Feynman Rules for Quantum Electrodynamics (QED)

Richard Feynman made significant contributions to quantum electrodynamics (QED), particularly in the development of Feynman diagrams and Feynman rules.

Feynman rules provide a systematic way to calculate probabilities and amplitudes for particle interactions in quantum field theory.

Here are the Feynman rules for QED:

1. Propagator:

• For a photon (gauge boson of electromagnetism): The propagator is given by the Feynman propagator for a massless particle:

$D_F^{\mu\nu}(k)$ = $-\frac{ig^{\mu\nu}}{k^2 + i\epsilon}$

where k is the four-momentum of the photon, $g^{\mu\nu}$ is the Minkowski metric tensor, and \epsilon is a small positive number used to regulate divergences.

 

• For an electron (fermion):

The propagator is given by the Dirac propagator:

S(p) = $\frac{i(\gamma^\mu p_\mu + m)}{p^2 – m^2 + i\epsilon}$

where p is the four-momentum of the electron, \gamma^\mu are the Dirac gamma matrices, and m is the mass of the electron.

 

2. Vertex:

Photon-electron vertex: The interaction between a photon and an electron is described by the vertex factor:

$-ie\gamma^\mu$

where e is the electric charge of the electron and \gamma^\mu is a gamma matrix.

 

3. Coupling constant:

The coupling constant for QED is given by the elementary charge e, which determines the strength of the electromagnetic interaction.

 

4. Conservation of momentum and charge:

• At each vertex, momentum must be conserved. This means that the sum of the incoming momenta is equal to the sum of the outgoing momenta.
• At each vertex, charge must also be conserved. This means that the sum of the charges of the incoming particles is equal to the sum of the charges of the outgoing particles.

 

5. Diagrammatic representation:

• Feynman diagrams represent different particle interactions. Lines represent particles, and vertices represent interactions.
• External lines correspond to incoming or outgoing particles, while internal lines represent virtual particles.
• Each line is associated with a propagator, and each vertex is associated with a vertex factor.

By using these Feynman rules, one can construct and evaluate Feynman diagrams to calculate scattering amplitudes and transition probabilities for various QED processes involving photons and electrons.

These rules provide a powerful framework for performing calculations in quantum field theory.