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

Gravity, Geometry and the Quantum 363not need their explicit forms.) Our task in quantum theory is three-folds: i)Elevate these constraints (or their ‘exponentiated versions’) to well-definedoperators on the kinematical Hilbert space H; ii) Select physical statesby asking that they be annihilated by these constraints; iii) introduce aninner-product on the space of solutions to obtain the final Hilbert spaceH final , isolate interesting observables on H final and develop approximationschemes, truncations, etc to explore physical consequences. I would like toemphasize that, even if one begins with Einstein’s equations at the classicallevel, non-perturbative dynamics gives rise to interesting quantum corrections.Consequently, the effective classical equations derived from the quantumtheory exhibit significant departures from classical Einstein’s equations.This fact has had important implications in quantum cosmology.How has loop quantum gravity fared with respect to these tasks? Sincethe canonical transformations generated by the Gauss and the diffeomorphismconstraints have a simple geometrical meaning, it has been possibleto complete the three steps. For the Hamiltonian constraint, on the otherhand, there are no such guiding principles whence the procedure is moreinvolved. In particular, specific regularization choices have to be made andthe final expression of the Hamiltonian constraint is not unique. A systematicdiscussion of ambiguities can be found in reference [24]. At the presentstage of the program, such ambiguities are inevitable; one has to consider allviable candidates and analyze if they lead to sensible theories. A key openproblem in loop quantum gravity is to show that the Hamiltonian constraint—either Thiemann’s or an alternative such as the one of Gambiniand Pullin— admits a ‘sufficient number’ of semi-classical states. Progresson this problem has been slow because the general issue of semi-classicallimits is itself difficult in any background independent approach. f However,a systematic understanding has now begun to emerge and is providing thenecessary ‘infra-structure’. 24,38 Recent advance in quantum cosmology, describedin Section 3.2, is an example of progress in this direction. Thesymmetry reduction simplifies the theory sufficiently so that most of theambiguities in the definition of the Hamiltonian constraint disappear andthe remaining can be removed using physical arguments. The resulting theoryhas rich physical consequences. The interplay between the full theoryand models obtained by symmetry reduction is now providing crucial inputsto cosmology from the full theory and useful lessons for the full theoryfrom cosmology.f In the dynamical triangulation 18,27 and causal set 22 approaches, for example, a greatdeal of care is required to ensure that even the dimension of a typical space-time is 4.