Quantum Field Theory Course

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

Part IV, Chapter 5

Elementary QED Processes II

Møller scattering, crossing symmetry, and the power of Feynman diagrams

🔗Course Connections

📚Prerequisites

QFT Part IV
QED Processes I
Basic QED calculations
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Video Lecture

Lecture 25: Elementary Processes in QED (II) - MIT 8.323

Advanced QED processes and crossing symmetry (MIT QFT Course)

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

5.1 Møller Scattering (e⁻e⁻ → e⁻e⁻)

Electron-electron scattering has both t and u channel diagrams. The electrons are identical fermions, so we must antisymmetrize!

$$\mathcal{M} = \mathcal{M}_t - \mathcal{M}_u$$

The minus sign comes from fermion exchange (Pauli exclusion principle).

Differential cross section:

$$\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4s}\left[\frac{s^2 + u^2}{t^2} + \frac{s^2 + t^2}{u^2} + \frac{2u^2}{st}\right]$$

5.2 Crossing Symmetry

Crossing relates different physical processes by analytic continuation in Mandelstam variables:

  • e⁺e⁻ → μ⁺μ⁻ (s-channel)
  • e⁻μ⁻ → e⁻μ⁻ (t-channel)
  • e⁻μ⁺ → e⁺μ⁻ (u-channel)

Same amplitude function M(s,t,u) describes all three! Just evaluate in different kinematic regions.

5.3 Photon-Photon Scattering

Photons don't directly couple to each other (ℒint = -eψ̄γμψAμ), but they scatter via virtual electron loop!

This is a one-loop process - quantum correction to classical EM.

💡 Quantum Effects

Photon-photon scattering is pure quantum - no classical analog! It demonstrates:

  • Virtual particles have real physical effects
  • Nonlinear corrections to Maxwell's equations
  • Vacuum polarization (vacuum acts like a medium)

🎯 Key Takeaways

  • Møller scattering: Fermion antisymmetrization (Mt - Mu)
  • Crossing symmetry: Related processes from same amplitude
  • Photon-photon scattering: One-loop quantum effect
  • Virtual particles: Contribute to real physical processes
  • QED precision: Tree level + loops match experiments
  • Next: Loop corrections and renormalization!