Approaches to Quantum Gravity

196 T. Bankswe have postulated are the analog of the Dirac–Schwinger commutation relationsfor the Wheeler–DeWitt operator in canonical approaches to Quantum Gravity.There is, at the present time, only one known solution to these conditions. Ifwe insist that the time dependent Hamiltonian of a given observer, is chosen independentlyat each instant from a certain random ensemble [18], 10 and choose theoverlap so that O(x, y, t) = H(x, t − D), where D is the minimal lattice pathlength between x and y, then all the consistency conditions are satisfied, and all thescaling relations of the flat FRW space-time with equation of state p = ρ are satisfiedby this rather explicit quantum system. This is the correct quantum descriptionof the dense black hole fluid that was postulated in [19; 20; 21]. By construction, itis a cosmology that saturates the covariant entropy bound at all times.The heuristic picture of a dense black hole fluid is based on the idea that at anygiven time, all the degrees of freedom in a horizon volume have coalesced to forma single black hole. An instantaneous distribution of black holes at relative separationsof order of their Schwarzschild radii, have the energy/entropy relation of ap = ρ fluid. If they continually coalesce to make larger, horizon filling black holes,always separated by about a horizon scale, then we indeed have an equilibrium systemwith equation of state p = ρ. The random Hamiltonian model described in theprevious paragraph is an explicit quantum system which has many of the propertiesderived from this heuristic picture.The concept of an observer does not make sense in the p = ρ background,because all degrees of freedom in any causal diamond are always in intense interaction,and there are no isolated sub-systems with a large number of semi-classicalobservables. The idea of holographic cosmology is that the universe we live inbegan as close as possible to the p = ρ system, consistent with the observerphilicprinciple: it is the maximally entropic solution of the consistency conditions wehave outlined, which does not quickly collapse back into the p = ρ phase, andallows for the existence of what we have called observers over very long time periods.Of course, we might want to strengthen our demands, and insist on some sortof criterion that guaranteed the existence of intelligent living organisms – observersin the more colloquial sense. Such restrictions are fine as long as we do not makeclaims that go beyond our abilities to actually do the calculations involved in guaranteeingor ruling out life. It’s also obvious that we want to make the weakestassumptions of this kind that give correct answers. It may be that the SU(1, 2, 3)gauge group of the standard model is only explainable because it leads to the type10 The existence and nature of this ensemble is based on the properties of quadratic fermionic systems withrandom one body Hamiltonians. In this way, the choice of holographic pixels as the fundamental variables ofQuantum Gravity, enters directly into the formulation of holographic cosmology.