Part V: Gauge Field Theories • Chapter 1

Gauge Symmetry Principles

From global to local symmetries: The origin of forces

📺 Video Lectures

For comprehensive video lectures on gauge symmetry and gauge theories, see:

• Susskind's Theoretical Minimum:Lectures 5-6 on Gauge Fields and Symmetry
Excellent intuitive explanation of gauge principle
• Tobias Osborne QFT 2016:YouTube Playlist (18 lectures)
Covers both Abelian and non-Abelian gauge theories
• David Tong QFT Lectures:Perimeter Institute (PIRSA)
Lecture notes available at www.damtp.cam.ac.uk/user/tong/qft.html

The Gauge Principle

The gauge principle is one of the most profound ideas in modern physics: forces arise from demanding that local symmetries be preserved.

This remarkable statement means that the electromagnetic, weak, and strong forces are not independent phenomena, but rather necessary consequences of requiring that the laws of physics remain invariant under local phase transformations.

Central Insight

Global symmetry + locality requirement → gauge fields (force carriers) must exist

1. Global vs Local Symmetries

Global U(1) Symmetry

Consider a complex scalar field with Lagrangian:

ℒ = (∂_μφ)^†(∂^μφ) - m²φ^†φ

This is invariant under global U(1) transformation:

φ(x) → e^iα φ(x)

where α is a constant (same everywhere in spacetime). This symmetry leads to charge conservation via Noether's theorem.

Local U(1) Symmetry (Gauge Symmetry)

Now demand invariance under local transformations:

φ(x) → e^iα(x) φ(x)

Problem: The derivative term is NOT invariant:

∂_μφ → e^iα(x)(∂_μφ + iφ ∂_μα)

The extra term iφ ∂_μα breaks the symmetry!

The Solution: Introduce a Gauge Field

Key idea: Replace the ordinary derivative with a covariant derivative:

D_μ = ∂_μ + ieA_μ(x)

where A_μ(x) is a new gauge field that transforms as:

A_μ(x) → A_μ(x) - (1/e) ∂_μα(x)

Then the covariant derivative transforms as:

D_μφ → e^iα(x) D_μφ

The gauge field A_μ exactly cancels the unwanted term, restoring gauge invariance!

2. Physical Interpretation

What is the gauge field A_μ?

For U(1) gauge symmetry, A_μ is the photon field (electromagnetic potential).

• Gauge invariance = electromagnetic gauge freedom:

The transformation A_μ → A_μ - ∂_μΛ is exactly the familiar gauge transformation of electromagnetism.

• The coupling constant e:

The parameter e in the covariant derivative is the electric charge (e² = 4πα ≈ 1/137 in natural units).

• Minimal coupling:

The replacement ∂_μ → D_μ = ∂_μ + ieA_μ is called minimal coupling - the simplest way to couple matter to gauge fields.

Profound Realization

The electromagnetic force is not an independent entity added to quantum mechanics. Rather, it is forced into existence by demanding local U(1) symmetry. The photon exists because we require gauge invariance!

3. The Gauge-Invariant Lagrangian

The complete gauge-invariant Lagrangian for a charged scalar field:

ℒ = (D_μφ)^†(D^μφ) - m²φ^†φ - (1/4)F_μνF^μν

where the field strength tensor is:

F_μν = ∂_μA_ν - ∂_νA_μ

Three pieces:

1. Matter kinetic term: (D_μφ)^†(D^μφ)
Contains φ propagation + interaction with A_μ
2. Mass term: -m²φ^†φ
Mass of the charged scalar field
3. Gauge field kinetic term: -(1/4)F_μνF^μν
Free propagation of the photon field (Maxwell's equations)

Key Property: Gauge Invariance

Under the transformations:

• φ(x) → e^iα(x) φ(x)
• A_μ(x) → A_μ(x) - (1/e) ∂_μα(x)

The entire Lagrangian ℒ remains unchanged. F_μν is automatically gauge-invariant because it involves differences of derivatives.

4. Equations of Motion

For the scalar field φ:

D_μD^μφ + m²φ = 0

The gauge-covariant Klein-Gordon equation.

For the gauge field A_μ:

∂_μF^μν = j^ν

where the current is:

j^μ = ie[(D^μφ)^†φ - φ^†(D^μφ)]

This is Maxwell's equation ∂_μF^μν = j^ν with source j^μ. Gauge invariance automatically ensures current conservation: ∂_μj^μ = 0.

5. Why Gauge Symmetry Matters

🔒 Ensures Consistency

Gauge symmetry prevents unphysical longitudinal photon modes. Only 2 transverse polarizations propagate, as required by massless spin-1 particles.

⚖️ Current Conservation

Gauge invariance implies (via Noether) that electric charge is conserved. ∂_μj^μ = 0 follows automatically from the gauge structure.

🎯 Unification

The same principle (local gauge symmetry) explains electromagnetism, weak force, and strong force under different gauge groups: U(1), SU(2), SU(3).

🧮 Renormalizability

Gauge symmetry constrains the theory so severely that it becomes renormalizable, making quantum corrections calculable to arbitrary precision.

6. Preview: Non-Abelian Gauge Theory

U(1) gauge theory (electromagnetism) is Abelian: transformations commute. The real power of gauge theory emerges with non-Abelian groups like SU(2) and SU(3).

Key differences:

• Multiple gauge fields:SU(2) has 3 gauge bosons (W¹, W², W³), SU(3) has 8 gluons
• Non-commuting transformations:Gauge transformations don't commute: [T^a, T^b] = if^abcT^c
• Self-interactions:Gauge bosons carry "charge" and interact with each other (gluons interact with gluons!)
• Asymptotic freedom:Coupling becomes weaker at high energies (QCD)

We'll develop non-Abelian gauge theory systematically in Chapter 3.

Summary

✓ Global symmetry φ → e^iαφ leads to conserved charge (Noether)
✓ Local symmetry φ → e^iα(x)φ requires introducing gauge field A_μ
✓ Covariant derivative D_μ = ∂_μ + ieA_μ transforms covariantly
✓ Field strength F_μν = ∂_μA_ν - ∂_νA_μ is gauge-invariant
✓ Minimal coupling ∂_μ → D_μ generates electromagnetic interactions
✓ Gauge invariance ensures current conservation and renormalizability
✓ This principle extends to SU(2)×U(1) (electroweak) and SU(3) (strong force)

Further Resources

• Peskin & Schroeder - Section 15.1 (The Abelian Gauge Theory)
• Srednicki - Chapters 54-56 (Gauge Symmetry)
• Zee - Part III (Chapters III.1-III.3)
• 't Hooft - "Gauge Theories of the Forces Between Elementary Particles" (Scientific American, 1980)

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