Part V: Gauge Field Theories • Chapter 3

Non-Abelian Gauge Theory

Lie groups, structure constants, and self-interacting gauge fields

📺 Video Lectures

For video lectures on non-Abelian gauge theory and Yang-Mills structure, see:

• Susskind's Theoretical Minimum:Lectures 2-4: QCD and Group Theory
SU(3) color symmetry, structure constants, gluon self-interactions
• Tobias Osborne QFT 2016:YouTube Playlist
Path integral quantization of non-Abelian gauge theories
• David Tong Gauge Theory Lectures:Lecture Notes (Cambridge)
Advanced treatment of Yang-Mills theory and non-Abelian structure

From U(1) to SU(N)

The step from Abelian (U(1)) to non-Abelian gauge theories is one of the most important conceptual advances in 20th-century physics. While electromagnetism is based on the commutative group U(1), the weak and strong forces require non-commutative Lie groups: SU(2) and SU(3).

The key difference: gauge bosons themselves carry "charge" and interact with each other. Gluons interact with gluons, W bosons interact with W bosons. This self-interaction is the source of confinement in QCD and the reason why the strong force behaves so differently from electromagnetism.

Central Difference

Abelian (U(1)): Photons don't interact with each other
Non-Abelian (SU(N)): Gauge bosons self-interact via 3-point and 4-point vertices

1. Lie Groups and Lie Algebras

Lie Group SU(N)

SU(N) = Special Unitary group of N×N matrices:

U^†U = 𝟙, det(U) = 1

• SU(2): 2×2 unitary matrices, 3 generators (Pauli matrices)

Examples: isospin symmetry, weak SU(2)L

• SU(3): 3×3 unitary matrices, 8 generators (Gell-Mann matrices)

Color symmetry in QCD

• SU(N): N²-1 generators

General case

Lie Algebra: Generators and Commutators

Elements near the identity can be written:

U = e^{iα^aT^a}

where T^a (a = 1, ..., N²-1) are the generators of the Lie algebra.

The generators satisfy the commutation relations:

[T^a, T^b] = if^abcT^c

where f^abc are the structure constants - completely antisymmetric tensors that encode the group's multiplication table.

Key Distinction: Structure Constants

U(1): Only 1 generator, so f^abc = 0 (Abelian - everything commutes)
SU(N), N≥2: f^abc ≠ 0 (Non-Abelian - generators don't commute)

Example: SU(2)

Generators are (1/2)×Pauli matrices:

T^a = σ^a/2, where σ^a are Pauli matrices

Commutation relations:

[T^a, T^b] = iε^abcT^c

So the structure constants are f^abc = ε^abc (Levi-Civita symbol). This is the same algebra as angular momentum!

2. Non-Abelian Gauge Transformations

Consider a field ψ transforming in the fundamental representation of SU(N):

ψ(x) → U(x) ψ(x) = e^{iα^a(x)T^a} ψ(x)

Problem: Just like in U(1), the derivative ∂_μψ does not transform covariantly:

∂_μψ → U(x) ∂_μψ + (∂_μU(x)) ψ

Solution: Introduce N²-1 gauge fields A_μ^a (one for each generator):

D_μ = ∂_μ + igA_μ^aT^a ≡ ∂_μ + igA_μ

(We write A_μ = A_μ^aT^a as shorthand)

The gauge fields transform as:

A_μ → U A_μ U^† - (i/g)(∂_μU) U^†

Infinitesimal Transformation

For small α^a(x), the gauge field transforms as:

A_μ^a → A_μ^a - (1/g)∂_μα^a - f^abcα^bA_μ^c

The last term (with f^abc) is new compared to U(1) - it arises from non-commutativity and means gauge fields transform inhomogeneously (they mix with each other).

3. The Non-Abelian Field Strength Tensor

Define the field strength tensor via the commutator of covariant derivatives:

[D_μ, D_ν] = igF_μν

Computing this commutator gives:

F_μν^a = ∂_μA_ν^a - ∂_νA_μ^a + gf^abcA_μ^bA_ν^c

⭐ The Crucial Extra Term

Compared to the Abelian case:

U(1) - Abelian:

F_μν = ∂_μA_ν - ∂_νA_μ

Linear in A_μ

SU(N) - Non-Abelian:

F_μν^a = ... + gf^abcA_μ^bA_ν^c

Nonlinear (quadratic) in A_μ

This gf^abcA_μ^bA_ν^c term is why gluons self-interact!

