Ivancevic_Applied-Diff-Geom

300 Applied Differential Geometry: A Modern IntroductionBesides its direct relevance, the quantization of the area and thee volumeis of interest because it provides a physical picture of quantum space–time.The states of the spin network basis are eigenstates of some area and volumeoperators. We can say that a spin network carries quanta of area along itslinks, and quanta of volume at its nodes. The magnitude of these quantais determined by the coloring. For instance, the half–integers j 1 . . . j n in(3.146) are the coloring of the spin network’s links that cross the givensurface. Thus, a quantum space–time can be decomposed in a basis of statesthat can be visualized as made by quanta of volume (the intersections)separated by quanta of area (the links). More precisely, we can view aspin network as sitting on the dual of a cellular decomposition of physicalspace. The nodes of the spin network sit in the center of the 3–cells, andtheir coloring determines the (quantized) 3–cell’s volume. The links of thespin network cut the faces of the cellular decomposition, and their color ⃗jdetermine the (quantized) areas of these faces via equation (3.146).3.10.4.3 Traditional Approaches to Quantum GravityDiscrete ApproachesDiscrete quantum gravity is the program of regularizing classical GR interms of some lattice theory, quantize this lattice theory, and then studyan appropriate continuum limit, as one may do in QCD. There are threemain ways of discretizing GR.Regge CalculusRegge introduced the idea of triangulating space–time by means of a simplicialcomplex and using the lengths l i of the links of the complex asgravitational variables [Regge (1961)]. The theory can then be quantizedby integrating over the lengths l i of the links. For a recent review andextensive references see [Williams and Tuckey (1992)]. More recent workhas focused in problems such as the geometry of Regge superspace [Hartleet. al. (1997)] and choice of the integration measure.Dynamical TriangulationsAlternatively, one can keep the length of the links fixed, and capturethe geometry by means of the way in which the simplices are glued together,namely by the triangulation. The Einstein–Hilbert action of Eu-