Approaches to Quantum Gravity

Questions and answers 157to hold), these considerations force us to replace the underlying manifold bysomething rather like a topos. The categorical structure of a causal site couldaccommodate that.On the other hand, if a subregion in one metric appeared to be insidea larger subregion of another to all observers, we could consider it to becontained.This picture leads to a causal site in which the ratio of numbers of regionsof different sizes could be quite different from the usual scaling property ina manifold. Perhaps Quantum Gravity leads to dimensional regularization atlower energies as well as a Planck scale cutoff. Quantum Field Theory wouldinclude contributions from Feynman diagrams with vertices in positions notsimultaneously meaningful.I think adhering to the traditional picture of an absolute backgroundmanifold in analysing this experiment would be very awkward. It would benecessary to pick some arbitrary correspondence between the observed spacetimeregions corresponding to each pair of classical metrics supported bythe quantum state. These correspondences would not affect the result of anyexperiment. Einstein’s principle suggests we discard them.• Q-D.Oriti-toO.Dreyer:1. If I understand correctly, in Volovik’s approach the non-zero mass of theeffective graviton and the failure to achieve full general covariance is the result,in the end, of the non-relativistic nature of the fundamental system he deals with,i.e. the fermionic gas/liquid; can you please clarify how exactly the presence ofa background absolute time in the fundamental system is associated to this lackof general covariance in the effective theory? Also, it seems to me that Volovik’sapproach relies on a fundamental time variable only because of the specificchoice of the physical system (here a non-relativistic and background dependentone) whose effective dynamics one studies, but that his general idea of spacetimeand General Relativity as emerging from some sort of condensed mattersystem in a specific phase does not really depend on this. If this is true, then hisapproach and ideas could be applied to fully background independent systemslike for example matrix models and group field theories where one could hopeof not ending up with any failure of general covariance in the condensed phase.What is your opinion on this?2. In your contribution, you didn’t mention explicitly the idea by Ted Jacobsonof the “Einstein’s equations as an equation of state”, that seems to me verymuch related to the type of ideas you nicely reviewed. Where would it fit withinyour scheme of “emergent gravity” approaches?3. What is the role of the background time or temperature (in a statisticalmechanics setting) of the spin system in your ’Internal Relativity’ model? It