Approaches to Quantum Gravity

418 Questions and answers2. The above applies also to your construction of causal dynamical triangulations,with the removal of baby universe configurations. Wouldn’t it be morenatural or satisfactorily to include such problematic configuration but havingthem confined to small (i.e. Planck size) volumes? Of course, this would requirethe presence of extra degrees of freedom on top of the combinatorial ones, forexample volume information associated to each d-simplex (in d dimensions).Have you considered such possibility?3. How would you modify your CDT construction to remove the gauge fixingcorreponding to the preferred foliation in T , assuming that it is indeed, as itbetter be, a gauge fixing? Is there already work going on in this direction?–A-J.Ambjørnet al.:1. Of course one could image a definition that suppresses topologies. The simplestmechanism is to leave them out by hand as we have suggested. Topologychanges do not appear very natural in a metric formulation of Einstein’s classicalgeneral relativity theory. That is one motivation for leaving them out.If one allows topology changes and then wants to suppress them, one hasto have a physically motivated mechanism for doing it. Such a mechanismmight exist, I am just not aware of one. The explicit example mentioned from3d Ponzano–Regge group field theory is in my opinion well understood andexplicitly non-physical. In fact it is in spirit very analogous to well studiedexamples in two-dimensional Quantum Gravity where one has been able toperform the summation over topologies and even obtain explicit analyticalresults. How does it work in 2d? You take 2d Euclidean Quantum Gravity,defined by some regularization, like dynamical triangulations, and you tryto sum over all topologies. You discover that the sum is factorial divergentin the genus of the 2d manifold, which is not surprising. Most perturbativeexpansions are. No obvious way suggests itself for a summation of the seriessince the coefficients are all positive: it is not Borel summable. There is aphysical reason for the coefficients being all positive: they are related to thecounting of different geometries of a fixed topology. This number grows factoriallywith the genus of the topology. Now one could get the marvelous ideato modify this counting of positive numbers by introducing a new “geometricunit” apart from the triangles: the square (say), but with negative weight. Atthis point we have no real idea what we are doing, but let us be courageousand blindly proceed. It is worth emphasizing the picture in terms of the socalledmatrix models which implement the explicit gluing of triangles, and thelarge-N expansion which gives the genus expansion. Starting with the gluingof the triangles we had an matrix model where the action was unbounded frombelow. Adding the squares produces an action which is bounded from below,and therefore well defined beyond perturbation theory, but the boundedness