Mathematics · Upper Division

Group Theory for Physics

Symmetry is the deepest principle in physics. Group theory is the mathematics of symmetry — it determines selection rules in spectroscopy, classifies elementary particles, and underlies the gauge theories of all fundamental forces.

Prerequisites

Linear algebra (Ch. LA) \cdot Quantum mechanics (Ch. 20) \cdot Basic matrix multiplication

Learning Goals

Verify that a set with a binary operation satisfies the four group axioms.
Construct the character table of a finite group and use it to decompose representations into irreps.
Apply Schur's lemma to relate symmetry to degeneracy in quantum systems.
Identify the Lie algebras su(2) and su(3) and connect their generators to physical observables.
Use selection rules derived from representation theory to determine which matrix elements vanish.

GT.1 Groups and Their Properties

Definition GT.1 — Group

A group

G is a set with a binary operation \cdot satisfying

1. Closure:

a, b ∈ G \to a\cdotb ∈ G

2. Associativity: (

a\cdotb)\cdotc = a\cdot(b\cdotc

3. Identity: ∃

e: e\cdota = a\cdote = a for all a

4. Inverses: ∀

a ∃a^{-1}: a\cdota^{-1} = e

Key examples: (ℝ, +), (GL(n,ℂ), \times), permutations S_{n}, rotation group SO(3), Lorentz group

, SU(N) gauge groups of the Standard Model.

A group is Abelianif a·b = b·a for all elements. SO(3) is non-Abelian (rotations don't commute). The order of a finite group is its number of elements.

GT.2 Representations

A representation of a group G is a homomorphism D: G → GL(V) — a map from group elements to invertible linear operators on a vector space V, preserving the group structure: D(g₁g₂) = D(g₁)D(g₂).

Irreducible representations (irreps) are representations with no invariant subspace. The dimension of an irrep equals the multiplicity of its character.

Theorem GT.1 — Schur's Lemma

If D_{1} and D_{2} are irreducible representations of G and M is an operator satisfying MD_{1}(g) =

D_{2}(g)M for all g ∈ G, then either M = 0 or M is an isomorphism (and D_{1} ≅ D_{2}). Corollary: f

r an irrep D and any operator commuting with all D(g): M = \lambdaI. Physical application: if [H

D(g)] = 0 for all symmetry transformations g, then energy eigenstates within an irrep are

egenerate.

The character of a representation is χ(g) = Tr D(g) — invariant under basis change and conjugation. The great orthogonality theorem:

\sum_g D^\alpha_{mn}(g)* D^\beta_{m'n'}(g) = (|G|/d_\alpha) \delta_\alpha\beta \delta_mm' \delta_nn'

(GT.1)

where d_α is the dimension of irrep α. Characters satisfy:

\sum_g \chi^\alpha(g)* \chi^\beta(g) = |G| \delta_\alpha\beta \qquad (orthogonality of characters)

(GT.2)

Example GT.1 — Representations of C₃ᵥ (Ammonia)

$The symmetry group of NH_{3} is C_{3v} =$ { $E, C_{3}, C_{3}^{2}, \sigma_v, \sigma_v', \sigma_v''$ } (order 6). Find all irreducible representations.

Classes:

3 conjugacy classes: E (1), 2C_{3} (2), 3\sigma_v (3). Number of irreps = number of classes = 3

Dimension constraint:

d_{1}^{2} + d_{2}^{2} + d_{3}^{2} = |G| = 6 \to only solution: 1,1,2

Character table:

C_{3v}: A_{1} (1,1,1): E=1, 2C_{3}=1, 3\sigma_v=1 (totally symmetric, z). A_{2} (1,1,-1): z-rotation, antis

ymmetric under \sigma_v. E (2,-1,0): degenerate pair (x,y), (Rx,Ry

Physical content:

The vibrational modes of NH_{3} transform as: 2A_{1} (symmetric stretch and umbrella) + 2E (dege

nerate stretch and bend). Only A_{1} and E modes are IR active (they transform like x,y,z). B

oth A_{1} and E are Raman active. This predicts the observed spectrum

GT.3 Lie Groups and Lie Algebras

A Lie group is a continuous group that is also a smooth manifold — elements are labeled by continuous parameters. The key examples in physics:

SO(3) = rotations in 3D (3 parameters: Euler angles). Doubly covered by SU(2) — spinors transform under SU(2), vectors under SO(3). This explains spin-½.

SU(2): 2×2 unitary matrices with det=1. Generators are Pauli matrices σᵢ/2. Algebra: [Jᵢ, Jⱼ] = iεᵢⱼₖ Jₖ — same as angular momentum!

SU(3): 3×3 unitary, det=1. 8 generators (Gell-Mann matrices). Used to classify hadrons (flavor SU(3)) and for QCD color (gauge SU(3)).

The Lie algebra g is the tangent space at the identity of the Lie group G, with the commutator as multiplication. For a compact Lie group, the irreps are labeled by the highest weight vector — the quantum numbers that maximize all commuting generators.

[J_{i}, J_{j}] = i\varepsilon_{ijk} J_{k} \qquad (su(2) algebra \to all angular momentum physics)

(GT.3)

GT.4 Applications: Selection Rules and Degeneracy

Selection rules: A matrix element ⟨ψ_f | O | ψ_i⟩ is zero unless the direct product of representations of ψ_f, O, and ψ_i contains the trivial (totally symmetric) representation. For electric dipole transitions (O transforms as vector): Δl = ±1, Δm = 0, ±1 — directly from the group theory of SO(3) irreps.

Degeneracy from symmetry:Energy levels transform as irreps of the Hamiltonian's symmetry group. The degeneracy of a level equals the dimension of its irrep. Accidental degeneracy (higher than expected) signals a hidden symmetry — the SO(4) symmetry of hydrogen gives the n² degeneracy.

Definition GT.2 — Common Traps

A group is defined by operations: the same set can form different groups under different operations.
Representations are matrices for abstract symmetries: the group itself is not the matrix choice.
Abelian and non-Abelian behavior differs: order matters in rotations and gauge groups.
Irreducible representations organize states: they explain degeneracy patterns and selection rules.

Exercises — GT.1–GT.4 Group Theory

State the order of the group C_{3v} and list its elements

Straightforward

How many irreducible representations does C_{3v} have, and what are their names

Straightforward

Write the multiplication table for D_{3} (symmetry of equilateral triangle). Find all subgrou

ps. Which are normal? What is the factor group D_{3}/C_{3}

Straightforward

4.Find all irreps of SU(2). What is the tensor product 1/2 ⊗ 1? How do Clebsch-Gordan coefficients arise from the representation theory?

Intermediate

5.Explain how SU(3) flavor symmetry classif

ies the meson and baryon multiplets (the Eightfold Way). What is the significance of the \Omega

Intermediate

6.Show that hydrogen has a hidden

SO(4) symmetry by constructing the Laplace-Runge-Lenz vector. How does this explain the n^{2}

Challenging

Key Takeaways

Group: closed, associative, has identity and inverses. Non-Abelian when order matters.
$Representation D(g): group \to matrices, preserving structure. Irreps are the building block$ s.
Schur's lemma: operators commuting with all irrep matrices are proportional to identity.
$Character \chi(g) = Tr D(g): class function, orthogonal for different irreps. Determines spec$ trum.
$Lie algebras [J_{i},J_{j}] = i\varepsilon_{ijk} J_{k}: su(2) ↔ all angular momentum. Exponential gives Lie group$ .
$Selection rules: ⟨f|O|i⟩ ≠ 0 only if trivial irrep appears in D_{f} ⊗ D_{O} ⊗ D_{i}$ .