Groups, Lie Algebras, and Representations

Groups encode composable transformations. Group actions say what those transformations act on, while representations realize them as linear operators on vector spaces. Lie groups add smooth parameters, and Lie algebras capture their infinitesimal generators. This hierarchy organizes quantum numbers, degeneracies, spin, angular momentum addition, selection rules, particle exchange, and canonical commutation relations.

The mathematical language lives here. Physical symmetry principles, conserved quantities, spin experiments, and angular-momentum spectroscopy remain canonical in Symmetry, Angular Momentum, and Spin.

Group, action, and representation

A group $G$ has an associative product, identity, and inverses. A left action on a set $X$ is a map

$$G\times X\longrightarrow X, \qquad (g,x)\longmapsto g\cdot x,$$

satisfying

$$e\cdot x=x, \qquad (g_1g_2)\cdot x =g_1\cdot(g_2\cdot x).$$

The orbit of $x$ records configurations reachable by the action; the stabilizer records transformations leaving $x$ fixed. The same abstract group can act differently on spatial points, fields, state vectors, or observables.

A representation is a homomorphism

$$D:G\longrightarrow GL(V), \qquad D(g_1g_2)=D(g_1)D(g_2).$$

The matrices $D(g)$ depend on a basis of $V$, while equivalence of representations is basis-independent. An invariant subspace is preserved by every $D(g)$. A representation is irreducible when it has no nontrivial invariant subspaces.

Groups, Group Actions, and Representations form the foundational route.

Unitary and projective structure

A unitary representation satisfies

$$D(g)^\dagger D(g)=I,$$

so the group action preserves inner products and transition probabilities. For compact groups, finite-dimensional representations can be made unitary by averaging an inner product over the group.

Quantum states are rays, so a physical symmetry may be represented projectively:

$$U(g_1)U(g_2) =e^{i\omega(g_1,g_2)}U(g_1g_2).$$

Passing to a covering group or central extension can turn some projective representations into ordinary representations. This is central to spin and the $SU(2)$ cover of $SO(3)$. Unitary Representations gives the Hilbert-space framework; projective and Wigner-theorem physics remain in the symmetry volume.

Lie groups and generators

A Lie group is both a smooth manifold and a group, with smooth multiplication and inversion. Its Lie algebra is the tangent space at the identity equipped with a bracket. For a unitary representation near the identity, this volume uses Hermitian generators $T_a$:

$$U(\boldsymbol\theta) =\exp(-i\theta^aT_a),$$

with

$$[T_a,T_b] =if_{ab}{}^cT_c.$$

Mathematics texts often use anti-Hermitian generators, moving the factor of $i$ into the definition. Structure constants, exponential signs, and normalization must therefore be translated together.

The exponential map captures transformations connected to the identity, but it need not describe every global component or be one-to-one. The Lie algebra determines local group structure; it does not by itself determine global topology. Read Lie Groups and Lie Algebras as a pair.

SO(3), SU(2), and spinors

$SO(3)$ is the group of real $3\times3$ matrices with

$$R^{\mathsf T}R=I, \qquad \det R=1.$$

$SU(2)$ is the group of complex $2\times2$ matrices with

$$U^\dagger U=I, \qquad \det U=1.$$

Using Pauli matrices, a spinor rotation is

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

Conjugation of $\mathbf r\cdot\boldsymbol\sigma$ by $U$ produces a three-dimensional rotation of $\mathbf r$. The map $SU(2)\to SO(3)$ is two-to-one: $U$ and $-U$ produce the same spatial rotation. Consequently,

$$U(2\pi)=-I, \qquad U(4\pi)=I$$

in the spin-$1/2$ representation. The sign after $2\pi$ is a global phase for an isolated spinor but can become observable relative to another path or sector.

Use SO(3), SU(2), and SU(2) versus SO(3) in that order.

