Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 403a 2D surface and the number (and labels) of 2-surfaces in the spin-foam that “puncture”it (as is sometimes claimed), this suggests a kind of fundamental discretenesson such surfaces. It also suggests an upper bound on degrees of freedom per unitvolume. But all this depends on the final form of the sum over triangulations in thatapproach, something not yet clarified.The status of local Lorentz invariance in spin-foam models remains controversial.As stated above, the causal set, with the above sprinkling-based discretecontinuumcorrespondence, is the only known Lorentz invariant discrete structure,and spin-foams are not of this type. But the real debate is over whether this impliesobservable Lorentz violation (if spin-foams models really do imply an upper boundon degrees of freedom per unit volume). It is sometimes claimed that, althoughan individual spin-foam cannot be said to satisfy LLI, a quantum sum over manyspin-foams may do (arguments from the closely related LQG program support this[54; 55]). An analogy is drawn with rotational invariance: in that case, the historiesmight only represent one component of the angular momentum of, say, an electron.In spite of this, the physics represented is in fact rotationally invariant.However, in standard theories, at least the macroscopic properties we observeare properties of each history in some (nonempty) set, and we should expectthe same for Quantum Gravity. Even these properties may fail to be present inthe case of Lorentz transformations, if the histories are not Lorentz invariantin the sense that a causal set is. It is possible that further thought along theselines could lead to quantitative predictions of Lorentz violation from spin-foammodels, giving an opportunity for observational support or falsification. A compromisebetween these views might be found in “doubly special relativity”, inwhich Lorentz transformations are deformed. In this case observational tests arestill possible.In the loop Quantum Gravity program, the spectra of certain operators (e.g. theareas of 2D surfaces) are claimed to be discrete, although as yet the physical Hilbertspace and operators have not been identified. Nevertheless, some arguments havebeen provided as to how the problems of spacetime singularities and black holeentropy might be solved in LQG. But without the physical observables, how thistype of discreteness could circumvent the arguments mentioned in the introductionor in the previous paragraph, or even whether it would exist in a completed formof loop Quantum Gravity, is not clear as yet.In dynamical triangulations, discreteness is used to solve the problems of definingthe path integral, and coming to grips with technical issues in a manageableway, notably the Wick rotation. However, in this approach the discreteness is notconsidered fundamental and a continuum limit is sought. As the “causal dynamicaltriangulations” program is in the happy situation of possessing a working model,it would be of great interest for the debate on discreteness to see what becomes of