Quantum Field Theory Course

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

Part V: Gauge Field Theories • Chapter 2

Abelian Gauge Theory

QED revisited: The prototype gauge theory

📺 Video Lectures

For video lectures on Abelian gauge theory and QED structure, see:

QED as Gauge Theory

Quantum Electrodynamics (QED) is the simplest gauge theory - based on the Abelian U(1) group. Having established the gauge principle in Chapter 1, we now examine QED's structure in depth, focusing on aspects that generalize to non-Abelian theories.

This chapter covers: photon propagator derivation, gauge fixing procedures (Lorenz, Coulomb, temporal), Ward-Takahashi identities ensuring gauge invariance of physical observables, and the deep connection between gauge symmetry and charge conservation.

1. The QED Lagrangian

The complete QED Lagrangian for a fermion field ψ:

ℒQED = ψ̄(iγμDμ - m)ψ - (1/4)FμνFμν

Breaking it down:

Fermion kinetic + interaction term

ψ̄(iγμDμ - m)ψ = ψ̄(i∂̸ - m)ψ - eψ̄γμψ Aμ

Free Dirac field + electromagnetic interaction (current jμ = eψ̄γμψ coupled to Aμ)

Photon kinetic term (Maxwell)

-(1/4)FμνFμν where Fμν = ∂μAν - ∂νAμ

Free photon propagation (contains no AμAν mass term - photon is massless)

Covariant derivative

Dμ = ∂μ + ieAμ

Ensures gauge invariance under ψ → eiα(x)ψ, Aμ → Aμ - (1/e)∂μα

2. Gauge Symmetry and Redundancy

The Problem of Redundant Degrees of Freedom

A vector field Aμ in 4D spacetime has 4 components. But a massless photon has only 2 physical polarizations (transverse).

Gauge transformation:

Aμ(x) → Aμ(x) - ∂μΛ(x)

Physics is unchanged under this transformation (gauge invariance). The function Λ(x) provides one degree of freedom worth of redundancy at each spacetime point.

Counting: 4 components - 1 gauge freedom - 1 constraint = 2 physical DOF

The constraint is ∂μFμ0 = j0 (Gauss's law), which determines A0 in terms of Ai and sources.

⚠️ Problem for Quantization

When we try to derive the photon propagator from the path integral, we encounter:

∫𝒟Aμ ei∫d⁴x ℒ

The integral over gauge-equivalent configurations leads to an infinite factor(integrating over all Λ(x)). We must "fix the gauge" to remove this redundancy.

3. Gauge Fixing Procedures

To quantize QED, we impose a gauge-fixing condition that picks out one representative from each gauge equivalence class.

a) Lorenz Gauge

∂μAμ = 0

Most common in relativistic calculations. Preserves Lorentz covariance explicitly.

Add gauge-fixing term to Lagrangian:

ℒGF = -(1/2ξ)(∂μAμ)²

ξ is the gauge parameter. Common choices: ξ = 1 (Feynman gauge), ξ → 0 (Landau gauge), ξ → ∞ (unitary gauge).

b) Coulomb Gauge (Radiation Gauge)

∇⃗·A⃗ = 0

Useful for non-relativistic QED and atomic physics. Explicitly shows transverse photons.

In this gauge, A0 is not an independent degree of freedom but determined by Gauss's law. The spatial components Ai satisfy ∂iAi = 0 (transversality).

c) Temporal (Weyl) Gauge

A0 = 0

Simplifies Hamiltonian formulation. Time component vanishes identically.

Useful for canonical quantization and Hamiltonian approaches to gauge theory.

4. The Photon Propagator

In Lorenz gauge with gauge-fixing parameter ξ, the photon propagator in momentum space is:

Dμν(k) = -i/(k² + iε) [ημν - (1-ξ) kμkν/k²]

Special cases:

Feynman Gauge (ξ = 1):

Dμν(k) = -i ημν/(k² + iε)

Simplest form - looks like scalar propagator with vector index. Most commonly used in QED calculations.

