Approaches to Quantum Gravity

String theory, holography and Quantum Gravity 191into black holes. We cannot take a limit where such objects become infinitely large,without going off to the boundaries of an infinite space. So we might expect a localformulation of Quantum Gravity to have an ineluctably approximate nature. Wewill see that this is the case.The clue to the nature of a local formulation of Quantum Gravity is the covariantentropy bound [12; 13; 14; 15; 16] for causal diamonds. A causal diamond in aLorentzian space-time is the intersection of the interior of the backward light coneof a point P, with that of the forward light cone of a point Q in the causal past of P.The boundary of the causal diamond is a null surface, and the holographic screenof the diamond is the maximal area space-like d − 2 surface on the boundary.The covariant entropy bound says that the entropy which flows through the futureboundary of the diamond is bounded by one quarter of the area of this surface, inPlanck units. For sufficiently small time-like separation between P and Q, this areais always finite, and its behavior as a function of the time-like separation is an indicatorof the asymptotic structure of the space-time. In particular, for a future asymptoticallyde Sitter space-time, with a Big Bang as its origin, the area approaches amaximal value, equal to four times the Gibbons–Hawking de Sitter entropy.In Quantum Mechanics, entropy is −trρlnρ, where ρ is the density matrix of thesystem. Infinite systems can have density matrices of finite entropy. However, thisis usually a consequence of the existence of special operators, like a Hamiltonian:the archetypal case being a thermal density matrix. Fischler and the present authorsuggested that generally covariant theories have no such canonical operators (theproblem of time) and that the only general assumption one could make about thedensity matrix implicit in the covariant entropy bound was that it is proportionalto the unit matrix. In other words, the number of quantum states associated witha small enough causal diamond is always finite. This conjecture is in accord withour intuition about simple stationary systems in asymptotically flat and AdS spacetimes.The maximal entropy configurations localized within a given area are blackholes. One cannot add more quantum states to a localized system without makingboth its mass and its area grow. In this case, the entropy bound counts the numberof states.A finite quantum system cannot contain machines which can make infinitelyprecise measurements on other parts of the system. Thus, the finite state spacehypothesis implies an irreducible ambiguity in the physics of a local region ofspace-time. We cannot expect its (generally time dependent) Hamiltonian, nor anyother operator to have a precise mathematical definition, 5 since there is no way,even in principle, to measure properties of the region with infinite accuracy.5 More properly: any precise mathematical definition will include elements which cannot be verified by measurement,and are thus gauge artifacts. There will be a class of Hamiltonians which give the same physics,within the ineluctable error associated with the finite size of the region.