Quantum Field Theory Course

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

Part IV, Chapter 3

Quantum Electrodynamics (QED)

The quantum theory of light and matter - physics' most precise theory

🔗Course Connections

📚Prerequisites

QFT Part I
Electromagnetic Field
Classical EM field Lagrangian
→
QFT Part I
Dirac Field
Spinor fields and gamma matrices
→
QFT Part II
Photon Quantization
Quantized electromagnetic field
→
▶️

Video Lecture

Lecture 22: Quantum Electrodynamics - MIT 8.323

Introduction to QED: the quantum theory of electrons and photons (MIT QFT Course)

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

3.1 The QED Lagrangian

QED describes electrons (Dirac fermions) interacting with photons (electromagnetic field):

$$\boxed{\mathcal{L}_{\text{QED}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi}$$

where:

  • Fμν = ∂μAν - ∂νAμ is the electromagnetic field strength tensor
  • Dμ = ∂μ + ieAμ is the gauge covariant derivative
  • ψ is the electron/positron Dirac field
  • Aμ is the photon field (4-vector potential)
  • e is the electric charge (e² = 4πα ≈ 1/137 in natural units)

The interaction term is:

$$\mathcal{L}_{\text{int}} = -e\bar{\psi}\gamma^\mu\psi A_\mu = -eJ^\mu A_\mu$$

This couples the electromagnetic current Jμ = ψ̄γμψ to the photon field!

💡Physical Interpretation

The interaction -eJμAμ means:

  • Vertex: Electron/positron emits or absorbs a photon
  • Coupling: Strength proportional to electric charge e
  • Current conservation: ∂μJμ = 0 → photons couple to conserved current

3.2 Gauge Invariance

QED has U(1) gauge symmetry under:

\begin{align*} \psi(x) &\to e^{ie\alpha(x)}\psi(x) \\ A_\mu(x) &\to A_\mu(x) - \partial_\mu\alpha(x) \end{align*}

The Lagrangian is invariant under these transformations! This is local gauge invariance.

3.3 Feynman Rules for QED

QED Feynman Rules (Momentum Space)

  1. Electron propagator:
    $\frac{i(\not{p} + m)}{p^2 - m^2 + i\epsilon}$
  2. Photon propagator (Feynman gauge):
    $-\frac{ig_{\mu\nu}}{k^2 + i\epsilon}$
  3. Vertex (e⁻γ interaction):
    -ieγμ with momentum conservation
  4. External lines:
    • Incoming electron: u(p)
    • Outgoing electron: ū(p)
    • Incoming positron: v̄(p)
    • Outgoing positron: v(p)
    • Photon: polarization vector εμ(k)

3.4 Dirac Spinors

Solutions to the Dirac equation (iγμ∂μ - m)ψ = 0:

\begin{align*} \text{Electron:} \quad \psi &= u(p)e^{-ip \cdot x}, \quad (\not{p} - m)u = 0 \\ \text{Positron:} \quad \psi &= v(p)e^{+ip \cdot x}, \quad (\not{p} + m)v = 0 \end{align*}

These satisfy orthogonality and completeness relations crucial for computing amplitudes.

🎯 Key Takeaways

  • QED Lagrangian: ℒ = -¼FμνFμν + ψ̄(iγμDμ - m)ψ
  • Interaction: -eψ̄γμψAμ couples current to photon
  • U(1) gauge symmetry: Local phase transformations
  • Feynman rules: Photon propagator, electron propagator, vertex -ieγμ
  • External states: Dirac spinors u(p), v(p) and polarization vectors εμ
  • Next: Computing actual QED processes!