  The idea that the existence of a consistent UV completion satisfying the
fundamental axioms of local quantum field theory or string theory may impose
positivity constraints on the couplings of the leading irrelevant operators in
a low-energy effective field theory is critically discussed. Violation of these
constraints implies superluminal propagation, in the sense that the
low-frequency limit of the phase velocity $v_{\rm ph}(0)$ exceeds $c$. It is
explained why causality is related not to $v_{\rm ph}(0)$ but to the
high-frequency limit $v_{\rm ph}(\infty)$ and how these are related by the
Kramers-Kronig dispersion relation, depending on the sign of the imaginary part
of the refractive index $\Ima n(\w)$ which is normally assumed positive.
Superluminal propagation and its relation to UV completion is investigated in
detail in three theories: QED in a background electromagnetic field, where the
full dispersion relation for $n(\w)$ is evaluated numerically for the first
time and the role of the null energy condition $T_{\m\n}k^\m k^\n \ge 0$ is
highlighted; QED in a background gravitational field, where examples of
superluminal low-frequency phase velocities arise in violation of the
positivity constraints; and light propagation in coupled laser-atom
$\L$-systems exhibiting Raman gain lines with $\Ima n(\w) < 0$. The possibility
that a negative $\Ima n(\w)$ must occur in quantum field theories involving
gravity to avoid causality violation, and the implications for the relation of
IR effective field theories to their UV completion, are carefully analysed.
