Arising from: Jacobson, T., Liberati, S. & Mattingly, D. Nature 424, 1019–1021 (2003)

Quantum gravity may cause the vacuum to act as a non-trivial medium (space–time foam), which alters the standard Lorentz relation between the energy and momentum of matter particles, thereby modifying their dispersion relations. Jacobson, Liberati and Mattingly1 argue that synchrotron radiation from the Crab nebula imposes a stringent constraint on any modification of the dispersion relation of the electron that might be induced by quantum gravity, but their analysis does not constrain any modification of the dispersion relation of the photon2,3. Such quantum-gravity effects need not obey the equivalence principle4 in the sense of being universal for all matter particles, as exemplified by quantum-gravity models in which photons are the only standard-model particles able to ‘see’ special quantum-gravity configurations that modify their dispersion relations. This implies that photons may be the only sensitive probe of quantum-gravity effects on particle dispersion relations, and the results of Jacobson et al. do not exclude all possible modifications of dispersion relations, even if they are suppressed by only a single power of the Planck mass (the characteristic quantum-gravity scale) — contrary to some subsequent interpretations of their results.

As pointed out previously4, there are theoretical models in which quantum gravity produces Lorentz invariance-violating effects for neutral particles, such as the photon, but not charged particles, such as the electron. One model of space–time foam3 suggests a linear modification of the dispersion relation for the photon: pγ=Eγ−(Eγ2/MQG), where pγ(Eγ) is the photon's momentum (energy) and MQG is some characteristic scale associated with quantum gravity, which may be of the same order as the Planck mass MP≈1019 GeV. However, this model3 predicts that there is no such modification of the dispersion relation for the electron4, and hence is compatible with the constraint1 from the Crab nebula. In such models, constraints on the electron and nucleon dispersion relations1,5,6 are irrelevant, leaving measurements on time profiles of very remote γ-ray bursts2,7 as the best approach for probing quantum-gravity effects.

The basic reason for this violation of the equivalence principle in the quantum-gravity model3 is its description of space–time foam by using quantum defects in space–time with vacuum quantum numbers, as in one interpretation of Liouville string theory8. These can be excited only by particles that are neutral under the gauge group of the standard model, such as photons, and such interactions give the vacuum a non-trivial refractive index for light of different frequencies (energies)2. Charged particles, such as electrons, cannot form such excitations, so do not ‘see’ the space–time foam at all, and hence obey the usual Lorentz kinematics. As a result of the excitation of the vacuum by an energetic photon, space–time is distorted and the photon travels with a velocity smaller than the (supposedly universal) speed of light in vacuo, c, as postulated in the special and general theories of relativity.

As the electron has no interaction with the quantum-gravitational vacuum medium in this approach, it emits no Čerenkov radiation, despite travelling faster than photons, thus avoiding the vacuum Čerenkov radiation constraint9, as well as the Crab nebula constraint derived by Jacobson et al.1 The model in ref. 3 also avoids the strong constraints described in ref. 10, as well as many other constraints on quantum-gravity effects11. Claims that modified dispersion relations for photons would result in phase incoherence of light, and thereby destroy diffraction patterns in images of extragalactic sources12, have been criticized by Ng13, who pointed out that the induced incoherent effects had been overestimated12 by a large factor. In the specific model of ref. 3, the re-emission of the photon by a space–time defect is accompanied by a random phase in its wave function, destroying any cumulative phase incoherence. Finally, we note that, as the nucleon is a bound state, it is more complex to analyse, but we also do not expect it to exhibit a linear modification of the normal Lorentz-invariant dispersion relation, avoiding other constraints5,6.

The strong bound of Jacobson et al.1 on the electron underlines the interest in probing directly the dispersion relation of the photon. The study of the arrival times of photons from γ-ray bursts2 still appears to be the best experimental probe of any possible refractive index for photons, and should be pursued further. It has already established a lower limit on MQG close to 1016 GeV (ref. 7), and current (HETE, INTEGRAL) and future (GLAST, AMS) high-energy space missions have the potential to reach the Planck scale for any linear quantum-gravity modification of the photon's dispersion relation.