Quantum Mechanics Course

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← Part I/Representations & Lie Algebras

7. Representations & Lie Algebras

Reading time: ~30 minutes | Pages: 6

Lie algebras are the infinitesimal structure of Lie groups. They provide the link between continuous symmetries and quantum observables, forming the foundation for understanding angular momentum, spin, and gauge theories.

From Lie Groups to Lie Algebras

For a continuous (Lie) group $G$, elements near the identity can be written as:

$$g(\epsilon) = e^{i\epsilon^a T_a} = I + i\epsilon^a T_a + O(\epsilon^2)$$

where $T_a$ are the generators and $\epsilon^a$ are infinitesimal parameters. The generators form a Lie algebra.

Definition: Lie Algebra

A Lie algebra $\mathfrak{g}$ is a vector space with a bilinear operation$[\cdot, \cdot]$ (the Lie bracket) satisfying:

  1. Antisymmetry: $[X, Y] = -[Y, X]$
  2. Bilinearity: $[\alpha X + \beta Y, Z] = \alpha[X,Z] + \beta[Y,Z]$
  3. Jacobi identity: $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$

In Quantum Mechanics

For quantum operators, the Lie bracket is the commutator divided by $i\hbar$:

$$[T_a, T_b] = \frac{1}{i\hbar}[\hat{T}_a, \hat{T}_b] = if_{ab}^c T_c$$

The constants $f_{ab}^c$ are called structure constants and completely define the Lie algebra.

The Angular Momentum Algebra so(3)

The Lie algebra of SO(3) is denoted $\mathfrak{so}(3)$. Its generators are the angular momentum operators $\hat{J}_x, \hat{J}_y, \hat{J}_z$:

$$[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$$

Structure Constants:

For $\mathfrak{so}(3)$: $f_{ij}^k = \epsilon_{ijk}$ (the Levi-Civita symbol)

Casimir Operator

The operator $\hat{J}^2 = \hat{J}_x^2 + \hat{J}_y^2 + \hat{J}_z^2$ commutes with all generators:

$$[\hat{J}^2, \hat{J}_i] = 0 \quad \text{for all } i$$

This is called a Casimir operator. By Schur's lemma, it's a multiple of the identity in each irrep, giving quantum number $j$: $\hat{J}^2|j,m\rangle = j(j+1)\hbar^2|j,m\rangle$.

The Algebra su(2)

The Lie algebra of SU(2) is $\mathfrak{su}(2)$, with generators often denoted $\hat{\sigma}_i/2$(Pauli matrices) or abstract $\hat{J}_i$:

$$[\hat{J}_i, \hat{J}_j] = i\epsilon_{ijk}\hat{J}_k$$

Key Fact:

$\mathfrak{su}(2) \cong \mathfrak{so}(3)$ as Lie algebras, but SU(2) is the universal cover of SO(3) as a group.

  • SO(3) representations: integer angular momentum $j = 0, 1, 2, \ldots$
  • SU(2) representations: half-integer and integer $j = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots$

This explains why spinors (spin-1/2) are projective representations of SO(3) but proper reps of SU(2)!

Representations of Lie Algebras

A representation of a Lie algebra $\mathfrak{g}$ is a linear map:

$$\rho: \mathfrak{g} \to \mathfrak{gl}(V)$$

that preserves the Lie bracket: $\rho([X,Y]) = [\rho(X), \rho(Y)]$.

Irreducible Representations of su(2)

Spin-$j$ Representation

Dimension: $(2j+1)$, labeled by $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$

States: $|j,m\rangle$ with $m = -j, -j+1, \ldots, j-1, j$

Ladder Operators

Define $\hat{J}_\pm = \hat{J}_x \pm i\hat{J}_y$:

$$\hat{J}_\pm|j,m\rangle = \hbar\sqrt{j(j+1) - m(m\pm 1)}|j,m\pm 1\rangle$$

Examples

  • $j=0$: Trivial (scalar), dimension 1
  • $j=\frac{1}{2}$: Spinor (spin-1/2), dimension 2 — $\mathbb{C}^2$
  • $j=1$: Vector (spin-1), dimension 3 — used for photons
  • $j=\frac{3}{2}$: Rarita-Schwinger (spin-3/2), dimension 4

