Approaches to Quantum Gravity

206 T. Banksit has given us a construction of these models as quantum field theories on theconformal boundary of space-time. Remarkably, all the asymptotically flat modelsare exactly supersymmetric, and all the well understood AdS models with curvaturesmall enough for the SUGRA approximation to be valid have SUSY restoredasymptotically on the boundary of space-time. All these models are holographic inthat they describe space-time in terms of variables defined on a holographic screenat infinity.More realistic models of Quantum Gravity, which take into account cosmology,need a more flexible and local version of holography. General arguments show thatthe underlying variables of such a local formulation cannot be gauge invariant. Idescribed a proposal for a general quantum space-time as a network of Hilbertspaces and evolution operators. Each Hilbert space was to be thought of as therepresentation of physics in a particular causal diamond in space-time. The holographicprinciple is implemented by relating the dimension of the Hilbert space tothe area of the holographic screen of the causal diamond. This was made more preciseby describing the operator algebra in terms of operators representing pixels ofthe holographic screen. The Cartan–Penrose equation leads to a description of thesevariables as elements of the SO(d−2) spinor bundle over the screen, where d is thespace-time dimension encoded in the topology of the network of Hilbert spaces.We saw that quantization of these spinor variables identified the states of a pixelas the states of a massless super-particle. Compact dimensions of space could beincorporated by enlarging the algebra of spinor operators at each pixel to includecentral charges corresponding to Kaluza–Klein momenta, or brane wrapping numberson topological cycles of the internal manifold. This is precisely the dataabout compact geometry that is invariant under topology changing string dualities.Thus, the holographic formulation provides a rationale for not just gravity, butsupergravity, as the natural outcome of quantum geometry.The holographic formulation of Quantum Gravity provided an explicit model ofa quantum system corresponding to a classical cosmology: a flat FRW universewith equation of state p = ρ. This universe saturates the holographic entropybound at all times. It has a heuristic description as a dense black hole fluid, anddoes not resemble our universe. An heuristic description of our own universe as acollection of defects in the p = ρ background, maximizing the entropy subject tothe constraints of the existence of observers (in a fairly well-defined mathematicalsense) seems to account for many facts about cosmology. It also leads to the predictionthat the universe is future asymptotically de Sitter, with a de Sitter radiusas small as permitted by environmental constraints like the existence of galaxies.I also described the beginnings of a holographic theory of eternal de Sitter space,which might be the appropriate arena for discussing non-cosmological particlephysics. I proposed tentative identifications of black hole, and particle states in