= Solution
{c}
For a real local parameter $\alpha(x)$, the <Dirac adjoint> varies as $\delta\bar\psi=ie\alpha\bar\psi$. The matter kinetic term varies by
$$
\delta(\bar\psi i\gamma^\mu\partial_\mu\psi)=e(\partial_\mu\alpha)\bar\psi\gamma^\mu\psi.
$$
The <QED> interaction varies by $-e(\partial_\mu\alpha)\bar\psi\gamma^\mu\psi$, since the phases in $\bar\psi\gamma^\mu\psi$ cancel. These two variations cancel pointwise. The <mass> term is invariant for the same reason, and $\delta F_{\mu\nu}=\partial_\mu\partial_\nu\alpha-\partial_\nu\partial_\mu\alpha=0$.
Only the <gauge fixing> term varies. Put $B=\partial_\mu A^\mu$; then $\delta B=\partial^2\alpha$, giving
$$
\boxed{\delta\mathcal L_{\mathrm{QED}}=-\frac1\xi(\partial_\mu A^\mu)\partial^2\alpha.}
$$
For arbitrary local $\alpha$, the <gauge-fixed action> is therefore not invariant under this transformation of its dynamical fields. The restricted parameters satisfying $\partial^2\alpha=0$ give a residual invariance of the displayed density.
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