Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 301clidean gravity is approximated by a simple function of the total numberof simplices and links, and the theory can be quantized summing overdistinct triangulations (for a detailed introduction, see [Ambjørn et. al.(1998)]). There are two coupling constants in the theory, roughly correspondingto the Newton and cosmological constants. These define atwo dimensional space of theories. The theory has a nontrivial continuumlimit if in this parameter space there is a critical point correspondingto a second order phase transition. The theory has phase transitionand a critical point. The transition separates a phase with crumpledspace–times from a phase with ‘elongated’ spaces which are effectively2D, with characteristic of a branched polymer [Bakker and Smit (1995);Ambjørn et. al. (2001a)]. This polymer structure is surprisingly the sameas the one that emerges from loop quantum gravity at short scale. Nearthe transition, the model appears to produce ‘classical’ S 4 space–times, andthere is evidence for scaling, suggesting a continuum behavior.State Sum ModelsA third road for discretizing GR was opened by a celebrated paper by [Ponzanoand Regge (1968)]. They started from a Regge discretization of 3D GRand introduced a second discretization, by posing the so–called Ponzano–Regge ansatz that the lengths l assigned to the links are discretized as well,in half–integers in Planck unitsl = G j, j = 0,1, 1, . . . (3.147)2(Planck length is G in 3D.) The half integers j associated to the links aredenoted ‘coloring’ of the triangulation. Coloring can be viewed as the assignmentof a SU(2) irreducible representation to each link of the Regge triangulation.The elementary cells of the triangulation are tetrahedra, whichhave six links, colored with six SU(2) representations. SU(2) representationtheory naturally assigns a number to a sextuplet of representations:the Wigner 6−j symbol. Rather magically, the product over all tetrahedraof these 6 − j symbols converges to (the real part of the exponent of) theEinstein–Hilbert action. Thus, Ponzano and Regge were led to propose aquantization of 3D GR based on the partition functionZ ∼ ∑ ∏6 − j(color of the tetrahedron), (3.148)coloringtetrahedra