Important Lie Algebras in Physics

$\mathfrak{u}(1)$

Abelian (all generators commute): $[T, T] = 0$

Gauge symmetry of electromagnetism, phase rotations

$\mathfrak{su}(2)$

Generators: $\hat{J}_x, \hat{J}_y, \hat{J}_z$, $[\hat{J}_i,\hat{J}_j] = i\epsilon_{ijk}\hat{J}_k$

Spin, weak isospin in particle physics

$\mathfrak{su}(3)$

8 generators (Gell-Mann matrices $\lambda_a$)

Color symmetry in QCD, flavor symmetry (approximate)

Poincaré Algebra

Generators: $\hat{P}^\mu$ (translations), $\hat{M}^{\mu\nu}$ (Lorentz)

Spacetime symmetries in special relativity

$\mathfrak{so}(3,1)$ (Lorentz Algebra)

Generators: 3 rotations + 3 boosts

Lorentz transformations, relativistic angular momentum

Exponential Map: From Algebra to Group

The exponential map connects Lie algebra elements to group elements:

$$g = \exp(i\theta^a T_a) = I + i\theta^a T_a + \frac{(i\theta^a T_a)^2}{2!} + \cdots$$

Example: Rotation by angle $\theta$ about $\hat{z}$

$$\hat{R}_z(\theta) = e^{-i\theta\hat{J}_z/\hbar}$$

Acting on a state: $\hat{R}_z(\theta)|j,m\rangle = e^{-i\theta m}|j,m\rangle$

Adjoint Representation

Every Lie algebra has a special representation called the adjoint representation, where generators act on the algebra itself:

$$\text{ad}_{T_a}(T_b) = [T_a, T_b] = if_{ab}^c T_c$$

The adjoint representation has dimension equal to the number of generators. For example:

  • $\mathfrak{su}(2)$: adjoint is the $j=1$ representation (dimension 3)
  • $\mathfrak{su}(3)$: adjoint is the 8-dimensional representation (gluons in QCD!)

Cartan Subalgebra & Roots

Cartan Subalgebra

A maximal set of commuting generators $\{H_i\}$ such that $[H_i, H_j] = 0$. The number of such generators is the rank of the Lie algebra.

  • $\mathfrak{su}(2)$: rank 1 (e.g., $H = \hat{J}_z$)
  • $\mathfrak{su}(3)$: rank 2 (two commuting diagonal generators)

The remaining generators $E_\alpha$ are called raising and lowering operators, and $\alpha$ are called roots:

$$[H_i, E_\alpha] = \alpha_i E_\alpha$$

Applications in Quantum Mechanics

1. Classifying Quantum States

States are labeled by eigenvalues of Cartan generators (quantum numbers like $m, m_s$) and Casimir operators (like $j$).

2. Selection Rules

Lie algebra structure determines which matrix elements vanish, giving selection rules for transitions (e.g., $\Delta m = 0, \pm 1$ for dipole radiation).

3. Composite Systems

Adding angular momenta is decomposing tensor products of representations:$j_1 \otimes j_2 = |j_1-j_2| \oplus \cdots \oplus (j_1+j_2)$

4. Gauge Theories

Particle physics is built on gauge symmetries: U(1)×SU(2)×SU(3) for the Standard Model. Fields transform in representations of these Lie algebras.

5. Noether Currents

Each generator of a continuous symmetry corresponds to a conserved current/charge via Noether's theorem.

Summary

  • ✓ Lie algebras are infinitesimal structure of Lie groups, defined by commutation relations
  • ✓ Generators correspond to quantum observables (angular momentum, charges)
  • ✓ Irreducible representations classify quantum states and give quantum numbers
  • ✓ $\mathfrak{su}(2) \cong \mathfrak{so}(3)$ as algebras, but SU(2) allows half-integer spin
  • ✓ Casimir operators label irreps and give degeneracies
  • ✓ Exponential map: $e^{i\theta T}$ connects algebra to group transformations
  • ✓ Essential for particle physics, QFT, and understanding symmetries in nature
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