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

376 A. AshtekarReuter have presented significant evidence for a non-trivial fixed point forpure gravity in 4 dimensions. When matter sources are included, it continuesto exist only when the matter content and couplings are suitablyrestricted. For scalar fields, in particular, Percacci and Perini have foundthat polynomial couplings (beyond the quadratic term in the action) areruled out, an intriguing result that may ‘explain’ the triviality of such theoriesin Minkowski space-times. 36 Are there similar constraints coming fromloop quantum gravity? To address these core issues, at least four differentavenues are being pursued: the Gambini-Pullin framework based on (Vassiliev)invariants of intersecting knots 15 ; the spin-foam approach based onpath integral methods 20,25 ; and the discrete approach summarized in thechapter by Gambini and Pullin, and the ‘master constraint program’ pursuedby Dittrich and Thiemann 23 and the related ‘affine quantum gravity’approach of Klauder 21 .• Quantum cosmology. As we saw in section 3, loop quantum gravity hasresolved some of the long-standing physical problems about the nature ofthe big-bang. In quantum cosmology, there is ongoing work by Ashtekar, Bojowald,Willis and others on obtaining ‘effective field equations’ which incorporatequantum corrections. Thanks to recent efforts by Pawlowski, Singhand Vandersloot, numerical loop quantum cosmology has now emerged as anew field. By a suitable combination of analytical and numerical methods,it is now feasible to analyze in detail a large class of homogeneous modelswith varying matter content. These models serve as a ‘background’ whichcan be perturbed. Since the dynamics of loop quantum cosmology is deterministicacross the singularity, evolution of inhomogeneities can be studiedin detail. This is in striking contrast with, say the pre-big-bang scenario orcyclic universes where new input is needed to connect the classical branchesbefore and after the big-bang.• Quantum Black Holes. As in other approaches to black hole entropy,concrete progress could be made because the analysis does not require detailedknowledge of how quantum dynamics is implemented in full quantumtheory. Also, restriction to large black holes implies that the Hawking radiationis negligible, whence the black hole surface can be modelled by anisolated horizon. To incorporate back-reaction, one would have to extendthe present analysis to dynamical horizons. 28 It is now known that, in theclassical theory, the first law can be extended also to these time-dependentsituations and the leading term in the expression of the entropy is againgiven by a hor /4l 2 Pl. Hawking radiation will cause the horizon of a largeblack hole to shrink very slowly, whence it is reasonable to expect that the