Nyimbi Odero

Series 01 · a numbered run · kept in public

Part 3: Representation Theory and Physical Applications

Chapter: Lesson 6: Lie Groups and Lie Algebras

Understanding How Symmetries Act

When we observe a physical system with symmetry, we’re really seeing a representation of that symmetry group in action. Consider a hydrogen atom: its spherical symmetry means that its states must transform in specific ways under rotations. These transformation patterns are precisely what representation theory studies. It provides the bridge between abstract symmetry groups and their concrete manifestations in physical systems.

Let’s organize our exploration using the mnemonic “REPRESENT”:

Realization in vector spaces Equivariant mappings Physical states Reduction and decomposition Eigenvalue problems Spin and angular momentum Elementary particles Normal operators Tensor products

The Fundamentals of Representation Theory

A representation of a Lie group G (or its Lie algebra g) is a way of realizing the abstract symmetry transformations as concrete linear transformations on a vector space:

  1. Group Representations A representation is a homomorphism:
ρ:G→GL(V)\rho: G \to GL(V)

preserving group structure:

ρ(g1g2)=ρ(g1)ρ(g2)\rho(g_1g_2) = \rho(g_1)\rho(g_2)
  1. Lie Algebra Representations The infinitesimal version:
π:g→End(V)\pi: \mathfrak{g} \to \text{End}(V)

preserving Lie bracket:

π(\pi(

X,Y

)=π(X)π(Y)−π(Y)π(X)) = \pi(X)\pi(Y) - \pi(Y)\pi(X)
  1. Connection For matrix groups:
ρ(exp⁡X)=exp⁡(π(X))\rho(\exp X) = \exp(\pi(X))

This relates finite and infinitesimal transformations.

Fundamental Physical Examples

  1. Angular Momentum in Quantum Mechanics The representation of SO(3) on quantum states:

For spin-j representation: J_i,J_j

>iℏϵijkJk> i\hbar\epsilon_{ijk}J_k J2∣\j,m⟩=>j(j+1)ℏ2∣\j,m⟩J^2|\j,m\rangle = > j(j+1)\hbar^2|\j,m\rangle Jz∣\j,m⟩=>mℏ∣\j,m⟩J_z|\j,m\rangle = > m\hbar|\j,m\rangle

This explains:

  • Quantization of angular momentum

  • Magnetic quantum numbers

  • Selection rules in spectroscopy

  1. Particle Physics and SU(3) The quark model uses:
3⊗3⊗3=1⊕8⊕8⊕10\text{3} \otimes \text{3} \otimes \text{3} = \text{1} \oplus \text{8} \oplus \text{8} \oplus \text{10}

Leading to:

  • Hadron classification

  • Mass relations

  • Decay patterns

  1. Electron Spin and SU(2) Pauli matrices form representation:
σiσj=δij+iϵijkσk\sigma_i\sigma_j = \delta_{ij} + i\epsilon_{ijk}\sigma_k

Explaining:

  • Spin-1/2 behavior

  • Magnetic moments

  • Exclusion principle

Weight Space Decomposition

For a representation on V:

  1. Weights and Weight Vectors For v ∈ V, H ∈ h (Cartan subalgebra):
H⋅v=λ(H)vH\cdot v = \lambda(H)v

where λ is a weight.

  1. Weight Diagram Visual representation showing:

    • Weight multiplicities

    • Root system action

    • Representation structure

  2. Highest Weight Theory Every irreducible representation has:

    • Unique highest weight

    • Weight lattice structure

    • Dimension formula

Essential Physical Applications

  1. Atomic Spectroscopy Consider hydrogen atom states:

Principal quantum number n

Angular momentum l

Magnetic quantum number m

Spin s

All arise from representation theory.

  1. Particle Multiplets For isospin SU(2):

    • Proton-neutron doublet

    • Pion triplet

    • Delta quartet

Masses split by symmetry breaking.

  1. Conservation Laws Noether’s theorem in quantum form:

Symmetry → Conserved charge operator

H,QH,Q

= 0 → Conservation

Quantum numbers from weights

Practical Computation Methods

  1. Character Theory The character of a representation:
χρ(g)=Tr(ρ(g))\chi_\rho(g) = \text{Tr}(\rho(g))

Properties:

  • Class function

  • Orthogonality relations

  • Decomposition tool

  1. Clebsch-Gordan Series Decomposing tensor products:

Find highest weights

Generate full representations

Count multiplicities

  1. Young Tableaux Systematic way to:

    • Construct representations

    • Decompose tensor products

    • Calculate dimensions

Advanced Topics

  1. Induced Representations From subgroup H ⊂ G:

Start with H-representation

Extend to G

Use for particle states

  1. Plethysm Symmetric powers of representations:

Sym²(V) composition

Alternating powers

Physical composite states

  1. Branching Rules Restriction to subgroups:

SU(3) → SU(2) × U(1)

Physical symmetry breaking

Mass splitting patterns

Common Issues and Solutions

  1. Reducibility Problem: Determining if representation decomposes Solution: Use Casimir operators Example: Angular momentum decomposition

  2. Multiplicity Problem: Same weight appears multiple times Solution: Use raising/lowering operators Example: SU(3) 8-dimensional representation

  3. Physical Interpretation Problem: Connecting math to observables Solution: Focus on quantum numbers Example: Particle classification

Exercises for Understanding

  1. Basic Computation Decompose tensor product:
2⊗2=3⊕1\text{2} \otimes \text{2} = \text{3} \oplus \text{1}

for SU(2)

Show how this relates to adding spins.

  1. Physical Application For hydrogen atom:

    • Find angular momentum eigenvalues

    • Calculate allowed transitions

    • Explain selection rules

  2. Advanced Problem Prove that representations of compact groups are:

    • Completely reducible

    • Unitarizable

    • Finite dimensional

Looking Ahead

In our next section, we’ll explore:

  • Symmetry breaking

  • Quantum groups

  • Infinite-dimensional representations

These concepts will be crucial for:

  • Gauge theory

  • String theory

  • Quantum gravity

Remember: Representation theory isn’t just abstract mathematics — it tells us how symmetries manifest in the physical world. The patterns we’ve studied appear throughout nature, from atomic spectra to particle physics.

Next: Part 4 - Symmetry Breaking and Physical Systems