Approaches to Quantum Gravity

154 Questions and answers– A-R.Sorkin:1. It has never been clear to me whether or not these models really live up totheir claim to respect Lorentz invariance. Perhaps if I understood them better,I could decide, but the proponents of the idea seem to disagree among themselvesabout questions like whether the dispersion relations are even modifiedfor a single particle, say a photon. What is clear, I think, is that (contrary towhat you write above) these theories do not admit a 10-dimensional symmetry*group*. Instead they have a Hopf algebra, maybe a “quantum group”.Does this really entail the physical equivalence of different reference framesin the sense required by the Michelson–Morley experiment, etc?2. It’s striking that this same conclusion (nonlocality) has emerged from suchapparently different trains of thought. But why do you name discreteness asan input to the DSR models? Is the point that modified dispersion relationswouldn’t really arise except as an effect of an underlying discreteness? Inany case, in the causet case *three* length scales seem to arise: not only aUV discreteness scale and an IR scale (as Lambda), but (modulo the caveatsin my article) an intermediate nonlocality scale. A similar triplet of scales isseen in the “fuzzy sphere”.3. Agreed. As I wrote in the article, the story for a massless scalar field (theonly case under control so far) seems to be that the dispersion relations are*unchanged* from those of the continuum. However, this conclusion refers tocausets well approximated by Minkowski spacetime. It would indeed be veryinteresting to work out the dispersion relations in the presence of curvature,say in de Sitter, for starters.4. I’ve expected such “mixing” all along as a concomitant of the non-localityimplied by Lorentz invariance plus discreteness. One can see for example,that an IR cutoff of any sort sets an upper bound to the degree of boostingthat can have meaning (thus a maximum velocity very slightly less than thatof light). What the recent results on the D’Alembertian add is the implicationthat nonlocality might show up well before you reach the Planck length.5. The main differences spring from the spatio-*temporal* and causal characterof causets as opposed to the purely spatial and “topological” character ofthe graphs Fotini is working with. The nonlocality I’m talking about (“nonlocallinks”) predominates on microscopic scales to the extent that localityloses all meaning there. In contrast, Fotini’s graphs still have a microscopicform of locality if I understand correctly (a relatively small number of nearestneighbors), and the “nonlocal links” are meant as a small perturbation.Also, the causet nonlocality is present whether or not there’s any slightmismatch between the macroscopic light cones and the microscopic orderrelation,whereas the graphical nonlocality by definition violates macroscopic