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 Incoming links: QED Feynman rules

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 4 / Solution 2026-09-28

A gauge symmetry is a local redundancy in the fields used to describe one physical state. With metric signature

(+ - - -)

D_{μ} = \partial_{μ} + i e A_{μ}, F_{μν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} .

The quantum electrodynamics Lagrangian is

L_{QED} = - \frac{1}{4} F_{μν} F^{μν} + \overline{ψ} (i γ^{μ} D_{μ} - m) ψ = - \frac{1}{4} F^{2} + \overline{ψ} (i \neq \partial - m) ψ - e \overline{ψ} γ^{μ} ψ A_{μ} .

Under the local U(1) gauge symmetry

ψ \mapsto e^{- i eχ (x)} ψ, \overline{ψ} \mapsto \overline{ψ} e^{i eχ (x)}, A_{μ} \mapsto A_{μ} + \partial_{μ} χ,

one has

D_{μ} ψ \mapsto e^{- i eχ} D_{μ} ψ

and

F_{μν} \mapsto F_{μν}

. Both terms in the Lagrangian are therefore invariant.

In Feynman gauge, the momentum-space QED Feynman rules are:

an internal electron line of momentum $r$ : $i (\neq r + m) / (r^{2} - m^{2} + i ϵ)$ ;
an internal photon line of momentum $k$ : $- i η_{μν} / (k^{2} + i ϵ)$ ;
an electron-photon vertex: $- i e γ^{μ}$ , with four-momentum conserved;
incoming and outgoing electrons: $u_{s} (p)$ and $\overline{u}_{s} (p)$ ;
incoming and outgoing positrons: $\overline{v}_{s} (p)$ and $v_{s} (p)$ along the oriented fermion chain;
incoming and outgoing photons: $ϵ_{μ}^{(λ)} (k)$ and $ϵ_{μ}^{(λ) *} (k)$ .

One integrates each undetermined loop momentum, includes a factor

- 1

for each closed fermion loop, and imposes overall momentum conservation. None of the following tree diagrams contains a loop.

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