Approaches to Quantum Gravity

Generic predictions of quantum theories of gravity 557Quantum Gravity effects [42; 43]. These corrections introduce a fine structure intothe Hawking radiation, which is discussed in [44].28.3.4 Heat and the cosmological constantThere turns out to be a natural role for the cosmological constant, which is thatit parameterizes a quantum deformation of the algebra A. Forthecaseof3+ 1dimensions, this leads to A being SL q (2) with q = e k+2 2πıwhere the level k is givenby [33; 45; 46; 47]k = 6πG . (28.9)The quantum deformation of the symmetry algebra has a simple physical meaning,at least for >0. The ground state should be de Sitter spacetime, which hasa horizon with an areaA = 12π . (28.10)By the Bekenstein bound there should be a finite number of degrees of freedomobservable on the horizon, given byN =A4G = 3πG . (28.11)This relationship has been called the N-bound and has been conjectured by Banksand Fishler to be fundamental [48]. If an observer rotates they see the horizon rotatearound them, hence these degrees of freedom should fall into a single irreduciblerepresentation. But if the Bekenstein bound is a real limit, there should not be anyirreducible representation with more than N states in it. This is precisely true if therotational symmetry is quantum deformed by (28.9). Thus the N-bound is a consequenceof the quantum deformation of the symmetry induced by the cosmologicalconstant [47].A consequence of the quantum deformation of the label set is that the graphs areframed, so edges are represented by ribbons or tubes [33; 45; 46; 49; 50].A classic result of quantum field theory in curved spacetime is that QFTs on thebackground of de Sitter spacetime are thermal, with a temperature√3 . (28.12)T = 12πIt turns out that one can extend this to Quantum Gravity at the non-perturbativelevel using a simple argument based on the few facts we have already mentioned.The key is that de Sitter spacetime corresponds to the solution of the topologicalfield theory (28.6.). In terms of the configuration and momenta variables of