[Paper Review] `Superluminal' Photon Propagation in QED in Curved Spacetime is Dispersive and Causal

Timothy J. Hollowood, G.M. Shore|arXiv (Cornell University)|Jun 7, 2010

Quantum Electrodynamics and Casimir Effect4 citations

TL;DR

This paper demonstrates that photon propagation in quantum electrodynamics (QED) on curved spacetime is dispersive and causal, despite apparent superluminal phase velocities at low frequencies. By analyzing the refractive index via the Penrose plane wave limit and effective field theory, it shows causality is preserved through frequency-dependent dispersion relations and analyticity in the complex frequency plane, countering claims of non-causality in prior work.

ABSTRACT

It is now well-known that vacuum polarisation in QED can lead to superluminal low-frequency phase velocities for photons propagating in curved spacetimes. In a series of papers, we have shown that this quantum phenomenon is dispersive and have calculated the full frequency dependence of the refractive index, explaining in detail how causality is preserved and various familiar results from quantum field theory such as the Kramers-Kronig dispersion relation and the optical theorem are realised in curved spacetime. These results have been criticised in a recent paper by Akhoury and Dolgov arXiv:1003.6110 [hep-th], who assert that photon propagation is neither dispersive nor necessarily causal. In this note, we point out a series of errors in their work which have led to this false conclusion.

Motivation & Objective

• To resolve the apparent paradox of superluminal photon phase velocities in QED on curved spacetime while preserving causality.
• To demonstrate that vacuum polarization effects in curved spacetime are dispersive, not non-dispersive as claimed in a prior critique.
• To validate the use of the Penrose plane wave limit as an accurate approximation for capturing curvature-induced modifications to photon propagation.
• To establish the analytic structure of the refractive index in the complex frequency plane, ensuring causality via the Kramers-Kronig relations.
• To refute claims of non-causality in photon propagation by explicitly showing the Pauli-Jordan commutator vanishes outside the light cone.

Proposed method

• Utilizing the eikonal and weak curvature approximations to reduce the general curved spacetime background to its Penrose plane wave limit along a null geodesic.
• Applying the effective action for QED in curved spacetime, including curvature-coupled terms proportional to $ R_{\mu\nu}F^{\mu\lambda}F^{\nu}{}_{\lambda} $ and $ R_{\mu\nu\lambda\rho}F^{\mu\nu}F^{\lambda\rho} $, to derive the refractive index.
• Deriving the frequency-dependent refractive index $ n_{ij} $ in terms of curvature components $ R_{uu} $ and $ R_{uiuj} $, showing explicit dispersion.
• Employing the Van Vleck-Morette determinant and worldline path integral techniques to compute vacuum polarization corrections in the geodesic deviation framework.
• Constructing the Pauli-Jordan commutator $ iG_{\mu\nu}(x,x') $ explicitly in the Penrose limit and proving its support is confined to the light cone.
• Using the Kramers-Kronig dispersion relations and analyticity in the upper half-plane to confirm causality, with the refractive index satisfying $ \text{Im}[n(\omega)] \sim \delta(\omega) $ only when curvature gradients are included.

Experimental results

Research questions

• RQ1 Is photon propagation in QED on curved spacetime truly dispersive, or is the refractive index frequency-independent as claimed in [2]?
• RQ2 Can the Penrose plane wave limit accurately reproduce the vacuum polarization effects found in the full curved spacetime background?
• RQ3 How is causality preserved in the presence of superluminal phase velocities due to vacuum polarization?
• RQ4 What is the role of curvature gradients (e.g., $ \partial_u R $) in generating frequency-dependent imaginary parts of the refractive index?
• RQ5 Does the Pauli-Jordan commutator for the electromagnetic field vanish outside the light cone in curved spacetime with vacuum polarization?

Key findings

• The refractive index for photons in curved spacetime is explicitly dispersive, with a frequency dependence arising from curvature components $ R_{uu} $ and $ R_{uiuj} $, as shown in Eq. (3).
• The Penrose plane wave limit correctly captures the Drummond-Hathrell effect, contradicting the claim in [2] that it fails to do so due to a sign error in index contraction.
• Causality is preserved because the refractive index is analytic in the upper half complex frequency plane, ensuring the Kramers-Kronig relations hold.
• The imaginary part of the refractive index is not just $ \delta(q^2) $; it includes contributions proportional to $ k^\mu \partial_\mu R $, which are frequency-dependent and arise from derivatives of the delta function in momentum space.
• The Pauli-Jordan commutator $ iG_{\mu\nu}(x,x') $ vanishes outside the light cone, as shown by explicit construction in Eq. (33), proving causality in the full theory.
• The full dispersive refractive index reduces to the Drummond-Hathrell result in the low-frequency limit, confirming consistency with established effective field theory.