Approaches to Quantum Gravity

From quantum reference frames to deformed special relativity 511In section 26.2 I will quickly recall the construction of observables in QG, butalso ask a number of questions that should be answered to my mind, to understandthe QG physics. It is hard of course to do so in a QG theory like LQG, so I willillustrate the possible answers using a little toy model consisting in a universe ofspin 1 2 (qubits).In section 26.3 I want to describe what kind of flat semiclassical spacetime wecan expect to recover. For this I will start by recalling how a modification of themeasurement theory can be seen as implementing a deformation of the symmetries.A strong analogy holds with the toy model, an analogy that can be seen as anotherheuristic argument indicating that DSR is the right QG semiclassical limit. Thedeformation is usually done in the momentum space, that is the cotangent space.I will argue then that the geometry (that is, the tangent bundle picture) associatedto this flat semiclassical spacetime can be a described by a Finsler geometry [10].Since each type of deformed reference frame will correspond to a deformation ofthe symmetries, it is natural to ask if there is a global structure that allows one tounify the different constructions. I will show that indeed these different choices ofreference frames just correspond to different choices of gaugefixings (or choice ofobservers) in an extended phase space [11]. This allows us also to specify in anunambiguous way the symplectic form and the physical spacetime coordinates. Iwill conclude with some comments on the multiparticles states.26.2 Physics of Quantum Gravity: quantum reference frameThe symmetry group of General Relativity is the diffeomorphisms group. Invarianceunder this group means that the physics should not depend on the choice ofcoordinates. The coordinates x μ are parameters, they should not have any physicalmeaning. To understand that was an essential step in the GR construction. It alsoled to a long-standing misunderstanding. Indeed when doing physics it is natural touse coordinates systems: there exists a reference frame (clock, rulers) that allows usto measure a spacetime position, and so provide physical coordinates. The confusionarose since it seems that a coordinate system must be at the same time physicaland not physical. As so often, the answer to this paradox lies in its formulation: themeasured coordinates do not have the same status as the coordinates met in the GRmathematical definition.To define the physical coordinates, we must use some degrees of freedom [12]:the reference frame (that often can be confused with the measurement apparatus)is made of matter (clock and rulers) or gravitational degrees of freedom. This isa general feature: any physical quantity that is the outcome of some measurementquantifies the relation between two systems (the reference frame or apparatus andthe system under study).