Quantum Electrodynamics
Quantum Electrodynamics
QED is the quantum field theory of electrons and photons, one of the most successful theories ever constructed. Its key prediction — the running of the fine structure constant — has been verified at colliders to extraordinary precision.
Definition
QED Lagrangian: \(\mathcal{L} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi - eA_\mu\bar{\psi}\gamma^\mu\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}\). The coupling constant is \(\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137\).
Key Result
Running coupling: due to vacuum polarization, \(\alpha(Q^2)\) increases with energy. At the Z mass (91 GeV): \(\alpha \approx 1/128\), measurably different from the low-energy value 1/137.
Example 1
Lamb shift: QED predicts the 2S\(_{1/2}\) and 2P\(_{1/2}\) levels of hydrogen differ by 1057 MHz due to vacuum fluctuations and self-energy — confirmed by Lamb's 1947 experiment to 6 significant figures.
Example 2
Pair production threshold: a photon must have energy \(E_{\gamma} > 2m_ec^2 = 1.022\) MeV to produce an electron-positron pair. In a Coulomb field, the cross-section grows logarithmically with energy.
Loading qed-calculator...
Practice
- What is renormalization, and why is it needed in QED?
- Calculate the fine-structure constant from the anomalous magnetic moment formula.
- Why does QED break down at the Landau pole?
- Explain virtual particle creation and its effect on the Casimir force.
Show Answer Key
1. Loop diagrams in QED produce ultraviolet divergences (infinite integrals). Renormalization absorbs these infinities into redefined ('renormalized') physical parameters (mass, charge, field strength). The procedure: (1) regularize (cutoff/dimensional regularization), (2) compute counter-terms, (3) express observables in terms of renormalized parameters. QED is renormalizable: all divergences can be absorbed into a finite number of parameters.
2. The anomalous magnetic moment $a_e = (g-2)/2 = \frac{\alpha}{2\pi} + O(\alpha^2)$ (Schwinger, 1948). $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137.036$. So $a_e \approx 1/(2\pi\cdot137) \approx 0.00116$. Full QED calculation (to 5th order in $\alpha$) agrees with experiment to 12 significant figures — the most precisely tested prediction in physics.
3. At the Landau pole, the running coupling $\alpha(\mu)$ diverges at a finite energy scale $\mu_L \sim m_e e^{3\pi/\alpha} \sim 10^{286}$ GeV (far above the Planck scale). This signals that QED is not a fundamental theory — it's an effective field theory valid below $\mu_L$. In practice, QED merges with the weak force (electroweak unification) at $\sim 100$ GeV, well below the Landau pole.
4. Virtual particles are internal lines in Feynman diagrams — off-shell ($E^2 \neq p^2c^2+m^2c^4$) quantum fluctuations allowed by the energy-time uncertainty principle ($\Delta E\cdot\Delta t \sim \hbar$). The Casimir force: two conducting plates restrict the virtual photon modes between them (boundary conditions). The energy density between plates is lower than outside, producing an attractive force $F/A = -\pi^2\hbar c/(240d^4)$.