Approaches to Quantum Gravity

Questions and answers 577where this choice is not made and one uses the full non-compact Lorentz groupinstead, in which case the spectra of some geometric observables are continuousand not bounded from below (e.g. no minimal spacelike areas or lengthsexist), and no uniqueness result is, unfortunately, available to us. Some of thesemodels remain ultraviolet finite despite this, as you correctly mention, but thisseems to be a result of very specific models (more precisely, of a very specificchoice of quantum amplitudes for the geometric configurations one sums overin the spin foam setting) and not a generic feature of this class of theories. I fullyagree, of course, that the class of models you discuss remains truly “discrete”in the sense that it bases its description of spacetime geometry on discrete andcombinatorial structures (graphs and their histories) and local discrete evolutionmoves. The question is the following. In the model of emergent matter that youpresent, where matter degrees of freedom are encoded in the braiding of theframed graphs on which the theory is based, where does the mass of such mattercome from? Do you expect that this could be defined in terms of somethinglike the holonomy “around these braids”, when one endowes the graphs withgeometric data, e.g. a connection field or group elements, as in the couplingof particles in topological field theories and 3d Quantum Gravity? If so, wouldyou imagine a sort of coherent (noiseless) propagation of such “holonomy +braiding” degrees of freedom to encode the conservation of mass, or do youenvisage a sort of “variable mass” field theory description for the dynamics forthese matter degrees of freedom, in the continuum approximation?– A-L.Smolin:Regarding your first comment, this of course depends on whether we take theview that the theory is derived by quantization of GR or invented. If we takethe first view then my view is that the canonical theory is more fundamentalfor sorting out the quantum kinematics. The canonical theory leads to labelsin SU(2) which is compact and thus implies the discreteness of area and volume.At the very least the canonical theory and the path integral theory shouldbe related so that the path integral gives amplitudes for evolution or definesa projection operator for states in the canonical theory. It is unfortunately thecase that none of the spin foam models which have so far been well developeddo this, although I am told there is work in progress which remedies this. Inthe original papers of Reisenberger and Reisenberger and Rovelli as well as inthe first paper of Markopoulou the spin foam amplitudes are defined in termsof evolution of states in the canonical theory. This to me is the preferred wayas it is well defined and does not lead to ambiguities in choices of representationsor whether one sums over triangulations or not. When the path integralis defined from the canonical theory all faces in the spin foam are spacelikeand all should be labeled from finite dimensional reps of SU(2).