Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 299of the existence of exact solutions [Rovelli and Smolin (1988)], and ofthe possible finiteness of the theoryMatter. The old hope that QFT divergences could be cured by QG hasrecently received an interesting corroboration. The matter part of theHamiltonian constraint is well–defined without need of renormalization.Thus, a main possible stumbling block is over: infinities did not appearin a place where they could very well have appeared [Rovelli (1997)].Black hole entropy. The first important physical result in loop quantumgravity is a computation of black hole entropy [Krasnov (1997); Rovelli(1996a); Rovelli (1996b)].Quanta of geometry. A very exciting development in quantum gravity inthe last years has been by the computations of the quanta of geometry.That is, the computation of the discrete eigenvalues of area and volume.In quantum gravity, any quantity that depends on the metric becomesan operator. In particular, so do the area A of a given (physically defined)surface, or the volume V of a given (physically defined) spatial region.In loop quantum gravity, these operators can be written explicitly. Theyare mathematically well defined self–adjoint operators in the Hilbert spaceH. We know from quantum mechanics that certain physical quantities arequantized, and that we can compute their discrete values by computing theeigenvalues of the corresponding operator. Therefore, if we can computethe eigenvalues of the area and volume operators, we have a physical predictionon the possible quantized values that these quantities can take, atthe Planck scale. These eigenvalues have been computed in loop quantumgravity. Here is for instance the main sequence of the spectrum of the areaA ⃗ j = 8πγ G ∑ √ji (j i + 1). (3.146)i⃗j = (j 1 , . . . , j n ) is an n−tuplet of half–integers, labeling the eigenvalues,G and are the Newton and Planck constants, and γ is a dimensionlessfree parameter, denoted the so–called Immirzi parameter [Immirzi (1997)],not determined by the theory. A similar result holds for the volume. Thespectrum (3.146) has been rederived and completed using various differenttechniques [DePietri and Rovelli (1996)]. These spectra represent solidresults of loop quantum gravity. Under certain additional assumptions onthe behavior of area and volume operators in the presence of matter, theseresults can be interpreted as a corpus of detailed quantitative predictionson hypothetical Planck scale observations.