Approaches to Quantum Gravity

26From quantum reference frames to deformedspecial relativityF. GIRELLI26.1 IntroductionQuantum Gravity (QG) theory was thought for a long time to be just a mathematicaltheory since it was relevant only at extreme energies: for example right after theBig Bang or very close to the singularity in a black hole. It is hard to probe any ofthe physics happening there for obvious reasons. This situation changed recently:the possible existence of extra dimensions lowers the QG typical energy scale andso could make it possible to see QG effects in the new particle accelerators (see forexample [1] and references therein). Even without extra dimensions, it was proposedthat some extreme astrophysical situations might provide ways to probe thequantum (more exactly the semiclassical) structure of spacetime, see for exampleAmelino-Camelia’s contribution to this book (chapter 22).In the context of loop quantum gravity (LQG) and spinfoams, different modelsexist and it is unclear if they are equivalent or not. If one were able to construct asemiclassical limit for those, one would be able to make predictions for the differentmodels and wait for the forthcoming experiments to falsify some of them. QGwould be about to become true physics!More explicitly one should calculate the full partition function∫S = dφ M dge i ∫ L M (φ M ,g) + L GR (g) ,where the φ M represent all the matter and interactions fields other than gravitationalwhich are encoded in the metric g,andL M (φ M , g), L GR (g) are respectivelythe Lagrangian for matter and gravity. To make valuable predictions for the nextexperiments we would like to integrate out all the QG degrees of freedom aroundthe flat metric η (assuming that the cosmological constant is zero) to obtain aneffective action for matter encoding the QG physics:∫S = dφ M e i ∫ ˜L M (φ) .Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.