Causality and Micro-Causality in Curved Spacetime

TL;DR

This paper investigates how causality and micro-causality are affected in quantum electrodynamics within curved spacetime, revealing non-analytic behaviors in the photon propagator and implications for causality principles.

Contribution

It introduces a non-perturbative calculation of vacuum polarization in curved spacetime using novel world-line methods, showing effects like superluminal phase velocities but preserving causality.

Findings

Photon propagator exhibits non-analytic behavior due to vacuum polarization.

Refractive index shows superluminal phase velocities, dispersion, and bi-refringence.

Wavefront velocity remains at c, ensuring causality is not violated.

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 \gamma \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.