Causality and Micro-Causality in Curved Spacetime

TL;DR

This work addresses how causality and micro-causality are realized for QED in curved spacetime by uncovering non-analytic features in the photon propagator arising from vacuum polarization. It introduces a nonperturbative, world-line sigma-model approach combined with the Penrose plane-wave limit to obtain the complete frequency dependence $\Pi_{ij}(\omega)$ and shows that conjugate points along null geodesics induce branch points that violate the conventional Kramers-Kronig dispersion relation, while still yielding a wavefront speed $v_{\rm wf}=c$ in the high-frequency limit. The results reveal superluminal low-frequency phase velocities, dispersion, absorption (via $\gamma\to e^+e^-$), and birefringence, with Type I and Type II plane-wave backgrounds exhibiting distinct analytic structures; the on-shell analysis signals possible violations of micro-causality, though a full off-shell treatment is needed to draw definitive conclusions. Overall, the study highlights a UV-IR linkage through the global geometry of null geodesics and suggests careful reconsideration of micro-causality in curved spacetime, especially near horizons or singularities. $$n_i(\omega) = 1 + \frac{1}{\omega^2} \Pi_{ii}(\omega)$$, with $\Pi_{ij}$ non-analytic in the upper half-plane and $v_{\rm wf}=c$ as $\omega\to\infty$.

Abstract

We consider how causality and micro-causality are realised in QED in curved spacetime. The photon propagator is found to exhibit novel non-analytic behaviour due to vacuum polarization, which invalidates the Kramers-Kronig dispersion relation and calls into question the validity of micro-causality in curved spacetime. This non-analyticity is ultimately related to the generic focusing nature of congruences of geodesics in curved spacetime, as implied by the null energy condition, and the existence of conjugate points. These results arise from a calculation of the complete non-perturbative frequency dependence of the vacuum polarization tensor in QED, using novel world-line path integral methods together with the Penrose plane-wave limit of spacetime in the neighbourhood of a null geodesic. The refractive index of curved spacetime is shown to exhibit superluminal phase velocities, dispersion, absorption (due to γ\to e^+e^-) and bi-refringence, but we demonstrate that the wavefront velocity (the high-frequency limit of the phase velocity) is indeed c, thereby guaranteeing that causality itself is respected.

Figures (8)

• Figure 1: Points $p$ and $q$ on a geodesic $\gamma$ are conjugate if they can be joined---at least in an infinitesimal sense---by a family of geodesics close to $\gamma$. The existence of conjugate points means that the classical saddle-point solution with the photon connected at $p$ and $q$ to a degenerate $e^+e^-$ loop squashed onto the geodesic $\gamma$ is deformable into a non-degenerate loop as shown.
• Figure 2: The $e^+e^-$ loop $x^\mu(\tau)$ with insertions of photon vertex operators at $\tau_1$ and $\tau_2$.
• Figure 3: The behaviour of the congruence of null geodesics for the (a) Type I and (b) Type II plane wave metrics.
• Figure 4: The $n=1$ (red) and $n=2$ (blue) zero modes for $\xi=\tfrac{1}{2}$. The points $u=\pm u_0$ are conjugate points for $\gamma$.
• Figure 5: The behaviour of $\text{Re}\,n_i(\omega)-1$ in units of $\alpha R/(2\pi m^2)$, as a function of $\tfrac{1}{2}\log\omega^2 R/m^4$ for (a) Type I conformally flat case ($n_1=n_2$) and (b) Type II Ricci flat case ($i=1$ red, $i=2$ green). Note that in both cases $n_i(\omega)$ approaches 1 from below as $\omega \to \infty$.