Approaches to Quantum Gravity

248 T. Thiemannthat Ĉ(N) cannot be defined on H Diff because it must not leave that space invariantin any non-anomalous representation. Now consider the Master constraint∫M := d 3 C 2x √ . (13.20)|det(E)|Owing to the judicious choice of the “matrix” K ∝|det(E)| −1/2 the function M isspatially diffeomorphism invariant. It therefore can be represented directly on theHilbert space H Diff and can be solved by the direct integral method of the previoussubsection.Thus we arrive at the physical Hilbert space H Phys , which, however, is ratherimplicitly defined via the spectral resolution of the operator ̂M. The operator ̂Mis rather complicated as one might expect and hence its spectrum cannot be determinedin closed form, although simple, normalisable (in the inner product of H Diff )solutions are already known.13.3.2 Outstanding problems and further resultsIn the previous subsection we have restricted our attention to the gravitationaldegrees of freedom but similar results also hold for the matter content of the (supersymmetricextension of) the standard model. In order to perform calculations ofphysical interest and to make contact with the well established framework of QFTon curved spacetimes (e.g. the physics of the standard model at large physicalscales) it is mandatory to develop approximation schemes both for the physicalinner product and for the Dirac observables that are in principle available as displayedin section 13.2. Also it is possible that what we have arrived at is a theorywhose classical limit is not GR but rather a completely different sector, similar tothe different phases that one can get in statistical physics or Euclidean QFT. Henceit is necessary to develop semiclassical tools in order to establish the correct classicallimit. There is work in progress on both fronts: the spin foam models [33] thathave been intensively studied can be viewed as avenues towards approximationschemes for the physical inner product. Furthermore, coherent (minimal uncertainty)states for background independent theories of connections have alreadybeen constructed at the level of H Kin [34; 35; 36; 37; 38; 39; 40; 41] and onenow has to lift them to the level of H Diff and H Phys respectively.Next, within LQG it has been possible to identify a black hole sector [42]which encompasses all black holes of astrophysical interest (Schwarzschild–Reissner–Nordstrom–Kerr–Newman family) and a careful analysis has identifiedthe microscopic origin of the black hole entropy as punctures of the knots labellingphysical states (plus the labels D) with the horizon. The entropy counting for largeblack holes results in the Bekenstein–Hawking value if the parameter ι assumes