Coulomb potential has nonlocal equal-time commutator

Choose a nonzero point $z_0$ on the $x_1$ axis. The delta term vanishes away from the origin, and by [F2] the real function ${\mathrm{\partial }}_1^{2}G$ is continuous and strictly positive near $z_0$. Thus the commutator kernel $-i{\mathrm{\partial }}_1^{2}G$ is continuous and nowhere zero on a sufficiently small neighborhood of $z_0$.

Choose nonnegative smooth tests $f,g$ from [F3], supported in small disjoint balls centered at points whose difference is $z_0$, with the difference of their supports contained in that neighborhood and both positive on smaller balls. The delta term in [F1] pairs to zero because the supports are disjoint. The remaining pairing is $-i$ times the integral of the strictly positive real kernel ${\mathrm{\partial }}_1^{2}G$ against $f(x)g(y)$, which is a positive real number before multiplication by $-i$ and hence is nonzero. Therefore the commutator pairing is nonzero for disjoint supports, so its kernel is not supported on the diagonal. The potential is nonlocal; the separate field-strength locality result is unaffected.