Approaches to Quantum Gravity

564 L. Smolinnodes so that no large areas or volumes are changed. In fact, we can do this Mtimes, at least so long as M ≪ 2N, without changing any large areas or volumes.Each of these M new links connects two randomly chosen nodes of Ɣ 0 , making anew graph Ɣ ′ . The corresponding state |Ɣ ′ > is still a semiclassical state for themetric qab 0 and will reproduce it when sufficiently coarse grained observables aremeasured. But it provides an example of Markopoulou’s observation that microand macro notions of locality need not coincide even in the low energy limit [68].We may call this phenomena, disordered locality.At first sight it seems as if disordered locality would kill the theory, becausethere would be macroscopic violations of locality in the low energy limit. But itturns out that this need not the case, if the disagreement between micro and macrolocality is rare enough. For example, suppose that the probability that a node hasa non-local edge, p = M 2N is on the order of 10−100 . This would still mean thereare on the order of 10 80 random non-local edges within the Hubble volume (seeFigure 28.7). Could we do any measurements to tell that the quantum geometry ofour universe was based on Ɣ ′ rather than Ɣ?It would be very unlikely that any two nodes within the earth are connectedby one of the non-local edges, so it would be very hard to directly detect nonlocality.Moreover, since the defects were at the Planck scale, the amplitude for lowenergy quanta to jump across a non-local link would be suppressed by lPl 2 E 2 .Sowe are unlikely to see fermions appearing and disappearing across the links. Moreoversince the whole universe is in thermal equilibrium at the same temperaturethe transfers of energy through the non-local links would also be hard to observe.Studies have been done of the thermodynamics of spin systems on networks withdisordered locality and the main effect for small p is to raise the Curie temperatureby an amount of order p without strongly affecting the correlation functions [69].Would dynamics suppress such non-local links? The answer is that the dynamicscannot. The reason is that the dynamics is micro-local, and hence defined by theconnectivity of the graph Ɣ ′ . The local moves that generate the dynamics cannotFig. 28.7. A lattice with disordered locality from a contamination of non-local links.