100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravity, Geometry and the Quantum 375• Quantum geometry. From conceptual considerations, an important issueis the physical significance of discreteness of eigenvalues of geometricoperators. Recall first that, in the classical theory, differential geometrysimply provides us with formulas to compute areas of surfaces and volumesof regions in a Riemannian manifold. To turn these quantities into physicalobservables of general relativity, one has to define the surfaces and regionsoperationally, e.g. using matter fields. Once this is done, one can simply usethe formulas supplied by differential geometry to calculate values of theseobservable. The situation is similar in quantum theory. For instance, thearea of the isolated horizon is a Dirac observable in the classical theoryand the application of the quantum geometry area formula to this surfaceleads to physical results. In 2+1 dimensions, Freidel, Noui and Perez haverecently introduced point particles coupled to gravity. 34 The physical distancebetween these particles is again a Dirac observable. When used in thiscontext, the spectrum of the length operator has direct physical meaning.In all these situations, the operators and their eigenvalues correspond tothe ‘proper’ lengths, areas and volumes of physical objects, measured inthe rest frames. Finally sometimes questions are raised about compatibilitybetween discreteness of these eigenvalues and Lorentz invariance. As wasrecently emphasized by Rovelli, there is no tension whatsoever: it suffices torecall that discreteness of eigenvalues of the angular momentum operatorĴ z of non-relativistic quantum mechanics is perfectly compatible with therotational invariance of that theory.• Quantum Einstein’s equations. The challenge of quantum dynamics inthe full theory is to find solutions to the quantum constraint equations andendow these physical states with the structure of an appropriate Hilbertspace. We saw in section 3.2 that this task can be carried to a satisfactorycompletion in symmetry reduced models of quantum cosmology. For thegeneral theory, while the situation is well-understood for the Gauss and diffeomorphismconstraints, it is far from being definitive for the Hamiltonianconstraint. It is non-trivial that well-defined candidate operators representingthe Hamiltonian constraint exist on the space of solutions to the Gaussand diffeomorphism constraints. However there are many ambiguities 24 andnone of the candidate operators has been shown to lead to a ‘sufficient numberof’ semi-classical states in 3+1 dimensions. A second important openissue is to find restrictions on matter fields and their couplings to gravityfor which this non-perturbative quantization can be carried out to asatisfactory conclusion. As mentioned in section 2.1, the renormalizationgroup approach has provided interesting hints. Specifically, Luscher and