Approaches to Quantum Gravity

Questions and answers• Q - L. Crane - to C. Rovelli:You say in your paper that we need to think about replacing the classicalspacetime continuum. The GFT picture seems to suggest that many paralleldiscrete “leaves” of spacetime exist, and that their effects superimpose in anobserver dependent way. Have you thought along these lines? (Mathematicianscall such a structure a “site”, and constructing objects over a site is called topostheory.)– A-C.Rovelli:No, I haven’t. But I take this as a very interesting suggestion. I fully agreethat the GFT picture is strongly suggestive, and points to “something”. I alsounderstand that categories and topoi might be a valuable language here, but Ido not have the expertise needed to take full advantage of these, I think.• Q - D. Oriti - to C. Rovelli:You mention the possibility of a quantum granularity of space as a consequenceof a proper quantum mechanical treatment of the gravitational field,and the fact that this sort of granularity is indeed realised in the spectrum ofsome geometric observables in loop quantum gravity, and also hinted at in somestring theory models. However, it is not obvious to me what sort of discretenesswe should really expect from a quantum theory of spacetime just by lookingat the quantum mechanical systems we know of. On the one hand, in fact, wehave systems like the hydrogen atom with its discrete energy spectrum, whileon the other hand we have quantum field theories, where spectra of observablesare continuous but the quantum discreteness is present in the availability of aFock space description of their state space, i.e. in the possibility of describingthem as made out of discrete fundamental constituents. In the case of QuantumGravity therefore one can equally well expect to obtain granularity in theform of either discrete spectra for geometric observables or in the form of somesort of fundamental “quanta or atoms” of space, whose related observables arehowever continuous. The first scenario seems to be realised in SU(2)-based Loop150