The bare Lagrangian is not wrong, it is unmeasurable

Lesson 3.4 ended with a divergence and refused to do anything about it. Lesson 3.5 computed two more and refused again. Here is what to do about them.

Start by being precise about what went wrong, because the usual description is wrong in a way that makes the fix look like a swindle. The usual description is that the theory predicts infinity, and renormalization subtracts the infinity off. If that were what happened it would be indefensible.

What actually happened is smaller and stranger. The Lagrangian

\mathcal{L} = -\tfrac14 F_{0\,\mu\nu}F_0^{\mu\nu} + \bar\psi_0\left(i\gamma^\mu\partial_\mu - m_0\right)\psi_0 - e_0\,\bar\psi_0\gamma^\mu\psi_0 A_{0\,\mu}

is written in terms of $e_0$ , $m_0$ and the fields $\psi_0$ , $A_0^\mu$ . Ask what $e_0$ is, numerically, and there is no answer — because nothing measures it. An experiment measures a cross section, or a spectral line, or a force between two charges at some separation. Each of those is a number you compute from $e_0$ , and every one of those computations includes loops.

So $e_0$ is not the charge. It is a symbol in an intermediate expression, and its relationship to anything measurable runs through exactly the divergent integrals that are causing the trouble.

That reframes the whole problem. The divergence is not in a prediction. It is in the dictionary between the parameters you wrote down and the parameters you can measure — and a dictionary is not something anybody observes.

What follows is a change of variables.