Approaches to Quantum Gravity

Quantum Gravity phenomenology 429Tasks one and two really are preparatory work. The “fun” begins immediatelyafter these first two tasks, when the relevant data are actually collected, possibledepartures from conventional theories are looked for, and the theories that could befalsified by those data are falsified.22.1.2 Task one accomplished: some effects introduced genuinelyat the Planck scale could be seenOver the past few years several authors have shown in different ways and for differentcandidate Planck-scale effects that, in spite of the horrifying smallness ofthese effects, some classes of doable experiments and observations could see theeffects. Just to make absolutely clear the fact that effects genuinely introduced atthe Planck scale could be seen, let me exhibit one very clear illustrative example.The Planck-scale effect I consider here is codified by the following energymomentum(dispersion) relation( )m 2 ≃ E 2 −⃗p 2 + η ⃗p 2 E 2, (22.1)where E p denotes again the Planck scale and η is a phenomenological parameter.This is a good choice because convincing the reader that I am dealing with an effectintroduced genuinely at the Planck scale is in this case effortless. It is in fact wellknown (see, e.g., Ref. [4]) that this type of Ep−2 correction to the dispersion relationcan result from discretization of spacetime on a lattice with Ep −1 lattice spacing. 2If such a modified dispersion relation is part of a framework where the laws ofenergy-momentum conservation are unchanged one easily finds [5; 6; 7; 8] significantimplications for the cosmic-ray spectrum. In fact, the “GZK cutoff”, a keyexpected feature of the cosmic-ray spectrum, is essentially given by the thresholdenergy for cosmic-ray protons to produce pions in collisions with CMBR photons.In the evaluation of the threshold energy for p+γ CMBR → p+π the correction termη ⃗p 2 E 2 /Ep 2 of (22.1) can be very significant. Whereas the classical-spacetime predictionfor the GZK cutoff is around 5.10 19 eV, at those energies the Planck-scaleE 2 p2 The idea of a rigid lattice description of spacetime is not really one of the most advanced for Quantum Gravityresearch, but this consideration is irrelevant for task one: in order to get this phenomenology started we firstmust establish that the sensitivities we have are sufficient for effects as small as typically obtained from introducingstructure at the Planck scale. The smallness of the effect in (22.1) is clearly representative of the typeof magnitude that Quantum Gravity effects are expected to have, and the fact that it can also be obtained froma lattice with Ep−1 spacing confirms this point. It is at a later stage of the development of this phenomenology,much beyond task one, that we should become concerned with testing “plausible Quantum Gravity models”(whatever that means). Still it is noteworthy that, as discussed in some detail in Section 22.3, some modernQuantum Gravity-research ideas, such as the one of spacetime noncommutativity, appear to give rise to thesame type of effect, and actually in some cases one is led to considering effects similar to (22.1) but with aweaker (and therefore more testable) Planck-scale correction, going like E −1 prather than E −2 p .