Approaches to Quantum Gravity

From quantum reference frames to deformed special relativity 521the physical coordinates. In fact by adding two extra dimensions to phase space,we can do more, that is, see all the different deformations as different gaugefixingsor different choices of non-equivalent observers.Since we have an extra fundamental mass parameter M P inthegamewecanrewrite the MDR (26.5) asE 2 − p 2 − M(p) 2 =−M P 2 , with M(p) 2 = m 2 + F(p,μ,M P ) − M P 2 ,such that it looks like a five dimensional mass shell conditionP μ P μ − P 2 4 =−M P 2 . (26.7)), we do the scale transformationL P → L P χ = L C that can be chosen to keep the speed of light c fixed as well as. G becomes however, variable G → χ 2 G: in the particle units we have a fixedmass but a variable G. Since the mass becomes a variable (in Planck units) encodingthe QG fluctuations, it is natural to extend the configuration space to include itas a true variable. This goes naturally as encoding G as a new universal constant[33], since it allows us to transform a mass into a length. We consider now ourextended phase space as given by a configuration space (y A = y μ , y 4 = G xc 2 4 ),where x 4 has mass dimension, and the momentum space given by P A .A DSR particle will be described by the actionP4 2 = M(p)2 can be interpreted as having a variable mass. This is something naturalfrom the GR point of view. For example an extended object has a varying massin a curved spacetime [28]. Since a quantum particle cannot be localized, curvaturemight introduce some slight variations to its mass. Note that now, momentumspace is identified with the de Sitter space, so that we are out of the usual geometricalscheme, the cotangent bundle is not a vector fiber bundle anymore. This willhave direct consequences on the addition of momenta as we shall see in the nextsection.From the QG point of view, one can expects the Newton constant G to berenormalized to encompass quantum corrections [31]. Instead of considering fluctuationsin G and a fixed mass, we can describe the theory in an effective wayas a fixed G, with a fluctuating mass: G(x) m → Gm(x). In fact all this isrelated to the choice of units. The notion of a variable mass in terms of unitshas already been studied in detail by Bekenstein [32]. The Planck units system(M P , L P , T P ), is independent of any particle data. All the different fundamentalconstants can be expressed in terms of these quantities, and in these units arefixed. Now consider a particle with a variable mass, that is expressed in the Planckunits we have m = χ M P . If one moves to the particle unit, for example the Comptonunit (M C = m, L C = mc , T C = cL −1C∫S 5d = dy A P A − λ 1 (P A P A + M 2 P ) − λ 2 (P 4 − M),