Landau Gauge (ξ = 0):

Dμν(k) = -i/(k² + iε) [ημν - kμkν/k²]

Manifestly transverse: kμDμν = 0. Useful for proving Ward identities.

✓ Key Property: Gauge Dependence

The photon propagator depends on the gauge choice (through ξ). However, all physical observables (S-matrix elements, cross sections) are gauge-independent! This is guaranteed by Ward-Takahashi identities.

5. Ward-Takahashi Identities

The Ward-Takahashi (WT) identities are the quantum analogue of gauge invariance. They relate different Green's functions and ensure that physical predictions are gauge-independent.

Basic Ward Identity:

kμ Γμ(p', p) = e[S-1(p') - S-1(p)]

where Γμ is the vertex function and S is the fermion propagator.

At tree level:

kμ γμ = k̸ (when contracted with external photon momentum)

Physical Consequence

When you compute an amplitude with an external photon and contract with kμ instead of the polarization vector εμ, the result vanishes (at least for on-shell fermions).

This ensures that unphysical photon polarizations (longitudinal and timelike) decouple from physical processes.

Examples of WT Identities in Action:

  • • Charge conservation:WT identities imply ∂μjμ = 0 at the quantum level
  • • Gauge independence of cross sections:σ computed in Feynman gauge = σ in Landau gauge = σ in Coulomb gauge
  • • Cancellation of IR divergences:Virtual + real photon contributions cancel in inclusive observables
  • • Photon mass protection:Quantum corrections cannot generate a photon mass (gauge symmetry forbids it)

6. QED Feynman Rules (Momentum Space)

External fermion lines:

  • • Incoming fermion: u(p)
  • • Outgoing fermion: ū(p)
  • • Incoming antifermion: v̄(p)
  • • Outgoing antifermion: v(p)

External photon lines:

εμ(k, λ) for polarization λ = 1, 2 (transverse)

Internal fermion propagator:

i(p̸ + m)/(p² - m² + iε)

Internal photon propagator (Feynman gauge):

-i ημν/(k² + iε)

Vertex:

-ieγμ

(with momentum conservation at each vertex)

Additional rules:

  • • Integrate ∫d⁴k/(2π)⁴ over each undetermined loop momentum
  • • Factor of (-1) for each fermion loop
  • • Overall factor: i for each disconnected diagram
  • • Symmetry factor 1/S for identical particles

7. Connection to Classical Electromagnetism

In the classical limit (ℏ → 0, or equivalently treating the photon field classically while keeping fermions quantum), QED reduces to the Dirac equation with external electromagnetic field:

(iγμDμ - m)ψ = 0, with Dμ = ∂μ + ieAμ

and Maxwell's equations with source:

∂μFμν = jν = eψ̄γνψ

Tree-level = Classical

Tree-level Feynman diagrams in QED correspond to classical electromagnetic interactions. Loop corrections represent genuine quantum effects (vacuum polarization, vertex corrections, etc.).

Summary

  • ✓ QED Lagrangian: ℒ = ψ̄(iD̸ - m)ψ - ¼FμνFμν
  • ✓ Gauge redundancy: Aμ has 4 components but only 2 physical (transverse photons)
  • ✓ Gauge fixing: Needed to define propagator - common choices: Lorenz, Coulomb, temporal
  • ✓ Photon propagator: Depends on gauge parameter ξ, but physics doesn't
  • ✓ Ward-Takahashi identities: Ensure gauge invariance of S-matrix, protect photon masslessness
  • ✓ Feynman rules: Vertex -ieγμ, propagators standard QFT forms
  • ✓ Abelian structure: U(1) group means photons don't self-interact (linear Maxwell equations)

Further Resources

  • • Peskin & Schroeder - Sections 4.7-4.8 (QED), 9.5 (Ward identity)
  • • Weinberg Vol. I - Chapter 8 (Electrodynamics)
  • • Srednicki - Chapters 58-62 (Gauge fixing and propagators)
  • • Schwartz - Chapter 8 (Spin 1 and gauge invariance)