Approaches to Quantum Gravity

524 F. GirelliWe can define different types of addition using the mathematical structuresat hand. In the first case, since P lives on the de Sitter space dS ∼SO(4, 1)/SO(3, 1), we can use the coset structure to define the addition justas in Special Relativity where one uses the coset structure of the hyperboloidH ∼ SO(3, 1)/SO(3) to define the speeds addition. This definition has somepeculiar drawbacks: the addition is in general non-commutative but also nonassociative,properties which are clearly due to the coset structure. For examplein the Snyder case, a coset element is given by e iPμ J 4μ∼ e iPμ x μand the additionis constructed frome iPμ 1 J 4μe iPμ 2 J 4μ= (P 1 , P 2 )e i(P 1⊕P 2 ) μ J 4μ,where is a Lorentz transformation, encoding a Lorentz precession. The additionis clearly non-commutative, non-associative. The bicrossproduct case correspondsto the parameterization of the coset e iP0 J 40e iPi ˜J 4i, with ˜J 4i = J 0i − J 4i ,thatgivesa(non-Abelian) group structure to the coset:e iP0 1 J 40e iPi 1 ˜J 4ie iP0 2 J 40e iPi 2 ˜J 4i= e i(P0 1 ⊕P0 2 )J 40e i(Pi 1 ⊕Pi 2 ) ˜J 4i.The addition is then non-commutative but associative, a natural feature since thisconstruction arises using quantum groups.This construction has, however, a further physical draw back: P lives on dS andis bounded by the Planck mass (either the rest mass in the Snyder case or the 3dmomentum in the bicrossproduct case). The sum of momenta being defined on thede Sitter space is then still bounded by the Planck mass: there can be no object withrest mass or 3d momentum bigger than the Planck mass. This is of course a contradictionwith everyday experience, therefore this addition seems to be ill defined.This problem has been called the soccer ball problem by Amelino-Camelia. Apossible way out is to consider interacting particles or fields as suggested byFreidel [35].Another way out is to argue that the physical momentum to add is the 5d momentumP [34]. It is easy to add since it is a linear momentum, carrying the linearrepresentation of the 5d Poincaré group ISO(4, 1). In this case the sum is triviallyP tot = P 1 + P 2 ,and the new representation of ISO(4, 1) is given by a new parameter κ, which canbe for example κ = 2M P . In this way we have a rescaling of the radius of the deSitter space and therefore of the maximum mass as we would have expected. Inthis way we escape the soccer ball problem. This argument can be also extendedto the case where P actually represents the intrinsic momentum, so that P is theactual physical momentum. Indeed the P addition induces a non-linear addition onP, commutative and associative, free of the soccer ball problem. For this we use