Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 309The Barret Crane model can virtually be obtained also from loop quantumgravity. This is an unexpected convergence of two very different linesof research. Loop quantum gravity is formulated canonically in the frozentime formalism. While the frozen time formalism is in principle complete,in practice it is cumbersome, and anti-intuitive. Our intuition is four dimensional,not three dimensional. An old problem in loop quantum gravityhas been to derive a space–time version of the theory. A space–time formulationof quantum mechanics is provided by the sum over histories. A sumover histories can be derived from the Hamiltonian formalism, as Feynmandid originally. Loop quantum gravity provides a mathematically well definedHamiltonian formalism, and one can therefore follow Feynman stepsand construct a sum over histories quantum gravity starting from the loopformalism. This has been done in [Reisenberger and Rovelli (1997)]. Thesum over histories turns out to have the form of a sum over surfaces.More precisely, the transition amplitude between two spin networkstates turns out to be given by a sum of terms, where each term can be representedby a (2D) branched ‘colored’ surface in space–time. A branchedcolored surface is formed by elementary surface elements carrying a label,that meet on edges, also carrying a labelled; edges, in turn meet in vertices(or branching points, see Figure 3.11). The contribution of one such sur-Fig. 3.11A branched surface with two vertices.