Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 415• Q - C. Rovelli - to J. Ambjørn et al.1. As far as I understand lattice gauge theory, a meaningful continuous theoryis defined only if there is second order phase transition. This is because only ina second order phase transition the correlations functions of the discrete theorydiverge in such a way that they give finite correlations functions in the continuumlimit. If there is no second order phase transition, a continuum limitmay still exist, but all the correlation functions in the continuum limit are trivialor diverge. As far as I understood the old dynamical triangulations program,this was, indeed, the main issue. That is, after having identified an interestingphase transition in the discrete model, the issue was to prove that it is secondorder. Now, I do not see this is the present approach. You focus on the transitionbetween the crumpled and the smooth phase but you do not discuss if it is secondorder or not. Have you solved the problem? Circumvented it? Understoodthat it was a false problem?2. The sum over triangulations you study can be viewed as a Feynman sum overgeometries, written in the time gauge, and weighted with the classical action.If this defines a consistent quantum theory, its classical limit is the field theorydefined by the Einstein–Hilbert action for geometries in the time gauge. Well,this is not general relativity: one equation is missing. For the same reason that ifyou pose A 0 = 0 in the Maxwell action, you lose the equation divE = 0. Theequation you loose is precisely the Hamiltonian constraint, which in a senseis where the core of the story is. It is well known, indeed, that to implementthis key equation one has, so to speak, to integrate over all lapse functions, or allproper times. And, as far as I understand, you do not do that. If so, the theory youare studying is not general relativity, the theory that works so well empirically.What am I missing?–A-J.Ambjørnet al.:1. In the framework of statistical mechanics you refer to one imagining acritical surface where the correlation length is infinite. If you are not on thecritical surface for some value of the coupling constants you have to fine-tunethe coupling constants such that you approach that surface. An example is thefine-tuning of the temperature in magnetic systems to a second order phasetransition between a magnetized phase and a phase where the magnetizationis zero. The spin–spin correlation length will diverge and the long distancephysics of the spin system can for a number of materials be described by athree-component φ 4 theory close to the Fisher–Wilson fixed point. The distancefrom the critical surface as related to the mass of the particle if we usefield-theoretical language. For continuum theories with a mass gap, and thisalso includes theories like non-Abelian gauge theories, we will always haveto stay a little away from the critical surface in a precise fine-tuned way such