Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part V: Gauge Field Theories

The foundation of the Standard Model: From symmetries to forces

Overview

Gauge field theories represent one of the most profound achievements in theoretical physics. The discovery that forces arise from local symmetries unified our understanding of electromagnetism, weak interactions, and strong interactions into a single conceptual framework.

In this part, we develop the mathematical machinery of gauge theories from first principles, starting with the simple U(1) gauge symmetry of electromagnetism and building up to the non-Abelian SU(3)×SU(2)×U(1) structure of the Standard Model.

Central principle: Local gauge invariance requires the introduction of gauge fields (photons, W/Z bosons, gluons) that mediate interactions between matter fields.

Fundamental Concepts

🔄 Gauge Symmetry

Global symmetry: ψ → eiαψ (same α everywhere)

Local (gauge) symmetry: ψ → eiα(x)ψ (spacetime-dependent)

⚡ Covariant Derivative

Ordinary derivative: ∂μψ (not gauge invariant)

Covariant derivative: Dμ = ∂μ + igAμ (gauge invariant)

🌊 Field Strength Tensor

Abelian (QED): Fμν = ∂μAν - ∂νAμ

Non-Abelian: Faμν = ∂μAaν - ∂νAaμ + gfabcAbμAcν

🎯 Yang-Mills Action

S = -¼ ∫d4x FaμνFaμν + ∫d4x ψ̄(iD̸ - m)ψ

Pure gauge + matter coupling

Chapters

1. Gauge Symmetry Principles

8 pages

From global to local symmetries. Why gauge invariance requires force carriers. The gauge principle and minimal coupling.

Topics: Global vs local symmetry • Gauge transformations • Minimal coupling • U(1) gauge theory

2. Abelian Gauge Theory (QED Structure)

7 pages

Deep dive into quantum electrodynamics as the prototype gauge theory. Photon propagator, Ward identities, and gauge fixing in QED.

Topics: QED Lagrangian • Photon propagator • Coulomb vs Lorenz gauge • Ward-Takahashi identities

3. Non-Abelian Gauge Theory

10 pages

Lie groups and Lie algebras. Structure constants and commutation relations. How non-commutativity leads to gluon self-interactions.

Topics: SU(N) groups • Lie algebra • Adjoint representation • Covariant derivative for non-Abelian

4. Yang-Mills Theory

12 pages

The complete Yang-Mills action and equations of motion. Three and four-gluon vertices. Classical solutions and instantons.

Topics: Yang-Mills Lagrangian • Gluon self-coupling • Equations of motion • BPST instantons

5. Gauge Fixing & Faddeev-Popov Ghosts

10 pages

Path integral quantization of gauge theories. The ghost field necessity. Faddeev-Popov determinant and BRST symmetry.

Topics: Gauge redundancy • Faddeev-Popov procedure • Ghost fields • BRST invariance

6. Quantum Chromodynamics

13 pages

SU(3) color gauge theory of the strong force. Asymptotic freedom and confinement. Running coupling constant and QCD phenomenology.

Topics: Color charge • Asymptotic freedom • Confinement • Running αₛ • Parton model

7. Electroweak Theory

10 pages

SU(2)×U(1) unification of electromagnetic and weak interactions. Spontaneous symmetry breaking and the Higgs mechanism.

Topics: SU(2)L×U(1)Y structure • W±, Z bosons • Higgs mechanism • Electroweak unification

Historical Development

1918 - Hermann Weyl:

First proposal of gauge invariance (for scale transformations)

1954 - Yang & Mills:

Non-Abelian gauge theory formulation (SU(2) isospin symmetry)

1967-1968 - Weinberg, Salam, Glashow:

Electroweak unification via spontaneous symmetry breaking

1973 - Gross, Wilczek, Politzer:

Discovery of asymptotic freedom in non-Abelian gauge theories (QCD)

1999 - 't Hooft & Veltman:

Proof of renormalizability of gauge theories

Prerequisites

Before starting Part V, you should be comfortable with:

Suggested Learning Path

  1. Week 1-2: Gauge Symmetry Principles + Abelian Gauge Theory (review QED from new perspective)
  2. Week 3-4: Non-Abelian Gauge Theory (study Lie groups in parallel if needed)
  3. Week 5-6: Yang-Mills Theory (work through derivations carefully)
  4. Week 7-8: Gauge Fixing & Faddeev-Popov (most technical chapter)
  5. Week 9-11: Quantum Chromodynamics (connect to experimental QCD)
  6. Week 12-14: Electroweak Theory (culmination of all previous concepts)

Note: This is the most mathematically sophisticated part of the course. Take time to work through calculations and consult supplementary resources.

Connections

← Builds Upon

  • • Noether's theorem (Part I)
  • • QED calculations (Part IV)
  • • Path integral quantization (Part III)
  • • Renormalization (Part IV)

→ Leads To

  • • Standard Model (Part VI)
  • • Beyond Standard Model physics
  • • Grand Unification Theories
  • • String Theory formulations