Approaches to Quantum Gravity

Quantum Gravity phenomenology 445is best illustrated contemplating the possibility that such a dispersion relation beintroduced within a DSR framework. First of all let us notice that any theorycompatible with the DSR principle will have stable massless particles, so that bylooking for massless-particle decay one could falsify the DSR idea. A thresholdenergyrequirement for massless-particle decay (such as the E γ ≫ (m 2 e E p/|η|) 1/3mentioned above) cannot of course be introduced as an observer-independent law,and is therefore incompatible with the DSR principles.An analysis of the stability of massless particles that is compatible with the DSRprinciples can be obtained by combining the modification of the dispersion relationwith an associated modification of the laws of energy-momentum conservation.The form of the new law of energy-momentum conservation can be derived fromthe requirement of being compatible both with the DSR principles and with themodification of the dispersion relation [25; 26], and in particular for the a → b + ccase that I am considering one arrives at E γ ≃ E + + E − − λ ⃗p +·⃗p − , ⃗p γ ≃ ⃗p + +⃗p − − λE + ⃗p − − λE − ⃗p + . Using these in place of ordinary conservation of energymomentumone ends up with a result for cos(θ) which is still of the form (A+B)/Abut now with A = 2E + (E γ − E + ) + λE γ E + (E γ − E + ) and B = 2m 2 e . Evidentlythis formula always gives cos(θ) > 1, consistently with the fact that γ → e + e − isforbidden in DSR.22.5.3 Threshold anomaliesAnother opportunity to investigate Planck-scale departures from Lorentz symmetryis provided by certain types of energy thresholds for particle-production processesthat are relevant in astrophysics. This is a very powerful tool for Quantum Gravityphenomenology, and in fact at the beginning of this chapter I chose the evaluationof the threshold energy for p + γ CMBR → p + π as a key example.Numerous Quantum Gravity-phenomenology papers (see, e.g., Refs.[5; 6; 7; 8])have been devoted to the study of Planck-scale-modified thresholds, so the interestedreaders can find an abundance of related materials. I should stress here that,for the purpose of the point I am trying to convey in this section, the study of thresholdanomalies is not different from the study of the stability of massless particles:once again in the case in which the modified dispersion relation is combined withthe unmodified law of energy-momentum conservation one finds a striking effect.But the size of this effect can change significantly if one also allows a modificationof the law of energy-momentum conservation.22.5.4 Time-of-travel analysesA wavelength dependence of the speed of photons is obtained from a modifieddispersion relation, if one assumes the velocity to be still described by v = dE/dp.