Approaches to Quantum Gravity

String theory, holography and Quantum Gravity 195slices of space-time (recall always that our formalism is constructed in a fixed butarbitrary physical gauge). We postulate that this topology does not change withtime. 8 We attach a sequence of observer Hilbert spaces and evolution operators toeach point of the lattice. In the Big Bang cosmology, which we are using as anexample, it is convenient to choose “equal area time slicing", where the dimensionof the kth Hilbert space at each point x is the same.For each pair of points on the lattice and each time, we define an overlap Hilbertspace, O(x, y, t), which is a tensor factor of both H(t, x) and H(t, y), and requirethat the dynamics imposed on this tensor factor by the two individual observers isthe same. 9 The idea behind this condition comes from a geometrical notion. Theintersection of two causal diamonds is not a causal diamond, but it does contain amaximal causal diamond. The physics in that maximal diamond should not dependon whether it is observed at some later time by one or the other of the favoredobservers in our gauge. We insist that for nearest neighbor points on the lattice, attime t, the overlap Hilbert space has dimension (dimK) t−1 . This defines the spacingof our lattice such that moving over one lattice spacing decreases the overlap byone unit of area. Note that this is the same as the time spacing and, like it, thespatial resolution goes to zero as the area grows.Consider a point x on the lattice, and a path emanating from it, whose lattice distancefrom x increases monotonically. We require that the dimension of the overlapHilbert space at fixed time, decrease monotonically along the path. We also insistthat, as our notation suggests, the dimension of the overlap depends only on theendpoints of a path, not on the path itself.These conditions are incredibly complicated, but seem to incorporate a minimalsort of framework for a unitary theory of Quantum Gravity. We have, in effect,constructed a quantum version of a coordinate system on a Lorentzian manifold,built from the trajectories of a group of time-like observers. The two rules thatwe use are equal area time-slicing, and space-time resolution defined in terms of aminimal difference in the size of the holographic screens at nearest neighbor spacetimepoints. Given any Big Bang space-time whose expansion continues forever,we could set up such a coordinate system. The complicated consistency conditions8 This may disturb readers familiar with claims for topology change in string theory. Here we are discussingthe topology of non-compact dimensions of space-time. I believe that the real lesson about topology changecoming from the duality revolution is that the topology of compact manifolds is entirely encoded in quantumnumbers, which are measured in scattering experiments in the non-compact dimensions. In different limits ofthe parameter space, the quantum numbers can be interpreted in terms of the topologies of different compactspace-times. The way to incorporate that lesson into the present formalism is to complicate the algebra ofoperators for a given pixel, to incorporate information about the compactification. That is, the holographicscreen in the non-compact dimensions, contains operators which describe the compactification. This is the waythe compact factors X in AdS d × X are described in AdS/CFT.9 It may be sufficient to require that the two sequences of evolution operators are related by a unitarytransformation on O.