nLab electron-photon interaction

Idea

In quantum electrodynamics the interaction between the Dirac field $\Psi$, whose quanta are electrons, and the electromagnetic field $A$, whose quanta are photons, is encoded by the interaction Lagrangian density

$$\mathbf{L}_{int} \;=\; i (\Gamma^\mu)^\alpha{}_\beta \overline{\psi}_\alpha \psi^\beta a^\mu \, dvol_\Sigma$$

(with notation as as used at A first idea of quantum field theory, see this example).

For $g_{sw} \in C^\infty_{cp}(\Sigma)$ a bump function on spacetime thought of as an adiabatically switched coupling constant, the corresponding interaction action functional is the local observable

$$\begin{aligned} S_{int} & \coloneqq i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, \overline{\mathbf{\Psi}}_\alpha(x) \cdot \mathbf{\Psi}^\beta(x) \cdot \mathbf{A}_\mu(x) \, dvol_\Sigma(x) \\ & = i \underset{\Sigma}{\int} g_{sw}(x) \, (\Gamma^\mu)^\alpha{}_\beta \, : \overline{\mathbf{\Psi}}_\alpha(x) \mathbf{\Psi}^\beta(x) \mathbf{A}_\mu(x) : \, dvol_\Sigma(x) \end{aligned} \,,$$

where in the first line we have the integral over a pointwise product (this def.) of three field observables (this def.), which in the second line we write equivalently as a normal ordered product, by the discusssion at Wick algebra (this def.).

(e.g. Scharf 95, (3.3.1))

The corresponding Feynman diagram is

The square of the coupling constant

$$\alpha \coloneqq \tfrac{1}{4 \pi} g^2$$

is called the fine structure constant.

Related concepts

• Lamb shift

• phi^n interaction

References

Discussion in the context of causal perturbation theory is in

• Günter Scharf, section 3.3 of Finite Quantum Electrodynamics – The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition