Superluminality and UV completion

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 vph(0)vph(0) exceeds c  . It is explained why causality is related not to vph(0)vph(0) but to the high-frequency limit vph(∞)vph(∞) and how these are related by the Kramers–Kronig dispersion relation, depending on the sign of the imaginary part of the refractive index Imn(ω)Imn(ω) 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(ω)n(ω) is evaluated numerically and the role of the null energy condition Tμνkμkν⩾0Tμνkμkν⩾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 Λ  -systems exhibiting Raman gain lines with Imn(ω)<0Imn(ω)<0. The possibility that a negative Imn(ω)Imn(ω) 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.