Approaches to Quantum Gravity

446 G. Amelino-CameliaFor the dispersion relation here considered one finds that at “intermediate energies”(m < E ≪ E p ) the velocity law will take the formv ≃ 1 − m22E + η n + 1 E n. (22.3)2 2 EpnOn the basis of this formula one would find that two simultaneously emitted photonsshould reach the detector at different times if they carry different energy. Andthis time-of-arrival-difference effect can be significant [10; 11] in the analysis ofshort-duration gamma-ray bursts that reach us from cosmological distances. For agamma-ray burst it is not uncommon that the time travelled before reaching ourEarth detectors be of order T ∼ 10 17 s. Microbursts within a burst can have veryshort duration, as short as 10 −3 s, and this means that the photons that composesuch a microburst are all emitted at the same time, up to an uncertainty of 10 −3 s.Some of the photons in these bursts have energies that extend at least up to theGeV range, and for two photons with energy difference of order E ∼ 1GeV aE/E p speed difference over a time of travel of 10 17 s would lead to a differencein times of arrival of order t ∼ T E E p∼ 10 −2 s, which is significant (the timeof-arrivaldifferences would be larger than the time-of-emission differences withina microburst).It is well established that the sensitivities achievable [47; 48] with the next generationof gamma-ray telescopes, such as GLAST [47; 48], could allow us to testvery significantly (22.3) in the case n = 1, by possibly pushing the limit on |η|far below 1. And, while probably beyond the reach of photon astrophysics, for thecase n = 2 neutrino astronomy may lead to valuable insight [21; 22].Notice, however, that in some test theories it may be incorrect to combine thelimits obtained in this way with the limits obtained from the threshold analysesdiscussed above in these two different ways. As stressed above the threshold analysesrequire a test theory with certain forms of the law of energy-momentumconservation, whereas the law of energy-momentum conservation is not relevantfor the time-of-travel studies. On the other hand a test theory using thetime-of-travel analyses, in adopting v = dE/dp, should have a standard formof the Heisenberg commutator (so that x → ∂/∂p) and assign a standard role tothe Hamiltonian (so that v ∼[x, H]).22.5.5 Synchrotron radiationAs observed recently in Ref. [49], in the mechanism that leads to the productionof synchrotron radiation a key role is played by the special-relativistic velocitylaw v = dE/dp ≃ 1 − m 2 /(2E 2 ). And an interesting observation is obtained byconsidering the velocity law (22.3) for the case n = 1. Assuming that all otheraspects of the analysis of synchrotron radiation remain unmodified at the Planck