Approaches to Quantum Gravity

416 Questions and answersthat the lattice spacing times the correlation length measured in lattice unitsis constant (equal to the inverse physical mass). This is the way to recover thecontinuum limit of the lattice theory.However, suppose we are already at the critical surface. As an explicitexample consider a free massless scalar particle (in Euclidean spacetime).Put it on the lattice in the simplest possible way. The propagator is nowG(p) =1sin 2 (pa/2) → 1a 2 p 2 for a → 0.Except for a prefactor, we have directly the continuum propagator when thelattice spacing a → 0. No fine-tuning is needed. In other theories where masslessparticles can be put on the lattice in a natural way which does not generatea mass term, neither perturbatively nor non-perturbatively, we have the samesituation. An example is four-dimensional lattice U(1) theory. For the (lattice)coupling constant above the critical value, one has a confining lattice theorywithout a continuum limit, but for the coupling constant below the criticalvalue one is automatically in the Coulomb phase where a trivial rescaling ofthe lattice spacing and fields leads to the continuum free field theory of thephoton.In CDT we seem to have the same situation: for some range of the baregravitational coupling constant we obtain a lattice theory with no continuumlimit. For another range of the gravitational coupling constant we obtain acontinuum limit (to the extent one can trust the computer simulations) just bytaking the lattice spacing to zero. If one wants to use the analogy with theU(1) theory mentioned above, the interpretation would be that the gravitonhas been incorporated in a natural way which does not lead to a mass, soone is staying on the critical surface for a range of coupling constants. It isprobably a good thing.In the “old” Euclidean DT the situation was the following: for almostall values of the gravitational couplings constant the computer simulationsshowed just a lattice theory without any obvious continuum limit. Only nearthe phase transition between a pathologically crumpled phase and an equallypathologically “stretched” phase (where the geometry degenerated to socalledbranched polymers) was there a chance to obtain something whichwas not a lattice artifact. Unfortunately the phase transition turned out to bea (weak) first order transition and the separation between the two phaseswould be sharp with increasing spacetime volume. Had it been a secondorder transition one could have hoped it would have been possible to definea continuum limit, in particular that there was a divergent correlation length