Approaches to Quantum Gravity

510 F. GirelliThe new Lagrangian ˜L M (φ M ) describes in an effective way the QG fluctuations.A natural consequence is then an important modification of matter dynamics andspacetime concepts. For example, field evolution in this context might not be unitarysince we have integrated out some degrees of freedom. In the case of a simpleclassical relativistic particle, we would expect the dynamics to be described by amodified mass shell condition.The explicit calculation can currently be done explicitly only in a three dimensionalspacetime [2]. In this case a non-commutative spacetime emerges, as wellas a modified notion of multiparticles states.Unfortunately, deriving this semiclassical limit is still a challenge in the fourdimensional case. Instead of trying to derive it by brute force, one can try tocook up a theory describing the semiclassical spacetime. We intend to put at thekinematical level the QG fluctuations, to have an effective notion of flat semiclassicalspacetime. A modification of the Poincaré symmetries is then present. To myknowledge, deformed (or doubly) special relativity (DSR) is one of the best candidatesto describe this setting. There is a number of heuristic arguments to showhow DSR can be derived from a 4d QG theory [3; 4], but not yet any solid mathematicalargument as in 3d [5]. Under the name of DSR actually go many differentapproaches (Snyder’s approach [6], modified measurement [7; 8], quantum groupsapproach [9]), which are not clearly equivalent. They all have common features: ingeneral a deformation of the Poincaré symmetries, a non-commutative spacetimeand a modification of the multiparticles states. There are two ways to understandthe apparent freedom in the choice of DSR type: either there is only one physicaldeformation together with one set of physical phase space coordinates (this is whathappens in the 3d case), or all the different DSR structures can be unified in onegeneral new structure. This question needs to be clarified at least in the simplestexample, the “free” particle, before getting to quantum field theory and so on.Clearly, there are now two jobs to be done: on one hand to check that DSRis really the semiclassical limit of some QG theory; on the other, to understandits physics and be able to make predictions to confront it with the forthcomingexperiments.Here I would like to argue that understanding the physics of DSR can be relatedto understanding the QG physics. Indeed General Relativity is a constrained theory,which means that observables must be relational, and in particular constructedin terms of physical reference frames [12]. When moving to the QG regime, oneshould talk about quantum reference frames (QRF), therefore quantum coordinatesand so on. Moving on to the semiclassical limit, one should still feel the funny QGphysics. From this perspective DSR will arise as a modified measurement theory,owing to the modification of the notion of reference frame still bearing somequantum/gravitational features.