groups are one-dimensional, so symmetry need not force degeneracy;
• degeneracy can be accidental or dynamical rather than forced by an obvious symmetry;
• a symmetry may organize a degenerate subspace without making every vector in that subspace invariant.

The Hamiltonian side is developed further in Symmetry Constraints on Hamiltonians.

Example: Translations

The translation group of the line is abelian:

$$a,b\in\mathbb R, \qquad a+b=b+a.$$

Its unitary representation on wavefunctions is

$$(T(a)\psi)(x) = \psi(x-a),$$

or, in generator form,

$$T(a) = \exp\left( -\frac{iaP}{\hbar} \right).$$

Momentum eigenstates transform by characters:

$$T(a)\lvert p\rangle = e^{-iap/\hbar} \lvert p\rangle.$$

The phase factor is the one-dimensional representation label. This is the representation-theoretic meaning of momentum conservation in a translation-invariant problem.

Example: Rotations and Spin

Rotations in ordinary three-dimensional space form $SO(3)$. Quantum spinors are naturally represented by $SU(2)$, the double cover of $SO(3)$. This distinction permits half-integer spin representations, although it does not by itself predict which spins occur in nature.

For spin-$1/2$,

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i\theta}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma \right)$$

acts on a two-dimensional Hilbert space. This is the $j=1/2$ irrep of $SU(2)$.

For general angular momentum, the irrep label $j$ gives dimension $2j+1$. Integer $j$ representations descend to ordinary $SO(3)$ representations; half-integer $j$ representations are representations of $SU(2)$ and projective representations of $SO(3)$ on rays.

Example: Adding Angular Momenta

When two systems are combined, the symmetry acts on a tensor product. For angular momenta $j_1$ and $j_2$,

$$V_{j_1}\otimes V_{j_2} \cong \bigoplus_{J=\lvert j_1-j_2\rvert}^{j_1+j_2} V_J.$$

The sum advances in steps of one. It splits the product into irreducible total-angular-momentum sectors; there is just one sector if either factor has zero angular momentum. Clebsch–Gordan coefficients are the change-of-basis amplitudes between the uncoupled product basis and the coupled irrep basis.

The detailed construction belongs to Coupled and Uncoupled Bases and Clebsch–Gordan Coefficients.

Exercises

1. The parity group has two elements, $e$ and $\Pi$, with $\Pi^2=e$. What are the two one-dimensional unitary irreps?

Solution

In a one-dimensional unitary representation, $U(\Pi)$ is a phase and must satisfy

$$U(\Pi)^2 = U(e) = 1.$$

Thus $U(\Pi)=+1$ or $U(\Pi)=-1$. These are the even and odd parity irreps.

2. Decompose the tensor product of $j_1=1$ and $j_2=1/2$ into total-angular-momentum irreps and check dimensions.

Solution

The allowed total angular momenta are

$$J = \lvert 1-1/2\rvert,\ldots,1+1/2,$$

so

$$J=1/2,\;3/2.$$

The product dimension is

$$(2\cdot1+1)(2\cdot1/2+1)=3\cdot2=6.$$

The decomposed dimensions are

$$(2\cdot1/2+1)+(2\cdot3/2+1)=2+4=6.$$

3. Why is $m$ not the irrep label for a spin-$j$ multiplet?

Solution

For fixed $j$, the states with different $m$ values are basis vectors inside the same irrep. Rotations generally mix them. The label $j$ identifies the irrep type and its dimension $2j+1$; $m$ labels a chosen basis after selecting the $z$ axis.

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chapter 3.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003 reprint of the 1964 edition.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.