= Solution
{c}
The massive <Dirac propagator> is
$$
S_F(x)=\int\frac{d^Dk}{(2\pi)^D}\frac{i(\not k+m)e^{-ik\cdot x}}{k^2-m^2+i0}.
$$
Use the <gamma matrix trace identities> $\operatorname{tr}\gamma^\mu=0$ and $\operatorname{tr}(\gamma^\mu\gamma^\rho)=4g^{\mu\rho}$. They give
$$
\operatorname{tr}[\gamma^\mu S_F(0)]=4i\int\frac{d^Dk}{(2\pi)^D}\frac{k^\mu}{k^2-m^2+i0}=\boxed{0}.
$$
The integrand is odd under $k\mapsto-k$, and <dimensional regularization> preserves that symmetry. The same vanishing holds for $S=iS_F$. One must not assert that $S(0)$ itself is zero: its term proportional to $m\mathbf1$ generally is nonzero and divergent, but its trace against a single <gamma matrix> is zero. This one-photon <fermion loop> is also the simplest instance of <Furry's theorem>.
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