Approaches to Quantum Gravity

202 T. Banksof the dS–Schwarzschild metric with the P 0 eigenvalue. Of course, extant quantumfield theory calculations which demonstrate that we have a thermal ensemble ofordinary energies in de Sitter space, refer only to energies much smaller than themaximal black hole mass. We are led to conjecture the above relation between theentropy deficit (relative to the vacuum) of a p 0 eigenspace, and the eigenvalue, onlyto leading order in the ratio of the black hole mass to the Nariai mass. Remarkably,this prediction is valid [26]!It is easy to construct a Hamiltonian out of the fermionic pixel operators introducedabove, which reproduces the spectrum of black holes in dS space. One worksin the approximation where the vacuum eigenstates are all exactly degenerate, sothat black holes are stable. The vacuum density matrix is just the unit matrix.Black hole states are simply states in which we break the fermionic matrix ψiAinto four blocks, and insist that ψm D |BH〉=0, for matrix elements in the loweroff diagonal block. A clumsy but explicit formula for the Hamiltonian P 0 can beconstructed [26].Some insight into the Hamiltonian P 0 is gained by remembering that globalsymmetry generators in General Relativity are defined on space-like or null boundaries.The way in which dS space converges to Minkowski space is that the causaldiamond of a single observer approaches the full Minkowski geometry. The futureand past cosmological horizons of the observer converge to future and past infinityin asymptotically flat space. Our basic proposal for the definition of observables inde Sitter space [27] is that there is an approximate S-matrix, S R which, as R →∞,approaches the S-matrix of asymptotically flat space. S R refers only to localizableprocesses in a single horizon volume. As in any such limiting situation, we mayexpect that S R is not unique, and it is important to understand what aspects of itare universal for large R. We will argue later that for scattering processes whosecenter of mass energy is fixed as R →∞, the non-universal features may fall offlike e −(RM P ) 3/2 .The geometry of the future cosmological horizon is the v → 0 limit of:ds 2 = R 2 (−dudv + d 2 ),and the static Hamiltonian is associated with the Killing vector(u∂ u − v∂ v ).Here, d 2 is the round metric on the 2-sphere. By contrast, future infinity inasymptotically flat space, is the v → 0 limit ofds 2 =−dudv + d2v 2 .