Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 419of the action is directly linked to the fact that squares, viewed as geometricunits, have negative weights. If we had included the squares with proper,understandable weight, the action would still, after the addition of the squares,have been unbounded from below. The bounded action allows us to define anon-perturbative sum over all genus. It is seen that the construction here isword by word the same as the one used in the 3d Ponzano–Regge group fieldtheory referred to by Oriti. In the 2d case one can complete the analysis: itturns out that this contrived model has a decent interpretation: it representsa (2,5) minimal conformal field theory coupled to two-dimensional QuantumGravity. The main point is that new non-geometric, non-unitary degrees offreedom have been introduced in the model and they totally dominate thehigh genus part of it. In this way one has tamed to topology, burying it inthe dominant interactions of a non-unitary theory. It has (until now) provedimpossible to repeat the same trick with unitary models couple to 2d QuantumGravity for the simple reason that integrating out unitary matter alwaysgives positive weight factors in Euclidean space. The situation in two dimensionsis infinitely simpler than in three dimensions, not to speak about fourdimensions.To summarize: the suggestions for summation over topologies I have seenso far have in my opinion no chance to work. Of course this does not ruleout that one day one will (1) understand that one should really sum overtopologies and (2) understand how to do it.2. Concerning the inclusion of baby universes or exclusion of baby universes,it is difficult to see the motivation for including them, but confining them tobe of Planck size unless there is a natural mechanism which confines them tothis size. Anyway, if they were included that way one could presumably justintegrate them out again when one addresses physics at a slightly larger scale.Actually one can address this question in a precise way in two-dimensionalQuantum Gravity. As we have shown: if you start out with CDT and thenallow baby universes (of all sizes), then you recover standard Euclidean twodimensionalQuantum Gravity (as described by dynamical triangulations,matrix models or Liouville field theory). Conversely, if you start out withEuclidean two-dimensional Quantum Gravity and chop away baby universesyou obtain CDT. From the theory of Euclidean two-dimensional QuantumGravity you know the precise distribution of baby universe volumes (it isgoverned by the so-called susceptibility (or entropy) exponent γ ). The distributionis very strongly peaked at baby universes of cut-off scale, whichone in this model would identify with the Planck-scale. So the model almostsatisfies your requirement of having the baby universes confined to thePlanck-scale simply by entropy. However, the rare larger baby universes are