Ivancevic_Applied-Diff-Geom

310 Applied Differential Geometry: A Modern Introductionfaces to the sum over histories is the product of one term per each branchingpoint of the surface. The branching points represent the ‘vertices’ of thistheory, in the sense of Feynman. The contribution of each vertex can becomputed algebraically from the ‘colors’ (half integers) of the adjacent surfaceelements and edges. Thus, space–time loop quantum gravity is definedby the partition functionZ ∼ ∑surfaces∑colorings∏verticesA loop (color of the vertex) (3.155)The vertex A loop is determined by a matrix elements of the Hamiltonianconstraint. The fact that one obtains a sum over surfaces is not too surprising,since the time evolution of a loop is a surface. Indeed, the time evolutionof a spin network (with colors on links and nodes) is a surface (withcolors on surface elements and edges) and the Hamiltonian constraint generatesbranching points in the same manner in which conventional Hamiltoniansgenerate the vertices of the Feynman diagrams.Now, (3.155) has the same structure of the Barret–Crane model (3.151).To see this, simply notice that we can view each branched colored surface aslocated on the lattice dual to a triangulation. Then each vertex correspondto a 4-simplex; the coloring of the two models matches exactly (elementarysurfaces → faces, edges → tetrahedra); and summing over surfacescorresponds to summing over triangulations. The main difference is thedifferent weight at the vertices. The Barret–Crane vertex A BC can be readas a covariant definition a Hamiltonian constraint in loop quantum gravity.Thus, the space–time formulation of loop quantum GR is a simple modificationof a TQFT. This approach provides a 4D pictorial intuition of quantumspace–time, analogous to the Feynman graphs description of quantumfield dynamics. John Baez has introduced the term ‘spin foam’ for thebranched colored surfaces of the model, in honor of John Wheeler’s intuitionson the quantum microstructure of space–time. Spin foams are aprecise mathematical implementation of Wheeler’s ‘space–time foam’ suggestions.3.10.4.5 Black Hole EntropyA focal point of the research in quantum gravity in the last years hasbeen the discussion of black hole (BH) entropy. This problem has beendiscussed from a large variety of perspectives and within many differentresearch programs.