Use $D_\mu=\partial_\mu+ieA_\mu$. The two Lagrangians are
$$
\mathcal L_{\mathrm{SQED}}=(D_\mu\phi)^*D^\mu\phi-M^2\phi^*\phi-\frac14F_{\mu\nu}F^{\mu\nu}
$$
and
$$
\mathcal L_{\mathrm{QED}}=\bar\psi(i\gamma^\mu D_\mu-m)\psi-\frac14F_{\mu\nu}F^{\mu\nu}.
$$
Both are invariant under the <U(1) gauge symmetry>
$$
\phi\mapsto e^{-ie\alpha(x)}\phi,
\qquad
\psi\mapsto e^{-ie\alpha(x)}\psi,
\qquad
A_\mu\mapsto A_\mu+\partial_\mu\alpha.
$$
For the constant phase subgroup, normalized to unit matter charge, the <Noether currents> are
$$
j^\mu_{\mathrm{SQED}}=i\{\phi^*D^\mu\phi-(D^\mu\phi)^*\phi\},
\qquad
j^\mu_{\mathrm{QED}}=\bar\psi\gamma^\mu\psi.
$$
Their electric currents are $e j^\mu$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-301/3/a/solution