Angular momentum representations

The angular-momentum algebra is

$$[J_i,J_j] =i\hbar\epsilon_{ijk}J_k.$$

Its Casimir operator $J^2=J_x^2+J_y^2+J_z^2$ commutes with all generators. Finite-dimensional unitary irreducible representations are labeled by

$$j=0,\frac12,1,\frac32,\ldots,$$

with basis states satisfying

$$J^2|j,m\rangle =\hbar^2j(j+1)|j,m\rangle, \qquad J_z|j,m\rangle =\hbar m|j,m\rangle.$$

Ladder operators $J_\pm=J_x\pm iJ_y$ move $m$ by one while preserving $j$. Positivity of norms terminates the ladder and determines $m=-j,-j+1,\ldots,j$.

Angular Momentum Algebra summarizes the representation labels and commutators. Ladder Operators as Lie Algebra Tools extracts the general raising-and-lowering mechanism. Physical orbital and spin realizations remain canonical in the symmetry volume.

Tensor products and coupling

If $D_1$ and $D_2$ are representations, their tensor product acts by

$$(D_1\otimes D_2)(g) =D_1(g)\otimes D_2(g).$$

For $SU(2)$,

$$\mathcal H_{j_1}\otimes\mathcal H_{j_2} \cong \bigoplus_{j=\lvert j_1-j_2\rvert}^{j_1+j_2} \mathcal H_j.$$

The uncoupled basis diagonalizes $J_{1z}$ and $J_{2z}$; the coupled basis diagonalizes total $J^2$ and $J_z$. Clebsch–Gordan coefficients are the unitary change-of-basis amplitudes between them. They depend on basis-phase conventions and are not probabilities until their moduli are squared in a specified state expansion.

Tensor Product Representations owns the decomposition. Clebsch–Gordan Coefficients summarizes conventions and identities; the canonical angular-momentum derivation is in Addition of Angular Momentum.

Rotations and recoupling data

Wigner matrices are representation matrices of rotations:

$$D^j_{m'm}(R) =\langle j,m'|U(R)|j,m\rangle.$$

Their numerical form depends on active-versus-passive choice, Euler-angle order, generator sign, and basis phases. Wigner $3j$ symbols reorganize Clebsch–Gordan data into a more symmetric form. The $6j$ symbols transform between two coupling orders of three angular momenta, while $9j$ symbols compare coupling schemes for four.

Use Wigner D-Matrices for rotations and Wigner 3j, 6j, and 9j Symbols for recoupling. Tables are meaningful only with the same phase and normalization conventions as the calculation.

Permutations, canonical translations, and antiunitarity

Three further structures extend the main route.

• The Symmetric Group acts by permuting tensor factors. Its trivial and sign representations lead to symmetrizers and antisymmetrizers; the physical boson and fermion postulates belong in the identical-particles volume.
• The Heisenberg Group exponentiates canonical commutation relations into Weyl relations, making central phases and phase-space translations mathematically controlled.
• Antiunitary Symmetries, First Look treats conjugate-linear norm-preserving maps. For antiunitary $\Theta$,

$$\Theta(a|\psi\rangle+b|\phi\rangle) =a^*\Theta|\psi\rangle +b^*\Theta|\phi\rangle,$$

and

$$\langle\Theta\psi|\Theta\phi\rangle =\langle\phi|\psi\rangle.$$

Time-reversal physics and Kramers structure remain canonical in the symmetry volume.

