Promoting $\alpha$ to a spacetime-dependent parameter makes the scalar kinetic term vary by both terms linear and quadratic in the gauge field. Merely adding $eJ^\mu A_\mu$ accounts for the linear term but is not invariant because the scalar current itself changes under a local transformation.

With
$$
D_\mu\Phi=(\partial_\mu-2ieA_\mu)\Phi,
\qquad
A_\mu\mapsto A_\mu+\frac1e\partial_\mu\alpha,
$$
minimal coupling gives
$$
|D_\mu\Phi|^2
=|\partial_\mu\Phi|^2
+eA_\mu J_\Phi^\mu
+4e^2A_\mu A^\mu|\Phi|^2.
$$
The missing term is the <seagull vertex> $4e^2A^2|\Phi|^2$. This agrees exactly with replacing partial derivatives by <gauge covariant derivatives>.

The Dirac kinetic term is first order in derivatives:
$$
\overline\Psi i\gamma^\mu(\partial_\mu-ieA_\mu)\Psi.
$$
Its expansion is only linear in $A_\mu$, so the current coupling already completes the gauge-invariant fermion kinetic term and no $A^2\overline\Psi\Psi$ term arises.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/3/d/solution