Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 307invariant QFT has a finite number of degrees of freedom, unless the invarianceis somehow broken, for instance dynamically. This belief is wrong.The problem of quantum gravity is precisely to define a diffeomorphisminvariant QFT having an infinite number degrees of freedom and ‘local’excitations. Locality in a gravity theory, however, is different from localityin conventional field theory. This point is often source of confusion. Hereis Rovelli’s clarification [Rovelli (1997)]:• In a conventional field theory on a metric space, the degrees of freedomare local in the sense that they can be localized on the metric manifold(an electromagnetic wave is here or there in Minkowski space).• In a diffeomorphism invariant field theory such as general relativity, thedegrees of freedom are still local (gravitational waves exist), but theyare not localized with respect to the manifold. They are neverthelesslocalized with respect to each other (a gravity wave is three metersapart from another gravity wave, or from a black hole).• In a topological field theory, the degrees of freedom are not localized atall: they are global, and in finite number (the radius of a torus is notin a particular position on the torus).The first TQFT directly related to quantum gravity was defined by[Turaev and Viro (1992)]. The Turaev–Viro model is a mathematically rigorousversion of the 3D Ponzano-Regge quantum gravity model describedabove. In the Turaev–Viro theory, the sum (3.148) is made finite by replacingSU(2) with quantum SU(2) q (with a suitable q). Since SU(2) qhas a finite number if irreducible representations, this trick, suggested by[Ooguri (1992a); Ooguri (1992b)], makes the sum finite. The extension ofthis model to four dimensions has been actively searched for a while andhas finally been constructed by [Crane and Yetter (1993)], again followingOoguri’s ideas. The Crane–Yetter (CY) model is the first example of 4DTQFT. It is defined on a simplicial decomposition of the manifold. Thevariables are spins (‘colors’) attached to faces and tetrahedra of the simplicialcomplex. Each 4–simplex contains 10 faces and 5 tetrahedra, andtherefore there are 15 spins associated to it. The action is defined in termsof the quantum Wigner 15 − j symbols, in the same manner in which thePonzano–Regge action is constructed in terms of products of 6−j symbols.Z ∼ ∑ ∏15 − j(color of the 4 − simplex), (3.151)coloring4−simplices(where we have disregarded various factors for simplicity). Crane and Yet-