Quantum Electrodynamics (QED)

Table of Contents

1. Introduction
2. Historical Background and Development
3. Classical Electrodynamics Recap
4. The Need for a Quantum Theory of Electromagnetism
5. QED as a Quantum Field Theory
6. The QED Lagrangian
7. Gauge Symmetry and U(1) Invariance
8. Interaction Terms and Feynman Rules
9. Quantization of the Electromagnetic Field
10. Dirac Field in QED
11. Feynman Diagrams and Perturbation Theory
12. Photon Propagator
13. Electron Propagator
14. Vertex Factors and Interaction Vertices
15. Scattering Amplitudes and Cross Sections
16. Renormalization in QED
17. Physical Predictions and Precision Tests
18. Anomalous Magnetic Moment
19. Running of the Fine Structure Constant
20. Applications and Legacy
21. QED in Modern Physics
22. Conclusion

1. Introduction

Quantum Electrodynamics (QED) is the quantum field theory that describes the interaction between light (photons) and matter (electrons and positrons). It is the first and most successful quantum gauge theory, forming the basis of the Standard Model and achieving extraordinary precision in its predictions.

2. Historical Background and Development

• Developed between 1927–1949 by Dirac, Feynman, Schwinger, Tomonaga, and Dyson.
• First theory to unify special relativity, quantum mechanics, and electromagnetism.
• Pioneered methods in perturbation theory, renormalization, and Feynman diagrams.

3. Classical Electrodynamics Recap

Maxwell’s equations describe classical electromagnetism, governing electric and magnetic fields. The interaction between charges is mediated by these fields, but the theory fails at atomic scales and in relativistic quantum contexts.

4. The Need for a Quantum Theory of Electromagnetism

Problems with classical theory:

• Cannot explain atomic spectra
• No mechanism for photon emission/absorption
• Violates Heisenberg uncertainty at small scales
• Incompatible with quantum mechanics

QED resolves these by treating both light and matter quantum mechanically.

5. QED as a Quantum Field Theory

In QED:

• The photon is the quantum of the electromagnetic field
• The electron is described by the Dirac field
• The interaction is mediated via exchange of virtual photons

QED is a U(1) gauge theory, meaning it is invariant under local phase transformations.

6. The QED Lagrangian

The QED Lagrangian is:

\[
\mathcal{L}{QED} = \bar{\psi}(i\gamma^\mu D\mu – m)\psi – \frac{1}{4}F^{\mu\nu}F_{\mu\nu}
\]

Where:

• \( \psi \): Dirac spinor field for the electron
• \( D_\mu = \partial_\mu + ieA_\mu \): gauge covariant derivative
• \( F_{\mu\nu} = \partial_\mu A_\nu – \partial_\nu A_\mu \): electromagnetic field strength tensor

7. Gauge Symmetry and U(1) Invariance

QED is invariant under local \( U(1) \) transformations:

\[
\psi(x) \rightarrow e^{i\alpha(x)} \psi(x), \quad A_\mu(x) \rightarrow A_\mu(x) – \frac{1}{e}\partial_\mu \alpha(x)
\]

This local symmetry enforces charge conservation and dictates the interaction form.

8. Interaction Terms and Feynman Rules

The interaction term:

\[
\mathcal{L}{int} = -e \bar{\psi} \gamma^\mu \psi A\mu
\]

This term describes the coupling of the electron current \( \bar{\psi} \gamma^\mu \psi \) to the photon field \( A_\mu \). Feynman rules are derived from this term for QED processes.

9. Quantization of the Electromagnetic Field

Field expansion in the radiation gauge:

\[
A_\mu(x) = \sum_{\lambda=1,2} \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2k^0}} \left[ \epsilon_\mu^\lambda(k) a_k^\lambda e^{-ikx} + h.c. \right]
\]

Photons are massless spin-1 particles with two polarization states.

10. Dirac Field in QED

The electron field is quantized as:

\[
\psi(x) = \sum_s \int \frac{d^3p}{(2\pi)^3} \left[ a_{p,s} u_s(p) e^{-ipx} + b_{p,s}^\dagger v_s(p) e^{ipx} \right]
\]

Electrons and positrons arise as excitations of this field.

11. Feynman Diagrams and Perturbation Theory

QED processes are computed via Feynman diagrams:

• Vertices: \( -ie\gamma^\mu \)
• Internal lines: propagators
• External lines: spinors or polarization vectors

Perturbative expansion in powers of the coupling constant \( \alpha \approx 1/137 \).

12. Photon Propagator

In Feynman gauge:

\[
D_{\mu\nu}(k) = \frac{-i\eta_{\mu\nu}}{k^2 + i\epsilon}
\]

Describes virtual photon exchange.

13. Electron Propagator

\[
S_F(p) = \frac{i(\not{p} + m)}{p^2 – m^2 + i\epsilon}
\]

Describes virtual electron propagation between interactions.

14. Vertex Factors and Interaction Vertices

Each vertex contributes a factor \( -ie\gamma^\mu \) to the matrix element. The number of vertices determines the order of the diagram in \( \alpha \).

15. Scattering Amplitudes and Cross Sections

Observable quantities like cross sections and decay rates are calculated from matrix elements using Fermi’s golden rule:

\[
\Gamma = \frac{1}{2E_i} \int d\Pi_f |\mathcal{M}|^2
\]

Where \( \mathcal{M} \) is the amplitude, and \( d\Pi_f \) is the final state phase space.

16. Renormalization in QED

Bare quantities are divergent. Renormalization redefines parameters to absorb infinities:

• Charge renormalization
• Mass renormalization
• Wavefunction renormalization

Yields finite, predictive results.

17. Physical Predictions and Precision Tests

QED is the most precisely tested theory:

• Electron magnetic moment agrees with experiment to 12 decimal places
• Lamb shift in hydrogen spectrum
• Bhabha and Møller scattering

18. Anomalous Magnetic Moment

QED predicts a correction to the electron’s magnetic moment:

\[
a_e = \frac{g – 2}{2} = \frac{\alpha}{2\pi} + \cdots
\]

This agrees with experiments to extreme accuracy.

19. Running of the Fine Structure Constant

The coupling \( \alpha \) is not constant at high energies. QED predicts:

\[
\alpha(q^2) = \frac{\alpha(0)}{1 – \frac{\alpha(0)}{3\pi} \log\left(\frac{q^2}{m^2}\right)}
\]

This is a manifestation of vacuum polarization.

20. Applications and Legacy

• Foundation for quantum electrodynamics and QFT
• Led to development of quantum chromodynamics (QCD) and electroweak theory
• Basis of modern particle physics experiments (collider physics, atomic precision tests)

21. QED in Modern Physics

QED remains essential in:

• Quantum computing and quantum optics
• High-energy physics
• Condensed matter systems (graphene, topological insulators)
• Astrophysics and cosmology (e.g., vacuum birefringence)

22. Conclusion

Quantum Electrodynamics is a triumph of theoretical physics. As the first quantum field theory, it established the principles of gauge invariance, renormalization, and quantum gauge interactions. It continues to serve as a model of precision and consistency, inspiring developments across physics.