Gauge Transformation of F_μν

The field strength transforms covariantly (not invariantly!):

F_μν → U F_μν U^†

This means F_μν^a components mix under gauge transformations. To get gauge-invariant quantities, we need to form traces:

Tr(F_μνF^μν) is gauge-invariant

4. The Non-Abelian Bianchi Identity

The field strength satisfies a modified Bianchi identity:

D_μF_νλ + D_νF_λμ + D_λF_μν = 0

Or equivalently (using covariant derivative):

D_μ*F^μν = 0

where *F is the dual field strength. This is the homogeneous Yang-Mills equation.

Compare to Maxwell

In electromagnetism: ∂_μ*F^μν = 0 (ordinary derivative)
In Yang-Mills: D_μ*F^μν = 0 (covariant derivative - includes gauge field!)

5. Yang-Mills Equations of Motion

The inhomogeneous Yang-Mills equation (equation of motion for A_μ):

D_μF^{μν a} = j^{ν a}

where j^{ν a} is the conserved current from matter fields.

Expanding the covariant derivative:

∂_μF^{μν a} + gf^abcA_μ^bF^{μν c} = j^{ν a}

⚡ Self-Interaction Term

The term gf^abcA_μ^bF^{μν c} represents the self-interaction of gauge bosons.

This is fundamentally different from Maxwell's equations, which are linear in A_μ. Yang-Mills equations are nonlinear - gluons source other gluons!

Physical Consequences

📡 Electromagnetism (U(1))

• Linear equations
• Photons don't interact
• Superposition works
• Long-range force (1/r²)

🌈 QCD (SU(3))

• Nonlinear equations
• Gluons self-interact
• No superposition
• Confinement (short-range)

6. Representations and Color Charge

Fundamental Representation

Matter fields (quarks in QCD) transform in the fundamental representation: N-component vectors acted on by N×N matrices.

SU(3): ψ = (ψ_red, ψ_green, ψ_blue)^T (3 colors)

Adjoint Representation

Gauge fields transform in the adjoint representation: there are N²-1 of them, and they transform via the structure constants.

SU(3): A_μ^a with a = 1, ..., 8 (8 gluons)

Gluons carry color charge: each gluon is a color-anticolor combination (e.g., red-antiblue).

Why Gluons Self-Interact

Because gluons carry color charge, they couple to the gauge field A_μ just like quarks do. A red-antigreen gluon can emit another gluon and change to red-antiblue. This is forbidden in QED because photons are electrically neutral.

7. Casimir Operators and Group Invariants

Important group-theoretic quantities:

Quadratic Casimir C₂(R):

T^aT^a = C₂(R) 𝟙

For SU(N) fundamental: C₂(F) = (N²-1)/(2N)
For SU(N) adjoint: C₂(A) = N

Normalization constant T(R):

Tr(T^aT^b) = T(R) δ^ab

For SU(N) fundamental: T(F) = 1/2
For SU(N) adjoint: T(A) = N

These invariants appear in loop calculations and determine the relative strength of different Feynman diagrams (e.g., quark loops vs gluon loops in QCD).

Summary

✓ Non-Abelian groups: SU(N) with N²-1 generators satisfying [T^a, T^b] = if^abcT^c
✓ Multiple gauge fields: A_μ^a for a = 1, ..., N²-1
✓ Field strength: F_μν^a = ∂_μA_ν^a - ∂_νA_μ^a + gf^abcA_μ^bA_ν^c (nonlinear!)
✓ Self-interactions: Gauge bosons carry charge and couple to each other
✓ Yang-Mills equation: D_μF^{μν a} = j^{ν a} (nonlinear partial differential equation)
✓ Representations: Matter in fundamental (N-dim), gauge fields in adjoint (N²-1 dim)
✓ Physical consequence: Confinement, asymptotic freedom (covered in QCD chapter)

Further Resources

• Peskin & Schroeder - Section 15.2 (Non-Abelian Gauge Theory)
• Weinberg Vol. II - Chapter 15 (Non-Abelian Gauge Theories)
• Srednicki - Chapters 69-72 (Yang-Mills theory)
• Georgi - "Lie Algebras in Particle Physics" (comprehensive group theory reference)
• Mathematics: Lie Groups - detailed group theory background

Quantum Field Theory Course