Groups, Lie Algebras, and Representations

A group describes transformations that can be composed and reversed. A representation tells us how those transformations act on vectors. Together they explain why quantum states fall into spin multiplets, why generators satisfy commutator relations, and how several angular momenta combine.

Enter this chapter

Start with matrix multiplication, eigenvectors, and inner products from Linear Algebra. Read Groups before Group Actions and Representations. The same rotation can act on a spatial vector, a wavefunction, or a spinor; these are different representations of related symmetry groups.

You can learn the finite-dimensional examples before tackling unbounded generators. Unitary Representations explains why preserving the inner product matters for quantum probabilities.

Choose a route

• For spin and rotations: Lie groups → Lie algebras → SO(3) and SU(2) → SU(2) versus SO(3).
• For angular-momentum calculations: angular-momentum algebra → ladder operators → tensor-product representations → Clebsch–Gordan coefficients.
• For explicit rotated states and recoupling: Wigner D-matrices describes rotations in a fixed spin basis; Wigner symbols describes changes of coupling order.
• For exchange or translations: Symmetric Group explains permutations; Heisenberg Group explains the phase in composing position and momentum translations.
• For time reversal: Antiunitary Symmetries, First Look introduces the necessary conjugate-linear algebra.

Local generators and global rotations

Lie algebras describe transformations near the identity. They do not determine every global feature of a group. The most useful example is $SU(2)\to SO(3)$: the Lie algebras are isomorphic, but two spinor matrices $U$ and $-U$ produce the same rotation of ordinary vectors. Half-integer spins detect this distinction.

In a unitary representation, rotations use Hermitian generators through $U(\theta)=e^{-i\theta J/\hbar}$. Mathematics texts instead use the anti-Hermitian generator $-iJ/\hbar$. Translate that convention before comparing commutators.

Physical symmetry principles, spin experiments, and selection rules are developed in Symmetry, Angular Momentum, and Spin. The pages below supply mathematical tools for those calculations.

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?

References

• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015 — matrix groups and their representations.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — quantum rotations and angular momentum.