Superluminality and UV Completion

TL;DR

The work scrutinizes whether fundamental UV completion axioms impose positivity constraints on leading irrelevant operators in IR EFTs and how causality constraints manifest. By examining QED in constant electromagnetic fields, QED in curved spacetime, and laser–atom Λ-systems with EIT or Raman gain, it shows that causality in flat space enforces $v_{ m ph}(\infty)\le1$ and positive ${\rm Im}\,n(\omega)$, aligning with IR positivity. In curved spacetime, however, $v_{ m ph}(0)$ can exceed unity due to curvature couplings, and the usual KK relation may fail to forbid such behavior if ${\rm Im}\,n(\omega)$ can be negative or if stable causality is invoked; the Λ-system analogies demonstrate that gain can produce $v_{ m ph}(\infty) < v_{ m ph}(0)$. Overall, the link between IR EFT positivity constraints and UV completion remains subtle in the presence of gravity, highlighting a need for further study of high-frequency dispersion in curved spacetime and its implications for quantum gravity.

Abstract

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 $Ł$-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.