Approaches to Quantum Gravity

Loop quantum gravity 249a definite, universal value. This is possible because in LQG geometrical operatorssuch as volumes, areas and lengths of regions, surfaces (such as the horizon) andcurves have discrete spectrum [43; 44; 45; 46] with a gap away from zero (otherwisethe entropy would be infinite). 13 One may speculate whether the discretenessof the spectra hints at a combinatorical, distributional Planck scale structure ofspacetime with no meaning to notions like smoothness or metrics which shouldemerge only on large scales.Finally, certain minisuperspace approximations to LQG have been developed inorder to perform approximate quantum cosmology [48]. The results obtained aresubject to the usual restriction that a truncated model might not display the truebehaviour of the full theory due to artificial suppression of degrees of freedomwhich might have large fluctuations in the full theory. See e.g. [49; 50] where it isshown that (big bang) singularity avoidance of the models is due to a mechanismwhich is only available in those models. On the other hand, as shown, in the fulltheory singularity avoidance could possibly be obtained by a more subtle featureof LQG. In any case, the models indicate that LQG indeed might be able to resolvethe singularities of full GR.To summarise: LQG is a mathematically rigorous approach to Quantum Gravitywhich is conceptually clear and simple. It only uses the principles of General Relativityand Quantum Mechanics and no experimentally unverified assumptions. Itis fully background independent as every Quantum Gravity theory must be. Nowtools have have been developed that enable one to make contact with experimentand thus to falsify the theory.References[1] C. Rovelli, Quantum Gravity (Cambridge, Cambridge University Press, 2004).[2] T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge,Cambridge University Press, 2007),[3] A. Ashtekar, J. Lewandowski, “Background independent quantum gravity: a statusreport”, Class. Quant. Grav. 21 (2004) R53, [gr-qc/0404018].[4] L. Smolin, “Quantum gravity with a positive cosmological constant”,hep-th/0209079[5] C. Rovelli, “Loop quantum gravity”, Living Rev. Rel. 1 (1998) 1, gr-qc/9710008[6] T. Thiemann,“Lectures on loop quantum gravity”, Lecture Notes in Physics, 631(2003) 41–135, gr-qc/0210094[7] R. Haag, Local Quantum Physics, 2nd edn (Berlin, Springer Verlag, 1996).13 Actually, there are two unequivalent Volume Operators, one due to Rovelli and Smolin (RS) and the other onedue to Ashtekar and Lewandowski (AL) [43; 44; 45; 46] which are both derived using background independenttechniques from the fundamental flux operator. In a recent non-trivial consistency check [47] theRSandALvolume operators have been shown to be inconsistent and consistent respectively with the flux operator. Thisis a first example for an analysis which uses internal mathematical consistency in order to improve the degreeof uniqueness of LQG.