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Title: Is the photon dispersion relation modified at linear order in $E/E_{\mathrm{P}}$?
Plain: Some quantum gravity ideas make high-energy photons travel at slightly different speeds from low-energy ones. Gamma-ray bursts billions of light years away test this and already require any such linear effect to be weaker than Planck-scale estimates.
Precise: Parametrize photon propagation as v(E) = c [1 - s ((n+1)/2) (E/E_QG,n)^n] with $s = +1$ (subluminal) or $-1$ (superluminal). Measure or bound E_QG,1 and E_QG,2 for both signs, and determine whether any candidate UV completion predicts a nonzero linear term consistent with these bounds. An answer is a detection of an energy-dependent arrival delay, or a demonstration that a given completion forbids linear Lorentz violation.
Settled by: Time-of-flight measurements of variable high-energy sources at cosmological distances with sensitivity beyond E_QG,1 = 10 E_P, together with predictions from specific completions.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: LHAASO observations of GRB 221009A give, for subluminal propagation, E_QG,1 > 10 E_P and E_QG,2 > 6 x 10^-8 E_P at 95 percent confidence (2024); superluminal effects are bounded separately by the absence of photon decay.
Posed since: 1998