Page map

Page	Central question
Groups	Which axioms encode composable transformations?
Group Actions	What does the group transform, and what are its orbits and stabilizers?
Lie Groups	How do continuous symmetries combine smooth and group structure?
Lie Algebras	Which infinitesimal generators and brackets control local symmetry?
Representations	How does a group become linear operators on a vector space?
Unitary Representations	Which representations preserve Hilbert-space geometry?
SO(3)	How are proper spatial rotations represented in three dimensions?
SU(2)	How are spinor rotations generated by Pauli matrices?
SU(2) versus SO(3)	Why does one spatial rotation have two spinor lifts?
Angular Momentum Algebra	How do commutators and the Casimir determine $j$ and $m$?
Ladder Operators as Lie Algebra Tools	How do commutators generate discrete label shifts?
Tensor Product Representations	How do product representations decompose into irreducibles?
Clebsch–Gordan Coefficients	Which amplitudes connect coupled and uncoupled bases?
Wigner D-Matrices	What are rotation matrices in a spin-$j$ representation?
Wigner 3j, 6j, and 9j Symbols	How are coupling and recoupling transformations encoded invariantly?
Symmetric Group	How do permutations act on tensor factors?
Heisenberg Group	How do Weyl relations encode canonical translations and central phase?
Antiunitary Symmetries, First Look	How does a conjugate-linear ray symmetry differ from a unitary one?

Common mistakes

Mistake	Correction
Confusing an abstract group with one matrix representation	name the representation space and basis
Treating an action as automatically linear	only a representation on a vector space has that extra structure
Mixing Hermitian and anti-Hermitian generator conventions	translate exponential signs, factors of $i$, and structure constants together
Assuming the Lie algebra fixes the global group	check topology, covering maps, and disconnected components
Identifying $SU(2)$ with $SO(3)$	use the two-to-one covering map and track spinor phases
Treating Clebsch–Gordan coefficients as dynamics or probabilities	they are convention-dependent basis-change amplitudes
Combining Wigner tables from different Euler-angle or phase conventions	state and reconcile every convention first
Manipulating an antiunitary operator as though it were complex-linear	conjugate scalar coefficients explicitly

Exercises

1. The spinor 2π rotation

Use $(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=I$ to evaluate $U(\hat{\mathbf n},2\pi)$ and $U(\hat{\mathbf n},4\pi)$.

Solution

The exponential identity gives

$$U(\hat{\mathbf n},\theta) =\cos\frac\theta2 I -i\sin\frac\theta2 \hat{\mathbf n}\cdot\boldsymbol\sigma.$$

At $2\pi$, this is $-I$; at $4\pi$, it is $I$.

2. Coupling two spin halves

Decompose $\mathcal H_{1/2}\otimes\mathcal H_{1/2}$ into irreducible $SU(2)$ representations and check dimensions.

Solution

The allowed total angular momenta are $j=0$ and $j=1$:

$$\frac12\otimes\frac12 =0\oplus1.$$

The singlet has dimension one and the triplet has dimension three, so $1+3=4=2\cdot2$, matching the tensor-product dimension.

3. Ladder termination

Suppose $J_+|j,m_{\max}\rangle=0$. Use

$$J_-J_+ =J^2-J_z^2-\hbar J_z$$

to show that $m_{\max}=j$ or $m_{\max}=-j-1$, and explain which value belongs to the finite spin-$j$ representation.

Solution

Taking the norm gives

$$0 =\hbar^2 \left[j(j+1)-m_{\max}(m_{\max}+1)\right].$$

Thus $(j-m_{\max})(j+m_{\max}+1)=0$. The two algebraic roots are $m_{\max}=j$ and $m_{\max}=-j-1$. A highest weight in the finite spin-$j$ representation is $m_{\max}=j$; the other root lies below the entire allowed ladder and is incompatible with being its maximum.

4. Antiunitary scalar action

Let $\Theta$ be antiunitary. Show that $\Theta(i|\psi\rangle)=-i\Theta|\psi\rangle$ and that the norm is preserved.

Solution

Conjugate linearity gives

$$\Theta(i|\psi\rangle) =i^*\Theta|\psi\rangle =-i\Theta|\psi\rangle.$$

The antiunitary inner-product rule gives

$$\lVert\Theta\psi\rVert^2 =\langle\Theta\psi|\Theta\psi\rangle =\langle\psi|\psi\rangle.$$

References

• J. F. Cornwell, Group Theory in Physics, Volume I, Academic Press, 1984.
• W. Fulton and J. Harris, Representation Theory: A First Course, Springer, 1991.
• H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview Press, 1999.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.
• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.