Contents

# Contents

## 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.

## References

Discussion in the context of causal perturbation theory is in

Last revised on February 12, 2018 at 08:19:28. See the history of this page for a list of all contributions to it.