Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2007-01-19
Nucl.Phys.B778:219-258,2007
Physics
High Energy Physics
High Energy Physics - Theory
42 pages, 14 figures
Scientific paper
10.1016/j.nuclphysb.2007.03.034
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.
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