The Refractive Index of Curved Spacetime: the Fate of Causality in QED

TL;DR

This work computes the complete frequency-dependent refractive index $n(\omega)$ for QED in curved spacetime using the world-line sigma model and the Penrose plane-wave limit, showing that the high-frequency wavefront velocity $v_{\rm wf}=v_{\rm ph}(\infty)$ equals $c$ while the Kramers-Kronig dispersion relation is violated due to non-analyticities from conjugate points in null geodesic congruences. The analysis reveals two background classes (Type I and II) with distinct dispersion and absorptive properties, and demonstrates that micro-causality cannot be guaranteed within the on-shell calculation, prompting the need for off-shell investigations. The results imply a UV-IR interplay where high-frequency propagation probes global spacetime structure and have broad implications for EFT bounds and the foundations of quantum field theory in curved spacetime. Overall, the paper provides a nonperturbative, geometrically transparent resolution to the causality puzzle in quantum gravitational optics, while opening questions about analyticity and micro-causality in curved backgrounds.

Abstract

It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime. Assuming the validity of the Kramers-Kronig dispersion relation, this would imply a superluminal wavefront velocity and the violation of causality. Here, we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic. We find that the high-frequency limit of the phase velocity (i.e. the wavefront velocity) is always equal to c and causality is assured. However, the Kramers-Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane, whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points. This puts into question the issue of micro-causality, i.e. the vanishing of commutators of field operators at spacelike separated points, in local quantum field theory in curved spacetime.

Figures (14)

• Figure 1: Photons propagating in curved spacetime feel the curvature in the neighbourhood of their geodesic because they can become virtual $e^+e^-$ pairs.
• Figure 2: The loop $x^\mu(\tau)$ with insertions of photon vertex operators at $\tau_1$ and $\tau_2$.
• Figure 3: The classical saddle point solution consists of a squashed loop which follows the geodesic $\gamma$. The length of the loop is $\sim\omega/m^2$ which represents a potentially interesting UV-IR mixing effect.
• Figure 4: (a) Type I null congruence with the special choice $\sigma_1=\sigma_2$ and $a_1=a_2$ so that the caustics in both directions coincide as focal points. (b) Type II null congruence showing one focusing and one defocusing direction.
• Figure 5: The real (green) and imaginary (red) parts of $n(\omega)-1$ for a simple model of a single absorption band with $\omega_0=1$, $\omega_p=0.1$ and $\gamma=0.3$.